Three Viewpoints on
Semi-Abelian Homology
Julia Goedecke
Emmanuel College
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
August 2009

I would like to dedicate this thesis to my parents. Although it took them a
while to accept that anyone might want to study something without any
apparent application to industry, they have always supported me in whatever
I wanted to do.
I would also like to dedicate this thesis to my collaborator Tim Van der
Linden, without whose constant help and encouragement I would probably
have given up research long ago.
ii
Declaration of Originality
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except as detailed below and where
specifically indicated in the text.
Chapter 1 gives the background for the thesis and is known material. Chapter
2 contains more background material in the first two sections, but Section 2.3
presents my own work, generalising known results from the abelian context into
the semi-abelian setting. Chapter 3 is based on joint work with Tim Van der
Linden, published in the Journal of Homotopy and Related Structures in 2007.
I contributed roughly 50% to these results. Chapter 4 again presents known
material which is needed in Chapter 5. This chapter is based on another joint
paper with Tim Van der Linden, which will be published in the Mathematical
Proceedings of the Cambridge Philosophical Society. As before I contributed
50% to the paper; for the thesis I rewrote the results using the concept of
an axiomatic class of extensions introduced by Tomas Everaert, which makes
many of the statements and proofs easier. The results of Chapter 6 are my
own work, developing a suggestion given by Marino Gran.
iii
iv
Acknowledgements
I would like to thank my supervisor Peter Johnstone for taking me on as his
student even though I did not show much enthusiasm for working with toposes.
A very big thank you goes to my collaborator Tim Van der Linden, who is a
great pleasure to work with and who supported and advised me throughout
my time as a PhD student, and will hopefully carry on doing so in the future.
I am also indebted to George Janelidze for pointing Tim and me to his work on
satellites as an alternative approach to homology. Marino Gran showed great
interest in my work, and his suggestion that I should work on stem extensions
resulted in Chapter 6. I am grateful to Tomas Everaert for helpful discussions
and comments, and for providing the useful concept of an axiomatic class of
extensions, which makes Chapter 5 so much easier.
Furthermore I would like to thank Alexander Shannon, arguably the best proof-
reader in the world, for finding (hopefully) all the little mistakes in my thesis
(though I take full responsibility for any remaining ones). Thank you also
to Martin Hyland, Alexander Frolkin and James Griffin for helpful comments
and suggestions throughout my time as a PhD student, and to Ignacio Lopez
Franco, Enrico Vitale and Burt Totaro for pointing out some mistakes in my
examples which nobody else had picked up on.
I would also like to thank Emmanuel College and the Department of Pure
Mathematics and Mathematical Statistics for their help and support during my
time at Cambridge, and the EPSRC for funding the research towards my PhD.
All my family and friends have also supported me to get to this point through
many ups and downs. In particular I would like to thank Marcus Zibrowius
for managing to boost my work morale enough so that I could actually write
this thesis.
v
vi
Abstract
The main theme of the thesis is to present and compare three different viewpoints
on semi-abelian homology, resulting in three ways of defining and calculating homology
objects. Any two of these three homology theories coincide whenever they are both de-
fined, but having these different approaches available makes it possible to choose the most
appropriate one in any given situation, and their respective strengths complement each
other to give powerful homological tools.
The oldest viewpoint, which is borrowed from the abelian context where it was intro-
duced by Barr and Beck, is comonadic homology, generating projective simplicial resolu-
tions in a functorial way. This concept only works in monadic semi-abelian categories, such
as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic
homology can be viewed not only as a functor in the first entry, giving homology of objects
for a particular choice of coefficients, but also as a functor in the second variable, varying
the coefficients themselves. As such it has certain universality properties which single
it out amongst theories of a similar kind. This is well-known in the setting of abelian
categories, but here we extend this result to our semi-abelian context.
Fixing the choice of coefficients again, the question naturally arises of how the homol-
ogy theory depends on the chosen comonad. Again it is well-known in the abelian case
that the theory only depends on the projective class which the comonad generates. We
extend this to the semi-abelian setting by proving a comparison theorem for simplicial
resolutions. This leads to the result that any two projective simplicial resolutions, the
definition of which requires slightly more care in the semi-abelian setting, give rise to the
same homology. Thus again the homology theory only depends on the projective class.
The second viewpoint uses Hopf formulae to define homology, and works in a non-
monadic setting; it only requires a semi-abelian category with enough projectives. Even
this slightly weaker setting leads to strong results such as a long exact homology sequence,
the Everaert sequence, which is a generalised and extended version of the Stallings-
Stammbach sequence known for groups. Hopf formulae use projective presentations of
objects, and this is closer to the abelian philosophy of using any projective resolution,
rather than a special functorial one generated by a comonad. To define higher Hopf for-
mulae for the higher homology objects the use of categorical Galois theory is crucial. This
theory allows a choice of Birkhoff subcategory to generate a class of central extensions,
which play a big role not only in the definition via Hopf formulae but also in our third
viewpoint.
This final and new viewpoint we consider is homology via satellites or pointwise Kan
extensions. This makes the universal properties of the homology objects apparent, giving
a useful new tool in dealing with statements about homology. The driving motivation
behind this point of view is the Everaert sequence mentioned above. Janelidze’s theory of
generalised satellites enables us to use the universal properties of the Everaert sequence
to interpret homology as a pointwise Kan extension, or limit. In the first instance, this
allows us to calculate homology step by step, and it removes the need for projective objects
from the definition. Furthermore, we show that homology is the limit of the diagram
consisting of the kernels of all central extensions of a given object, which forges a strong
connection between homology and cohomology. When enough projectives are available, we
can interpret homology as calculating fixed points of endomorphisms of a given projective
presentation.
vii
viii
Contents
Introduction 1
1 The Semi-Abelian Context 7
1.1 Semi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Abelian objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Homology in semi-abelian categories . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Categorical Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Central extensions in the context of abelianisation . . . . . . . . . . . . . . 24
2 Comonadic Homology 31
2.1 Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Comonadic Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Hn(−, E)G as a functor in the variable E . . . . . . . . . . . . . . . . . . . 38
3 A Comparison Theorem for Simplicial Resolutions 47
3.1 Simplicial resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 The Comparison Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Comonads generating the same projective class . . . . . . . . . . . . . . . . 53
4 Homology via Hopf Formulae 61
4.1 Extensions and higher extensions . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Strongly (E-)Birkhoff subcategories . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 The Galois structures Γn, centralisation and trivialisation . . . . . . . . . . 68
4.4 Hopf formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Homology via Satellites 81
5.1 Satellites and pointwise satellites . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Hn+1(−, I)E as a satellite of Hn(−, I1)E1 . . . . . . . . . . . . . . . . . . . . 85
5.3 Hn+1(−, I)E as a satellite of In . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Hn(−, I)E as a right satellite of Hn(−, I1)E1 . . . . . . . . . . . . . . . . . . 90
5.5 Homology without projectives . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Homology with projectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Stem Extensions in the Context of Abelianisation 111
6.1 Stem extensions and stem covers . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Perfect objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 A natural isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References 123
Index 127
ix
x
Introduction
The main theme of this thesis is to present and compare three different viewpoints on
semi-abelian homology. As a motivation for the newest of these viewpoints, consider a
perfect group A. We know that it has a universal central extension, and the kernel of this
universal central extension is isomorphic to the second integral homology group H2(A,Z).
It would be nice if this kind of property could be extended to the homology of all groups,
not only perfect ones. One of the main results in this thesis makes this possible: it shows
that the homology H2(A,Z) is the limit of the diagram of kernels of all central extensions
of A. This is a natural generalisation of the perfect case, as the limit of a diagram with
an initial object is just this initial object. Similarly the higher homology groups form
limits of diagrams using kernels of higher central extensions. This result emphasises the
close connection of homology with central extensions, and thus makes a connection to
cohomology.
Semi-abelian categories
The results of this thesis are not limited to the category of groups, but take place in the
more general setting of semi-abelian categories. Classically, homological algebra is an area
which is studied in the context of abelian categories. Many of the diagram lemmas used
in homological algebra, such as the Five Lemma and the Snake Lemma, are proved in this
context, but they also hold in the category of groups and other settings close to it, such as
Lie algebras or crossed modules. Semi-abelian categories give a wider context for homolog-
ical algebra which includes these non-abelian examples as well as the traditional abelian
categories. Many of the homological algebra results which hold in abelian categories also
hold in semi-abelian ones, though not quite all, as can be seen most easily in the category
of groups. For example, products and coproducts do not coincide in general semi-abelian
categories, and not every monomorphism is a kernel, as for instance not every subgroup
is a normal subgroup. We use semi-abelian categories as a framework to study homology
in a non-abelian context.
Three viewpoints on semi-abelian homology
The oldest viewpoint on semi-abelian homology, which is borrowed from the abelian con-
text where it was introduced by Barr and Beck, is comonadic homology, generating pro-
jective simplicial resolutions in a functorial way. This concept only works in monadic
semi-abelian categories, such as semi-abelian varieties, which include the categories of
groups and Lie algebras. The second viewpoint uses Hopf formulae to define homology,
1
Three Viewpoints on Semi-Abelian Homology
and works in a non-monadic setting; it only requires a semi-abelian category with enough
projectives. Even this slightly weaker setting leads to strong results such as a long exact
homology sequence. Hopf formulae use projective presentations of objects, and this is
closer to the abelian philosophy of using any projective resolution, rather than a special
functorial one generated by a comonad. The final viewpoint we consider in this thesis is
homology via satellites or pointwise Kan extensions. This makes the universal properties of
the homology objects apparent, giving a useful new tool in dealing with statements about
homology. Any two of these three definitions of homology coincide whenever they are
both defined; but having these different viewpoints available makes it possible to choose
the most appropriate one in any given situation, and their respective strengths comple-
ment each other to give powerful homological tools. There is another viewpoint we do
not consider in this thesis, which uses kernel pairs and Galois groupoids. This viewpoint
introduced by Janelidze in [Jan2008] is perhaps closest to our second viewpoint using Hopf
formulae. We now describe our three viewpoints in slightly more detail.
Viewpoint 1: Comonadic homology
Comonadic homology can be viewed not only as a functor in the first entry, giving homol-
ogy of objects for a particular choice of coefficients, but also as a functor in the second
variable, varying the coefficients themselves. As such it has certain universality properties
which single it out amongst theories of a similar kind. This is well-known in the setting
of abelian categories, but here we extend this result to our semi-abelian context.
Fixing the choice of coefficients again, the question naturally arises of how the homol-
ogy theory depends on the chosen comonad. Again it is well-known in the abelian case
that the theory only depends on the projective class which the comonad generates. We
extend this to the semi-abelian setting by proving a comparison theorem for simplicial
resolutions. This leads to the result that any two projective simplicial resolutions, the
definition of which requires slightly more care in the semi-abelian setting, give rise to the
same homology. Thus again the homology theory only depends on the projective class.
Viewpoint 2: Hopf formulae
The Hopf formula
H2(A,Z) ∼= [P, P ] ∩K[p][K[p], P ]
for the second homology group is very well known in the context of integral group homol-
ogy. Here p : P −→ A is a projective presentation of the group A with kernel K[p], and
[P, P ] and [K[p], P ] are commutator subgroups of P . This formula can be generalised both
to a semi-abelian setting and also to higher Hopf formulae, which are then used to define
higher homology objects. A good way to summarise these higher Hopf formulae is to say
that homology measures the difference between the centralisation and the trivialisation
2
Introduction
of a projective presentation of a given object. As hinted above, using this definition any
short exact sequence
0 ,2 K  ,2 ,2 B
f  ,2 A ,2 0
gives rise to a long exact homology sequence
· · · ,2 Hn+1(A, I)
δn+1f ,2 K[Hn(f, I1)]
γnf ,2 Hn(B, I)
Hn(f,I) ,2 Hn(A, I) ,2 · · ·
· · · ,2 H2(A, I)
δ2f
,2 K[H1(f, I1)]
γ1f
,2 H1(B, I)
H1(f,I)
,2 H1(A, I) ,2 0
which we call the Everaert sequence. This sequence has a different appearance than its
abelian counterpart: instead of being functorial in the objects of the short exact sequence,
it is functorial in the extension f . It incorporates not only homology of objects, such
as Hn(A, I), but also homology of an extension Hn(f, I1), which is a higher-dimensional
version of the same theory. The lowest part of the Everaert sequence is the Stallings-
Stammbach sequence, which first appeared in the context of groups, but now it can be
extended to a full long exact sequence. Its universal properties play a very important role
in a large part of this thesis.
Birkhoff subcategories and abelian objects
The main ingredient of all approaches to semi-abelian homology is a Birkhoff subcategory
B of a semi-abelian category A, as the reflector I takes the role of coefficients of the
homology theory. Thus semi-abelian homology calculates the derived functors of this
reflector, using different approaches to do this. The most common examples such as
integral group homology or homology of Lie algebras use the subcategory of abelian objects
as the Birkhoff subcategory, which makes it easy to see that all homology objects are
abelian. But there are also many examples where the Birkhoff subcategory is not given
by the abelian objects, for instance the categories of nilpotent or solvable groups inside
the category of groups, the category of Lie algebras inside that of Leibniz algebras, or the
category of crossed modules inside that of precrossed modules. In these cases it can still
be shown that all homology objects from the second onwards are abelian, but as the first
homology just gives the reflection of an object into the Birkhoff subcategory, these will
not in general be abelian objects. In the Everaert sequence, this means that at the lower
end the objects stop being abelian. But the map δ2f from the last abelian object to the
first non-abelian object turns out to be central in the sense of Huq, meaning that its image
commutes with everything in its codomain. This nicely connects the abelian part of the
sequence with the non-abelian part, and does not seem to have been realised before.
3
Three Viewpoints on Semi-Abelian Homology
Viewpoint 3: Homology via satellites
The universal properties of the Everaert sequence are the driving motivation for defining
homology via satellites, as a pointwise Kan extension or limit. The connecting homo-
morphism δ in the Everaert sequence is exactly what makes the Kan extension work. In
the first instance, this allows us to calculate homology step by step, and it removes the
dependance on projective objects from the definition. For example, the (n+1)st homology
Hn+1(−, I) is the left satellite of the nth homology of extensions Hn(−, I1).
ExtA
cod
z


 Hn(−,I1)
$?
??
??
??
A
δn+1 +3
Hn+1(−,I) $
ExtA
kerz



A
For n = 1 this can be reformulated to give the result hinted at in the first paragraph of
this introduction: homology is the limit of a diagram of kernels of all central extensions
of a given object.
H2(A, I) = lim
f∈CExtAA
K[f ]
That is, we consider all central extensions of an object A, and the maps between them.
Taking kernels gives us a diagram in the category we are working in, and the second
homology H2(A, I) is the limit of this diagram.
K[f ]  &-
&-
B
f
TTTT
TT
 &-T
TTTT
H2(A, I)
4<ppppppppppppppp
,2
"*NN
NNN
NNN
NNN
NNN
N
...
...
K[f ′]  ,2 ,2
LR

B′
LR

f ′  ,2 A
B′′
LR
f ′′jjjjj
* 18jjjjj
K[f ′′]
LR
) 18
18
An analogous result using higher central extensions can be obtained for the higher homol-
ogy objects by concatenation of the original Kan extensions. In the case where enough
projectives do exist, this category of central extensions of a given object has a weakly
initial object induced by a projective presentation. In this case, homology can be viewed
as calculating the common fixed points of the endomorphisms of this weakly initial pre-
sentation.
H2(A, I) ,2 K[f ]
 ,2 ,2

CK B

	
AI
f  ,2 A
4
Introduction
Structure of the text
In Chapter 1 we set up the semi-abelian context we are working in and state known
results that we will use throughout the thesis. Chapter 2 introduces the first viewpoint
on comonadic homology and deals with the theory as a functor in the second variable of
coefficients. While the first two sections are known material, Section 2.3 contains my own
work, generalising a corresponding result from the abelian context. Chapter 3 is devoted
to the Comparison Theorem for simplicial resolutions and is joint work with Tim Van der
Linden. In Chapter 4 the established theory of homology via Hopf formulae is introduced,
which is then needed in Chapter 5. This chapter develops the theory of homology via
satellites and is based on another joint paper with Tim Van der Linden; I have rewritten the
results using the concept of an axiomatic class of extensions introduced by Tomas Everaert,
which simplifies many of the statements and proofs. Finally Chapter 6 explores special
cases of central extensions in the context of abelianisation, using the lowest part of the
Everaert sequence to give a natural correspondence between isomorphism classes of central
extensions of a perfect object B by a fixed abelian object K and maps H2(B, ab) −→ K.
The results of this chapter came into existence after a suggestion from Marino Gran.
5
Three Viewpoints on Semi-Abelian Homology
6
Chapter 1
The Semi-Abelian Context
As mentioned in the introduction, the framework for this thesis is formed by semi-abelian
categories. These include abelian categories as well as many non-abelian examples in
which homological algebra can be studied. A good survey of semi-abelian categories can
be found in [Bor2004], while the monograph [BB2004] gives a more detailed introduction
to semi-abelian categories and the weaker settings connected to them.
In this chapter, we give established definitions and results that will be used throughout
the thesis. Section 1.1 introduces semi-abelian categories and many classical results of
homological algebra, while Section 1.2 defines the concept of abelian objects, which play
a prominent role in our theory. Homology of chain complexes and the main ingredients of
semi-abelian homology are set up in Section 1.3. Section 1.4 introduces categorical Galois
theory, which forms a crucial base for the theory in Chapter 4 and Chapter 5. Finally
in Section 1.5 we compare several different notions of central extensions in the context of
abelianisation.
1.1 Semi-abelian categories
This section gives the main definitions that we need for homological algebra in semi-abelian
categories, and gathers some known results that will be used throughout the thesis. Two
of the first things we need in order to do homological algebra are kernels and cokernels.
1.1.1 Definition (kernels and cokernels): A category A is pointed when it has a
zero-object, i.e. an object 0 which is both initial and terminal. For any two objects A
and B, there is a unique map B −→ A that factors over the zero object; we call this a
zero map 0: B −→ A.
In a pointed category with finite limits we define the kernel of a morphism f : B −→ A
by the pullback
K[f ]  ,2
Ker f ,2

B
f

0 ,2 A
where we refer both to the object K[f ] and to the map Ker f as the kernel of f . Notice that
Ker f is a monomorphism, being the pullback of the monomorphism 0 −→ A. Equivalently
7
Chapter 1. The Semi-Abelian Context
K[f ] is the equaliser of f and the zero map. We say a monomorphism m : K −→ B is
normal when it is the kernel of some map, and write K  ,2 ,2B as in the diagram above.
In a pointed category with finite colimits we dually define the cokernel of a morphism
f : B −→ A by the pushout
B
f

,2 0

A
Coker f
 ,2Q[f ].
If f is a normal monomorphism, we also write Q[f ] = A/B. Any cokernel is a regular
epimorphism: it is the coequaliser of f : B −→ A and the zero map 0: B −→ A. We say a
regular epimorphism is normal when it is the cokernel of some map, and write A  ,2Q .
The definition of semi-abelian categories is built up out of many constituent parts,
which we introduce now.
1.1.2 Definition: A category A is called regular [Bar1971] when it is finitely complete,
every kernel pair in A has a coequaliser and the class of regular epimorphisms in A is
pullback-stable.
A category A is called Barr exact [Bar1971] when it is regular and every equivalence
relation in A is a kernel pair.
A pointed and regular category is called Bourn protomodular [Bou1991] when the
(regular) Short Five Lemma holds: for every commutative diagram
K[f ′]  ,2
Ker f ′ ,2
k

B′
f ′  ,2
b

A′
a

K[f ]  ,2
Ker f
,2 B
f
 ,2 A
such that f and f ′ are regular epimorphisms, if k and a are isomorphisms then b is an
isomorphism.
1.1.3 Definition: [JMT2002] A semi-abelian category is a category which is pointed,
Barr exact and Bourn protomodular, and has binary coproducts.
1.1.4 Example: All abelian categories are semi-abelian. The category Gp of groups is
one of the leading examples of semi-abelian categories. More generally, any variety of
Ω-groups is semi-abelian. This includes the categories of Lie algebras, Leibniz algebras,
non-unital rings and (pre)crossed modules. Other examples are the category of Heyting
semi-lattices, the dual of the category of pointed sets, and more generally the dual of the
category of pointed objects in any topos.
8
1.1 Semi-abelian categories
A semi-abelian category has all finite colimits as well as finite limits. Many results
relating to homological algebra that hold in abelian categories are also true in semi-abelian
ones, but not quite all of them. Semi-abelian categories are not enriched in abelian groups
(only in pointed sets), and finite products and coproducts do not coincide as in abelian
categories. In fact, a semi-abelian category with isomorphic binary products and coprod-
ucts is abelian. While K[f ] = 0 if and only if f is a monomorphism, the “dual” statement
only holds in one direction: if f is a regular epimorphism, then its cokernel is zero, but not
the other way around. As in abelian categories, every regular epimorphism is a cokernel
(of its kernel), but not every monomorphism is a kernel. For example in the category of
groups, not every subgroup is a normal subgroup. The weaker result holds that every
normal monomorphism is the kernel of its cokernel.
1.1.5 Remark: As every regular epimorphism is a cokernel, we shall mostly talk only
about regular epimorphisms and denote them by B  ,2A . We also call such a regular
epimorphism an extension (of A).
1.1.6 Definition (Image factorisation): [Bar1971] As a semi-abelian category is regu-
lar, any map f : B −→ A in A can be factored into a regular epimorphism followed by a
monomorphism.
B
f ,2
 %
BB
BB
BB
B A
I[f ]
9D Im f
9D}}}}}}}
We refer to both the object I[f ] and the map Im f as the image of f . This image
factorisation is stable under pullback, as both regular epimorphisms and monomorphisms
are. Notice that the regular epimorphism is a cokernel (as all regular epimorphisms are),
but the monomorphism need not be a kernel, as it is in abelian categories. If Im f is a
kernel, we call f a proper morphism. These play an important role in homological algebra
in semi-abelian categories.
1.1.7 Definition (Exact sequences): A sequence of morphisms
A
f ,2 B
g ,2 C
is called exact (at B) if Im f = Ker g.
A sequence 0 −→ A −→ B is exact if and only if the map A −→ B is a monomorphism,
and A −→ B −→ 0 is exact if and only if A −→ B is a regular epimorphism. Note that a
non-proper map can never occur as the first map of an exact sequence, so in a long exact
sequence all maps must be proper, except possibly the very last.
9
Chapter 1. The Semi-Abelian Context
As mentioned earlier, there are many weaker settings that are part of the whole semi-
abelian context. Later we will wish to consider an example which is not quite semi-
abelian, as it does not have coproducts, and many of our results do not need coproducts.
But instead of always stating the precise weakest setting in which a result holds, we will
generally work in semi-abelian categories and sometimes highlight results which do not
need the full strength of the definition. The monograph [BB2004] gives a very detailed
description of the weaker settings, some of which we now mention.
Results from homological algebra
A pointed regular and protomodular category is also called homological, and many results
of homological algebra hold already in this weaker setting. An example of a homological
category which is not semi-abelian is the category of topological groups, which is not exact.
Any homological category is Mal’tsev, which means that it has all finite limits and every
internal reflexive relation is an equivalence relation [CLP1991, CKP1993, Bou1996]. Also,
for any two objects A and B, the pair of morphisms
A
(1A,0) ,2 A×B B(0,1B)lr
is jointly strongly epic, which means that the category is unital [Bou1996, BB2004].
We will now give a collection of results that we will need repeatedly throughout the
thesis, and will often use without referring to them.
1.1.8 Lemma (pullback cancellation): [Bou1991, BB2004] In a homological category,
given a commutative diagram
A′

,2 B′
b
_
,2 C ′

A ,2 B ,2 C
where b is a regular epimorphism, if the outer rectangle is a pullback, the left hand square
is a pullback if and only if the right hand square is a pullback.
Notice that the direction “right hand square pullback implies left hand square pullback”
holds in general, even when b is not a regular epimorphism, but the other direction uses
protomodularity.
1.1.9 Lemma: [Bou1991] In a protomodular category, pullbacks reflect monomorphisms.
The following classical result will be used later on to show that a particular morphism
is regular epic.
10
1.1 Semi-abelian categories
1.1.10 Lemma: Let A be a regular category. A map y : Z0 −→ Y factors through the
image of a map f : X −→ Y if and only if there is a regular epimorphism z : Z −→ Z0 and
a map x : Z −→ X with yz = fx.
1.1.11 Remark: Of course, if we can show that every map y : Z0 −→ Y factors through
the image of a given map f : X −→ Y , this map f is a regular epimorphism.
The well-known Five Lemma is an easy consequence of the Short Five Lemma given
in the definition of pointed protomodular categories above.
1.1.12 Lemma (Five Lemma): [Bor2004] In a homological category, given a morphism
between five-term exact sequences
A′ ,2
a∼=

B′ ,2
b∼=

C ′ ,2
c

D′ ,2
∼=d

E′
∼=e

A ,2 B ,2 C ,2 D ,2 E
where the outer four morphisms a, b, d and e are isomorphisms, then c is also an isomor-
phism.
1.1.13 Lemma (3× 3 Lemma): [Bou2001, BB2004] In a homological category, given a
commutative diagram
0

0

0

0 ,2 K ′′

 ,2 ,2 B′′
b′

 ,2 A′′ ,2

0
0 ,2 K ′

 ,2 ,2 B′
b

 ,2 A′ ,2

0
0 ,2 K

 ,2 ,2 B

 ,2 A ,2

0
0 0 0
with short exact rows and b◦b′ = 0, if any two columns are short exact, then so is the third.
11
Chapter 1. The Semi-Abelian Context
1.1.14 Lemma (Snake Lemma): [Bou2001, BB2004] In a homological category, a com-
mutative diagram with exact rows as the diagram of solid arrows below,
K[a]
_



,2 K[b]
_



,2 K[c]
_



|
A′
a

,2 B′  ,2
b

C ′ ,2
c

0
0 ,2 A  ,2 ,2
_


 B
_



,2 C
_



Q[a] ,2 Q[b] ,2 Q[c]
where a, b and c are proper maps, gives rise to a six-term exact sequence
K[a] −→ K[b] −→ K[c] −→ Q[a] −→ Q[b] −→ Q[c]
which is natural in a, b and c. Moreover, if the first solid row is short exact, we can add a
zero to the front of this six term exact sequence, and if the second solid row is short exact
we can add a zero to the end.
A consequence of the 3× 3 Lemma which we shall be using is
1.1.15 Lemma (Noether’s Third Isomorphism Theorem): [BB2004] In a homo-
logical category, consider two normal subobjects A C C and B C C, with A ⊆ B. Then
• A is a normal subobject of B,
• B/A is a normal subobject of C/A,
• the isomorphism C/AB/A ∼= CB holds.
0 ,2 BA
 ,2 ,2 C
A
 ,2 C
B
,2 0
1.1.16 Lemma: [Bou1991, Bou2001] Given a morphism between short exact sequences in
a homological category as follows,
0 ,2 K ′
k

 ,2 ,2 B′  ,2
b

A′
a

,2 0
0 ,2 K  ,2 ,2 B  ,2 A ,2 0
(1) the left hand square is a pullback if and only if a is a monomorphism,
12
1.1 Semi-abelian categories
(2) the right hand square is a pullback if and only if k is an isomorphism.
Proof. The implications from right to left are easy diagram chases which hold in general,
but the opposite implication in (1) uses that K[f ] = 0 if and only if f is a mono (which
holds in a pointed protomodular category), while the one in (2) uses Lemma 1.1.8.
We will most often be using the easy implications of these two results, but give both
directions here for completeness. There is a dual result to the easy implication of (2)
which we shall also need.
1.1.17 Lemma: Let A be a pointed category. Consider the diagram
B′
f ′ ,2

A′  ,2

Q[f ′]

B
f
,2 A
 ,2Q[f ]
where Q[f ] and Q[f ′] are the cokernels of f and f ′ respectively. If the left square is a
pushout, then the induced map between the cokernels is an isomorphism.
An important concept later on in this thesis will be a regular pushout [Bou2003,
CKP1993], which is a commutative square of regular epimorphisms where the comparison
map to the pullback is also a regular epimorphism.
B′
f ′
 '
b
z"
r
 %
P
 ,2
_
A′
a
_
B
f
 ,2 A
A regular pushout in a semi-abelian category is always a pushout; in fact, a commutative
square of regular epimorphisms is a regular pushout if and only if it is a pushout [Bou2003,
CKP1993]. The following lemma is a consequence of this.
1.1.18 Lemma: [JMT2002] In a semi-abelian category, the direct image of a normal sub-
object along a regular epimorphism is again a normal subobject. That is, given a diagram
as below where f is regular epimorphism, b is a normal monomorphism and a◦f ′ is the
13
Chapter 1. The Semi-Abelian Context
image factorisation of f◦b,
B′
f ′  ,2
_
b

A′

a

B
f
 ,2 A
then a is also a normal monomorphism.
There is another definition of a regular pushout which is slightly more general than
the one above. We will need it in this more general form in Chapter 3.
1.1.19 Definition (generalised regular pushout): [VdL2006, VdL2008] Let A be a
semi-abelian category. A square in A with horizontal regular epimorphisms
B′
f ′  ,2
b

A′
a

B
f
 ,2 A
is a generalised regular pushout when the comparison map (b, f ′) : B′ −→ B ×A A′ to
the pullback B ×A A′ of a along f is a regular epimorphism. (The maps b and a are not
required to be regular epimorphisms.)
A generalised regular pushout is always a pushout, but a pushout need not be a
generalised regular pushout, differing from the first definition of regular pushout above.
1.2 Abelian objects
Just as one has abelian groups in the category of groups, we can define abelian objects in
any semi-abelian category. These will play an important role throughout the thesis.
1.2.1 Definition (Abelian objects): [Huq1968, Bor2004] An object A in a semi-abelian
category is called an abelian object when there exists a (necessarily unique) morphism
m : A×A −→ A satisfying
A
FF
FF
FF
FF
F
(1A,0)

A×A m ,2 A
A
xxxxxxxxx
(0,1A)
LR
called the multiplication or addition of A (see also [Bou2002] and Section 1.5).
14
1.2 Abelian objects
1.2.2 Remark: The multiplicationm is unique because the pair ((1A, 0), (0, 1A)) is jointly
epic (see Section 1.1).
An abelian object in the category of groups is an abelian group. We have a similar
result in any semi-abelian category, saying that the abelian objects are exactly the internal
abelian groups.
1.2.3 Lemma: [Bor2004, Bou2000] Let A be an object in a semi-abelian category. The
following are equivalent:
(1) A is an abelian object;
(2) the diagonal ∆: A −→ A×A is a normal subobject;
(3) A carries an internal abelian group structure.
Moreover, the abelian group structure on A is necessarily unique.
Abelian objects in a semi-abelian category allow us to establish connections with
abelian categories.
1.2.4 Lemma: [BG2002a, Bou2000, JMT2002, Bor2004] Let A be a semi-abelian category,
and denote the full subcategory of abelian objects by AbA. Then
• AbA is an abelian category.
• A is abelian if and only if all objects in A are abelian objects.
• A is abelian if and only if every subobject in A is normal.
• If A is semi-abelian and Aop is semi-abelian, then A is abelian.
The full subcategory of abelian objects will play an important role throughout this
thesis. The most important aspect of it is the reflector ab: A −→ AbA.
1.2.5 Lemma: [BB2004] Let A be a semi-abelian category. The inclusion AbA −→ A
admits a left adjoint. More explicitly, given an object A in A, the coequaliser of the pair
A
(1A,0) ,2
(0,1A)
,2 A×A
is the abelianisation abA of A.
Proof. Note that the construction of this abelianisation functor does need coequalisers.
15
Chapter 1. The Semi-Abelian Context
In [BB2004], an object with a multiplication as in Definition 1.2.1 is called commu-
tative, and this lemma is proved for the subcategory of those commutative objects. But
in a homological category, every commutative object is abelian (meaning it is an internal
abelian group), so we can still use this result.
1.2.6 Example (Abelianisation functors): The abelianisation functor ab: Gp −→ Ab
on groups takes a group G to abG = G/[G,G], where [G,G] is the commutator subgroup
of G, the normal subgroup generated by elements of the form aba−1b−1. Similarly for
ab: LieK −→ AbLieK , a Lie algebra g is sent to ab g = g/[g, g], where [g, g] now denotes
the ideal generated by elements of the form [x, y], the Lie bracket of x and y.
The following result can be a useful tool for determining whether an object is abelian.
1.2.7 Lemma: [Bou2005, Theorem 2.1] Let A be a semi-abelian category, and consider
two normal subobjects U C B and V C B of an object B in A. Let f : B −→ A be a
regular epimorphism. Then the quotient
f(U) ∩ f(V )
f(U ∩ V )
is always an abelian object.
1.3 Homology in semi-abelian categories
The first step towards any homology theory is usually homology of chain complexes, which
we introduce in this section. We also give the basic setting of semi-abelian homology.
As usual a chain complex is a sequence of maps (dn : Cn −→ Cn−1)n∈Z satisfying
dn−1◦dn = 0 for all n. A chain complex is called proper when all the maps dn are proper
maps.
1.3.1 Definition (Homology of chain complexes): [EVdL2004b] Let C be a proper
chain complex in a semi-abelian category. The nth homology object HnC is the cokernel
of d′n+1 : Cn+1 −→ Ker [dn], the factorisation of dn+1 : Cn+1 −→ Cn over kerdn.
Cn+1
d′n+1 (H
HH
HH
HH
HH
dn+1 ,2 Cn
dn ,2
Coker dn+1
 (
HH
HH
HH
HH
Cn−1
K[dn]
9 7B
Ker dn
7Byyyyyyy
Coker d′n+1_
Q[dn+1]
d′′n
5?ttttttttt
HnC ________ ________ KnC
_LR
Ker d′′n
LR
16
1.3 Homology in semi-abelian categories
For a proper chain complex this is isomorphic to the dually defined KnC, the kernel of
the factorisation of dn over the cokernel of dn+1.
It is easy to see from the definition that C is exact at Cn if and only if HnC = 0. Thus
homology detects failure of exactness as usual. When talking about homology, we only
consider proper chain complexes, because otherwise this property is false. Consider, for
instance, the following example in the category of groups. Recall that in Gp a monomor-
phism is normal if and only if it is the inclusion of a normal subgroup. Define a chain
complex by taking d1 to be the inclusion of A4 into A5, and all other objects to be zero.
Since A5 is simple, d1 is not proper, and all objects HnC are zero, but clearly C is not
exact at C0.
To calculate homology of an ordinary object A in A, the most common way is to
form some resolution of A which gives rise to a chain complex, of which we can then take
homology. But there are also other ways of calculating the homology of an object in a
semi-abelian category. This is the main theme of this thesis.
Semi-abelian homology is based on a semi-abelian category A with a Birkhoff subcat-
egory B and a reflector I : A −→ B. It calculates the derived functors of the reflector I.
1.3.2 Definition (Birkhoff subcategory): [JK1994] By a reflective subcategory B
of A we mean a full and replete subcategory for which the inclusion functor ⊆ : B −→ A
admits a left adjoint I : A −→ B, called the reflector, or sometimes the reflection.
A Birkhoff subcategory B of a Barr-exact category A is a reflective subcategory
which is closed under subobjects and regular quotients.
To understand this choice of name, recall the classical Birkhoff Theorem of universal
algebra, also called Birkhoff’s HSP theorem: a variety of algebras is a class of algebraic
structures of a given signature satisfying a given set of identities, or equivalently, via
Birkhoff’s theorem, a class of algebraic structures of the same signature which is closed
under taking homomorphic images, subalgebras and products. Notice that the inclusion
functor of a reflective subcategory is a right adjoint and so preserves limits, so that Birkhoff
subcategories as we have defined them are automatically closed under products as well.
1.3.3 Example: It is immediately clear from the explanation above that a Birkhoff sub-
category of a semi-abelian variety of universal algebras is the same as a subvariety. Exam-
ples are the category Ab of abelian groups inside Gp, the category of abelian Lie algebras
inside Lie algebras, and the category of crossed modules inside precrossed modules. For
any semi-abelian category A, the full subcategory of abelian objects AbA forms a Birkhoff
subcategory with reflector ab: A −→ AbA as in Lemma 1.2.5 (see [Gra2001]). The first
two examples given above are of this form (see Example 1.2.6), but the last is not. There
17
Chapter 1. The Semi-Abelian Context
are many more examples where the Birkhoff subcategory is not the subcategory of abelian
objects; we will mention three.
A group G is nilpotent of class m when LCmG = γm+1G = 0 (and LCm−1G 6= 0),
where LCmG is the (m+ 1)st term in the descending central series of G
LC1G = γ2G = [G,G]
LC2G = γ3G = [[G,G], G]
· · · = · · · = · · ·
LCmG = γm+1G = [LCm−1G,G].
(The unconventional notation with shifted numbering is used to keep the index of the
reflector and the kernel of the reflector consistent, see below. We write 0 for the one-
element group, as it is the zero-object of the category of groups, but we will later write 1
for the identity element inside a non-abelian group.) The subcategory Nilm of all nilpotent
groups of class at most m is a Birkhoff subcategory of Gp with reflector
Gp
nilm ,2 Nilm
G
 ,2 G/LCmG
for any m.Similarly, a group G is solvable of derived length m when (Dm−1G 6= 0 and)
DmG = 0, the mth term in the derived series of G
D1G = [G,G]
D2G = [[G,G], [G,G]]
· · · = · · ·
DmG = [Dm−1G,Dm−1G].
The subcategory Solm of all solvable groups of class at most m is also a Birkhoff subcate-
gory of Gp, with reflector
Gp
solm ,2 Solm
G
 ,2 G/DmG
for any m.
Another example is that of Lie algebras inside Leibniz algebras. A Leibniz algebra is
a vector space g over a field K with a bilinear bracket [·, ·] : g× g −→ g satisfying the
Leibniz identity
[x, [y, z]] = [[x, y], z]− [[x, z], y].
18
1.3 Homology in semi-abelian categories
When [x, x] = 0 for all x ∈ g, and thus [x, y] = −[y, x] by bilinearity, this Leibniz identity
becomes the Jacobi identity
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0
and g is a Lie algebra. Writing gAnn for the two-sided ideal generated by all elements of
the form [x, x], we have a reflector
LeibK
lie ,2 LieK
g  ,2 g/gAnn
and LieK is a Birkhoff subcategory of LeibK .
We usually write the unit of the reflection as η : 1A −→ I, leaving out the inclusion
functor. An alternative characterisation of a Birkhoff subcategory is the following.
1.3.4 Lemma: [JK1994] A reflective subcategory B of an exact category A is a Birkhoff
subcategory if and only if the following diagram is a regular pushout for every extension f .
B
ηB

f ,2 A
ηA

IB
If ,2 IA
Proof. In fact any reflective subcategory B is closed under subobjects if and only if each
ηA is a regular epimorphism, and it is also closed under regular quotients if and only if
the above square is a regular pushout.
1.3.5 Remark: Notice that this immediately implies that the functor I preserves exten-
sions.
Given this data, there are several different ways to compute semi-abelian homology,
which coincide in situations where they are all defined. The three viewpoints discussed in
this thesis are
(1) comonadic homology in a semi-abelian monadic category, in Chapters 2 and 3;
(2) homology via Hopf formulae in a semi-abelian category with enough projectives, in
Chapter 4;
(3) homology via Kan extensions or limits in a semi-abelian category, in Chapter 5.
19
Chapter 1. The Semi-Abelian Context
1.3.6 Example: Many examples of Birkhoff subcategories give well-known homology the-
ories in these three different ways above. For example, the subcategory Ab of abelian
groups in Gp gives rise to integral group homology, and the subcategory AbLieK of abelian
Lie algebras inside LieK gives rise to the Chevalley-Eilenberg homology of Lie algebras.
1.4 Categorical Galois theory
A Birkhoff subcategory B of a semi-abelian category A with its reflector I : A −→ B gives
rise to a Galois structure in the sense of Janelidze [Jan1991]. We will not go into the
whole theory of categorical Galois structures and its connection to the Galois theory of
field extensions; for this we refer the reader to the monograph [BJ2001] and the papers
[Jan1991, JK1994]. We will just recall as much of the theory as we will need.
1.4.1 Definition: (see [Jan1991]) A Galois structure Γ = (A,B,E,Z, I,H) consists of
two categories A and B, an adjunction
A ⊥
I ,2
B,
H
lr
and classes E and Z of morphisms in A and B respectively, such that:
(1) A has pullbacks along arrows in E,
(2) E and Z contain all isomorphisms, are closed under composition and are pullback-
stable,
(3) I(E) ⊂ Z,
(4) H(Z) ⊂ E,
(5) the counit ε is an isomorphism,
(6) each A-component ηA of the unit η belongs to E.
Maps in E and Z are called extensions, and we denote them by B  ,2A .
1.4.2 Example (Galois structures): Let A be the category Gp of groups, B the sub-
category Ab of abelian groups, H the inclusion functor and I the abelianisation functor
ab: G 7−→ G/[G,G]. Then choosing E and Z to be the surjective group homomorphisms
in Gp respectively in Ab gives a Galois structure (Gp,Ab,E,Z, ab,⊆).
A related Galois structure exists on any semi-abelian category A. We take B to be
the subcategory AbA of abelian objects in A, H again the inclusion functor and I the
abelianisation functor, and choose for E and Z all regular epimorphisms in A and in AbA
respectively. This gives a Galois structure (A,AbA,E,Z, ab,⊆).
20
1.4 Categorical Galois theory
An example of a Galois structure not coming from abelianisation is given by the cate-
gory of Lie algebras inside that of Leibniz algebras, as explained in Example 1.3.3. Taking
E and Z as the classes of regular epimorphisms in LeibK and LieK respectively, we have a
Galois structure (LeibK , LieK ,E,Z, lie,⊆) for a field K. Similarly, we have the structures
(Gp,Nilm,E,Z,nilm,⊆) and (Gp,Solm,E,Z, solm,⊆) for a fixed m ∈ N. Another Galois
structure not using abelianisation is (PXMod,XMod,E,Z, xmod,⊆), where xmod is the
reflector from precrossed modules into crossed modules, and E and Z are the appropriate
classes of regular epimorphisms.
In a Galois structure Γ, we define several different sorts of extensions.
1.4.3 Definition: [Jan1991, Jan1990] Let Γ be a Galois structure and f : B −→ A an
extension in E. We say that f is
(1) a trivial extension (or f is trivial) when the square below is a pullback,
B
f  ,2
ηB
_
A
ηA
_
IB
If
 ,2 IA
(2) a central extension when there exists a map a : A′ −→ A in E such that the pull-
back a∗f is trivial,
A′ ×A B  ,2
a∗f
_
B
f
_
A′ a
 ,2 A
(3) a normal extension when the first projection pi1 : R[f ] −→ B of the kernel pair of
f (or equivalently, the second projection pi2) is a trivial extension.
Clearly every normal extension is central. But when the category A is Mal’tsev, the
converse is also true: every central extension is normal (see Theorem 4.8 in [JK1994]).
Thus f is central with respect to Γ if and only if either of the two commuting squares in
the following diagram is a pullback.
R[f ]
pi1 ,2
pi2
,2
ηR[f ]

B
f  ,2
ηB

A
IR[f ]
Ipi1 ,2
Ipi2
,2 IB
21
Chapter 1. The Semi-Abelian Context
1.4.4 Example (central extensions): When A is the category of groups, B is the
Birkhoff subcategory of abelian groups, and E is the class of all surjections, the central
extensions are those whose kernel lies in the centre of the group. The trivial extensions
are exactly those surjections B −→ A where the restriction to the commutator subgroups
[B,B] −→ [A,A] is an isomorphism.
[B,B]  ,2 ,2
∼=

B
f
_
ηB  ,2 abB
ab f
_
[A,A]  ,2 ,2 A
ηA  ,2 abA
Similarly, for A = LieK and B = AbLieK , the central extensions are the classical central
extensions of Lie algebras, that is, the kernel K[f ] of f : b −→ a lies in the centre of b:
K[f ] ⊆ Zb = {z ∈ b | [z, b] = 0 for all b ∈ b}
However, when A = LeibK and B = LieK for a field K with charK 6= 2, the central
extensions are those extensions whose kernel lies in the Lie-centre
ZLie(b) = {z ∈ b | [b, z] = −[z, b] for all b ∈ b}
of b (for a proof, see e.g. [CVdL2009]).
A similar result holds for A = Gp and B = Nilm or B = Solm for a fixed m ∈ N.
Write lm(b1, b2, . . . , bm+1) for an element [[· · · [b1, b2], b3], . . .], bm+1] ∈ LCmB, and sim-
ilarly dm(b1, b2, . . . , b2m) = [dm−1(b1, . . . , b2m−1), dm−1(b2m−1+1, . . . , d2m)] for an element
of DmB, thus for example d2(b1, b2, b3, b4) = [[b1, b2], [b3, b4]]. It is clear that lm and
dm give the identity 1 of B whenever any entry is 1. We also write lm(K,B, . . . , B) and
dm(K,B, . . . , B) whenK is a normal subgroup of B, for example l2(K,B,B) = [[K,B], B].
Now define the m-nil-centre of B by
ZNilm(B) = {z ∈ B | lm(z, b2, . . . , bm+1) = 1 ∀ bi ∈ B}.
Notice the nesting
ZB = ZNil1(B) ⊆ ZNil2(B) ⊆ · · · ⊆ ZNilm(B) ⊆ · · ·
starting with the usual centre ZB of B. In fact, these subgroups exactly form the upper
central series
0 = Z0B C Z1B C · · · C ZmB C · · ·
of B defined by Zi+1B = {z ∈ B | [z, b] ∈ ZiB ∀ b ∈ B} with Z1B = ZB. Notice also
that K[f ] ⊆ ZNilmB if and only if lm(K[f ], B, . . . , B) = 0. A group extension f : B −→ A
is central with respect to Nilm if and only if lm(K[f ], B, . . . , B) = 0, and so if and only
22
1.4 Categorical Galois theory
if K[f ] ⊆ ZNilmB. This was proved by Everaert and Gran in [EG2006, Corollary 3.5].
Similarly a group extension f : B −→ A is central with respect to Solm if and only if
K[f ] ⊆ ZSolm(B) = {z ∈ B | dm(z, b2, . . . , b2m)∀ bi ∈ B}
if and only if dm(K[f ], B, . . . , B) = 0 [EG2006, Corollary 4.3]. For better understanding,
we give a sketch of our own proof of the result for nilpotent groups. The case of solvable
groups works analogously.
Consider the following diagram (here J = LCm):
JR[f ] ∩K[pi2]  ,2 ,2_

JR[f ]
∼= ,2
_

JB_

K[pi2]
 ,2 ,2 R[f ]
pi2  ,2
_
B
_
IR[f ]  ,2 IB
By definition, f is central if and only if JR[f ] ∼= JB, and this isomorphism is induced
by the diagonal ∆: B −→ R[f ]. This is the case if and only if JR[f ] ∩ K[pi2] = 0 (or
equivalently if JR[f ] ∩ K[pi1] = 0). Recall that we write 0 for the one-element group but
1 for the identity element inside a non-abelian group. So assume first that JR[f ] ∼= JB,
and let k ∈ K[f ]. Then (k, 1) and (bi, bi) are elements of R[f ], for any bi ∈ B. So we have
lm
(
(k, 1), (b2, b2), . . . , (bm+1, bm+1)
)
=
(
lm(k, b2 . . . , bm+1), 1
) ∈ LCmR[f ] = JR[f ].
But as JR[f ] ∼= JB via the diagonal, this means
lm(k, b2, . . . , bm+1) = 1
for any bi ∈ B, as claimed.
Conversely, suppose K[f ] ⊆ ZNilmB, and let
(
lm(b1, . . . , bm+1), lm(k1, . . . , km+1)
)
be an
element of JR[f ]∩K[pi2]. This means lm(k1, . . . , km+1) = 1 and each bi ·k−1i ∈ K[f ]. If we
can show that lm(b1, . . . , bm+1) = 1, then JR[f ] ∩K[pi2] = 0 and f is central. This can be
done by tedious calculations using lm(k1, . . . , km+1) = 1 and lm(bi · k−1i , g2, . . . , gm+1) = 1
for suitable gj ∈ B, as well as results such as [[B,B],K[f ]] ⊆ [[K[f ], B], B] and correspond-
ingly lm(B, . . . , B,K[f ]) ⊆ lm(K[f ], B, . . . , B) for any m (see [EG2006, Lemma 3.1]). For
higher m these calculations become too long to be done by hand, but for m = 2 and m = 3
I have checked all the details.
Every trivial extension is central. But some special central extensions also turn out
to be trivial, as the next lemma shows. Here a split central extension is just a central
extension which is split as an epimorphism.
23
Chapter 1. The Semi-Abelian Context
1.4.5 Lemma: [JK1994, Theorem 4.8] Let A be a semi-abelian category with a Galois
structure Γ where B is a Birkhoff subcategory of A. Then every split central extension is
trivial.
Proof. In fact every split central extension is trivial if and only if every central extension
is normal (Theorem 4.7 in [JK1994]).
1.5 Central extensions in the context of abelianisation
In Section 1.4 we have met central extensions with respect to a Galois structure. These
central extensions will become very important later on in the thesis, in Chapter 5. In the
special case of abelianisation, they coincide with other definitions of central extensions
which are more algebraic. Here we introduce two of these algebraic definitions, one using
centrality of congruences introduced by Smith [Smi1976] and generalised in [Ped1995,
CPP1992], and the other using the notion of central arrows first defined by Huq [Huq1968].
These two definitions of central extensions coincide in pointed protomodular categories (see
[BG2002b]), and we will also prove that they do indeed coincide with the Galois-theoretic
central extension for abelianisation. Then we can apply results about these algebraic
central extensions also to the Galois-theoretic ones we will be using. Even when the
Birkhoff subcategory is not that of abelian objects, central morphisms in the sense of Huq
appear in the homology theory, as can be seen in Section 4.4.
Huq centrality
We first introduce the concept of cooperating morphisms, using the terminology due to
Bourn [Bou2002].
1.5.1 Definition: Two morphisms with common codomain f : B −→ A and f ′ : B′ −→ A
are said to cooperate when there is a morphism φf,f ′ : B ×B′ −→ A such that the fol-
lowing diagram commutes.
B
(1B ,0)

f
#+PP
PPP
PPP
PPP
PPP
P
B ×B′
φf,f ′ ,2 A
B′
(0,1B′ )
LR
f ′
3;nnnnnnnnnnnnnnn
The morphism φf,f ′ is called the cooperator of f and f ′. We sometimes write φ instead
of φf,f ′ if the context makes clear which maps are meant.
In Huq’s terminology f and f ′ commute.
24
1.5 Central extensions in the context of abelianisation
1.5.2 Remark: Notice that as the pair of morphisms into A × A′ is jointly epic (see
Section 1.1), the cooperator is unique as soon as it exists. So for any pair of morphisms f
and f ′, having a cooperator is a property, not a structure.
1.5.3 Example: In the category of groups, two subgroups H ,2 ,2G and K ,2 ,2G
cooperate precisely when they commute. More generally, two group homomorphisms
f : H −→ G and h : K −→ G cooperate when their images in G commute, that is, for
all h ∈ H and k ∈ K, we have f(h)g(k) = g(k)f(h) in G.
If f and g have a cooperator φ : H ×K −→ G, then for all h ∈ H and k ∈ K we have
φ(h, k) = φ((h, 1) · (1, k)) = φ(h, 1) · φ(1, k) = f(h) · g(k)
and similarly φ(h, k) = φ((1, k) · (h, 1)) = g(k) · f(h). Conversely, if the images of f and
f ′ commute, we can define a cooperator φ by φ(h, k) = f(h) · g(k). This becomes a group
homomorphism precisely because the images of f and g commute.
From this we define the following notion of central morphism, and that of a central
extension.
1.5.4 Definition (Huq central): A morphism f : B −→ A is called central (in the
sense of Huq) when f and 1A cooperate. An extension f : B
 ,2A is called a central
extension when its kernel is central in the sense of Huq.
K[f ]
(1K[f ],0)

 (
Ker f
(H
HH
HH
HH
HH
K[f ]×B φ ,2 B
B
(0,1B)
LR uuuuuuuuuuu
1.5.5 Example: When A is the category of groups, central extensions defined as above
coincide with the traditional central extensions, extensions where the kernel lies inside the
centre of the group. In the categories LieK of Lie algebras or LeibK of Leibniz algebras,
central extensions in this sense are also those f : b −→ a where the kernel K[f ] lies in the
centre Zb = {z ∈ b | [z, b] = 0 for all b ∈ b} of b. Notice that the Huq central extensions in
LeibK are not those where K[f ] lies in the Lie centre of b, as seen in Example 1.4.4. As we
will see in Proposition 1.5.15, the central extensions in the sense of Definition 1.5.4 coincide
with Galois central extensions with respect to abelianisation. As the abelianisation of a
Leibniz algebra is not just a Lie algebra but already an abelian Lie algebra, we get the
same characterisation of central extensions in LeibK as in LieK in this case.
25
Chapter 1. The Semi-Abelian Context
1.5.6 Remark: Notice the relation of central morphisms to abelian objects as defined in
1.2.1: an object A is abelian when the identity 1A is central.
For more results about cooperating and central morphisms see for example Sections 1.3
and 1.4 of [BB2004]. In particular, we will use the following two results:
1.5.7 Lemma: [BB2004, Corollary 1.3.8] Consider the following commutative diagram
V
v ,2

U

u

V ′
v′lr

B
f ,2 A B′
f ′lr
where u is a monomorphism. If f and f ′ cooperate, then v and v′ cooperate as well.
1.5.8 Lemma: [Bou2002], also [BB2004, Proposition 1.3.20] If f is a central morphism,
then any morphism of the form f◦g is central.
Centrality of congruences
One can also define an equivalence relation to be central. This will have a close connection
to the central extensions above. We first introduce our notation for equivalence relations.
1.5.9 Notation: We write an equivalence relation R ,2 r ,2B ×B onB as R pi1 ,2
pi2
,2B or sim-
ply (R, pi1, pi2), and denote the subdiagonal which is induced by reflexivity by dR : B −→ R;
that is, pi1dR = pi2dR = 1B.
R
pi1  ,2
pi2
 ,2 B
lrdRlr
As we are in an exact category, every equivalence relation is the kernel pair of its
coequaliser (see Definition 1.1.2). Therefore we will often write an equivalence relation
(R, pi1, pi2) on an object B as (R[f ], pi1, pi2), where f is a regular epimorphism (the co-
equaliser of pi1 and pi2).
To any equivalence relation (R[f ], pi1, pi2) on an object B, we can associate a normal
subobject of B by taking the composite
K[pi1]
 ,2kerpi1 ,2 R[f ]
pi2  ,2 B.
26
1.5 Central extensions in the context of abelianisation
This composite coincides with the kernel of f and so is indeed a normal subobject of B.
K[f ] = K[pi1]
 ,2 kerpi1 ,2

R[f ]
pi1
_
pi2  ,2 B
f
_
0 ,2 B
f  ,2 A
There is another notion of a normal subobject of B, defined via equivalence relations.
1.5.10 Definition: [Bou2000] An arrow k : K −→ B is said to be normal to an equiv-
alence relation R on B when
• k−1(R) is the largest equivalence relation (K ×K,pi1, pi2) on K,
• there is a map k : K ×K −→ R making the diagram
K ×K k ,2
pi2

pi1

R
pi2

pi1

K
k
,2 B
commute, and moreover both of the commutative squares are pullbacks. That is,
the induced map (K × K,pi1, pi2) −→ (R, pi1, pi2) in the category EqA of internal
equivalence relations in A is a discrete fibration.
Such an arrow k is necessarily a monomorphism, and when A is protomodular, k can
be normal to at most one equivalence relation, making normality a property rather than a
structure. Every kernel is normal, and in a semi-abelian category, normal monomorphisms
in this sense and kernels coincide. As we will only be working in this easier situation, we
can read normal mono as kernel as we have been so far.
The association above gives a bijection between the normal subobjects of B and the
equivalences on B. For a given equivalence relation R on B, we call the subobject defined
above the associated normal subobject or normalisation kR. This bijection already
exists in the context of pointed protomodular categories (see [Bou2000]). The associated
normal subobject to R can be viewed as the equivalence class of zero under R.
1.5.11 Definition (Smith central): [Ped1995, CPP1992] Two equivalence relations
R and S on an object B centralise (in the sense of Smith) when there is a double
27
Chapter 1. The Semi-Abelian Context
equivalence relation C on R and S such that any commutative square in the diagram
C
p1 ,2
p2
,2
p1

p2

S
pi1

pi2

R
pi2
,2
pi1 ,2
B
is a pullback. When such a C exists, we call it the centralising double relation on R and
S.
An equivalence relation R on B is called central when R and B × B (the largest
equivalence relation on B) centralise.
Some references say R and S are connected when they centralise. This terminology
appears for example in [BG2002b].
There is a very useful connection between central equivalence relations in this sense
and central morphisms in the sense of Huq.
1.5.12 Proposition: [GVdL2008b, Proposition 2.2] Let A be a pointed protomodular
category. An equivalence relation R in A is central if and only if its associated normal
subobject kR is central in the sense of Huq.
It follows immediately that an extension f : B −→ A is a central extension if and only
if its kernel pair is central in the sense of Smith.
Coincidence of algebraic and Galois-theoretic definitions
Another useful characterisation of a central equivalence relation is given by the subdiagonal
expressing the reflexivity of the relation.
1.5.13 Proposition: Let B be an object in a semi-abelian category. The equivalence
relation (R, pi1, pi2) on B is central if and only if its subdiagonal dR : B −→ R is a kernel.
Proof. R is central if and only if it centralises with B×B, the largest equivalence relation
on B. The normal subobject associated to B × B is of course the identity on B. Then
Theorem 5.2 in [BG2002b] tells us that R and B × B are connected (or equivalently,
centralise) if and only if the composite B
kB×B
B
dR ,2R is normal, i.e., is a kernel.
We will now compare these two algebraic notions of central extensions to the Galois-
theoretic one arising from abelianisation. So let A be a semi-abelian category with sub-
category AbA of abelian objects, let ab: A −→ AbA be the abelianisation functor and let
E and Z denote the regular epimorphisms in A and AbA respectively.
28
1.5 Central extensions in the context of abelianisation
1.5.14 Proposition: [BG2002a, Proposition 3.1] An extension f : B −→ A is central with
respect to the Galois structure Γ = (A,AbA,E,Z, ab,⊆) if and only if the subdiagonal dR[f ]
of the kernel pair of f is a kernel.
R[f ]
pi1  ,2
pi2
 ,2 B
lrdlr
f  ,2 A
This enables us to compare the two notions of centrality.
1.5.15 Proposition: In a semi-abelian category A, an extension f : B −→ A is central
with respect to Γ = (A,AbA,E,Z, ab,⊆) if and only if it is a central extension in the sense
Definition 1.5.4.
Proof. This follows directly from Propositions 1.5.12, 1.5.13 and 1.5.14.
Central extensions in this sense have a useful property which we will need in Chapter 6.
1.5.16 Proposition: [GVdL2008b, Proposition 2.3] Let A be a semi-abelian category and
let f : B −→ A be a central extension with respect to abelianisation. Every subobject of the
kernel Ker f : K[f ] −→ B of f is normal in B.
29
Chapter 1. The Semi-Abelian Context
30
Chapter 2
Comonadic Homology
In this chapter, we introduce the first viewpoint on semi-abelian homology: comonadic
homology. This works in a slightly more general setting than that of the usual semi-
abelian homology theories introduced in Section 1.3. The concept of comonadic homology
was first introduced in an abelian context by Barr and Beck in [BB1969]. Everaert and Van
der Linden generalised it to semi-abelian categories in [EVdL2004b]. First we introduce
simplicial objects and their homology in Section 2.1, and then Section 2.2 shows how a
comonad creates a simplicial object out of any given object, which gives rise to comonadic
homology. Section 2.3 treats comonadic homology as a functor in the second variable,
that of coefficients. The results of this section are mine, and are mostly straightforward
generalisations of the corresponding abelian results in [BB1969].
2.1 Simplicial objects
In semi-abelian categories, simplicial objects play the role which is filled by chain complexes
or resolutions in abelian categories. The slightly more complex structure is needed here
as it is generally not possible to add or subtract maps in semi-abelian categories.
Given a simplicial object in A, we can form its normalised chain complex. Let us first
fix some notation for simplicial objects.
2.1.1 Notation: A simplicial object A = (An)n≥0 in A
· · ·
,2,2,2,2 A2
,2,2,2 A1
,2 ,2 A0
has face operators ∂i : An −→ An−1 for i ∈ [n] = {0, . . . , n} and n ∈ N>0, and de-
generacy operators σi : An −→ An+1, for i ∈ [n] and n ∈ N, subject to the simplicial
identities
∂i◦∂j = ∂j−1◦∂i if i < j
σi◦σj = σj+1◦σi if i ≤ j
∂i◦σj =

σj−1◦∂i if i < j
1 if i = j or i = j + 1
σj◦∂i−1 if i > j + 1.
31
Chapter 2. Comonadic Homology
In a semi-abelian category A, we can define the homology of a simplicial object A via
the Moore complex of A.
2.1.2 Definition (Moore complex): Let A be a simplicial object in a semi-abelian
category A. The Moore complex or normalised chain complex N(A) has as objects
N0A = A0, N−nA = 0 and
NnA =
n−1⋂
i=0
K[∂i : An −→ An−1] = K[(∂i)i∈[n−1] : An −→ Ann−1],
for n ≥ 1, and boundary maps dn = ∂n◦
⋂
iKer ∂i : NnA −→ Nn−1A for n ≥ 1. This
gives rise to a functor N: SA −→ ChA from the category of simplicial objects in A to the
category of chain complexes in A, called the normalisation functor.
The object of n-cycles is ZnA = K[dn] =
⋂n
i=0K[∂i : An −→ An−1] for n ≥ 1. We
write Z0A = A0.
The Moore complex of a simplicial object is always a proper chain complex [EVdL2004b,
Theorem 3.6]; thus we can define
HnA = HnN(A)
for a simplicial object A. In the abelian case, the homology of the Moore complex is the
same as the homology of the unnormalised chain complex C(A) of A, where CnA = An
and dn = ∂0 − ∂1 + · · ·+ (−1)n∂n.
Notice that the Moore complex and thus the homology of a simplicial object only
involve the face maps ∂i, and not the degeneracies σi. So we need only consider semi-
simplicial maps between simplicial objects, i.e. maps that commute with the ∂i but not
necessarily with the σi. This will be used in Chapter 3.
The homology objects obtained this way are special objects of A.
2.1.3 Lemma: [EVdL2004b, Theorem 5.5] Let A be a simplicial object in a semi-abelian
category A. For any n ≥ 1, the object HnA is an abelian object of A.
An important property of the normalisation functor is the following:
2.1.4 Lemma: [EVdL2004b, Proposition 5.6] Let A be a semi-abelian category. The
Moore normalisation functor N: SA −→ ChA is exact.
Proof. A slightly easier proof than that in [EVdL2004b] can be found in [Eve2007].
This, together with the Snake Lemma, implies the following result.
32
2.1 Simplicial objects
2.1.5 Lemma: [EVdL2004b, Corollary 5.7] A short exact sequence of simplicial objects
0 ,2 A ,2 B ,2 C ,2 0
in A gives rise to a long exact sequence
· · · ,2 HnA ,2 HnB ,2 HnC
rykkkk
kkkk
kkkk
kkkk
k
Hn−1A ,2 · · · ,2 H0C ,2 0
of homology objects which depends naturally on the given short exact sequence.
Later we will be considering simplicial resolutions, which are really augmented simpli-
cial objects.
2.1.6 Definition: An augmented simplicial object is a simplicial object A together with
a map ∂0 : A0 −→ A−1 satisfying ∂0◦∂0 = ∂0◦∂1, that is, it has equal composite with the
two face maps A1 −→ A0. A contraction of an augmented simplicial object A is a family
of maps hn : An −→ An+1, for n ≥ −1, which satisfy ∂0hn = 1An and ∂ihn = hn−1∂i−1 for
i > 0. A simplicial object that admits a contraction is called contractible.
· · ·
,2 ,2,2,2 A2
h2
T]
,2,2,2 A1
h1
U^
,2,2 A0
h0
U^
,2 A−1
h−1
U_
When computing the homology of a simplicial object, the following observation is often
useful.
2.1.7 Lemma: [EVdL2004b, Proposition 3.9] (cf. [Bou2001]) Given a diagram
A
∂1 ,2
∂0
,2 B
e ,2 C
where e◦∂0 = e◦∂1, suppose there is a common splitting t : B −→ A of ∂0 and ∂1, that is,
∂0◦t = 1B = ∂1◦t. Then e is the coequaliser of ∂0 and ∂1 if and only if e is the cokernel of
∂1◦Ker ∂0.
A consequence of this result is:
2.1.8 Lemma: [EVdL2004b, Proposition 3.11] A contractible augmented simplicial object
A has H0A = A−1 and HnA = 0 for n ≥ 1.
33
Chapter 2. Comonadic Homology
A simplicial object in the category of sets is commonly called a simplicial set. A
classical property simplicial sets may have is the Kan property. Kan simplicial sets are
exactly the fibrant ones (in the usual model structure on SSet) and may be described as
follows.
2.1.9 Definition (Kan property): Let S be a simplicial set and n ≥ 1, k ∈ [n] natural
numbers. An (n, k)-horn in S is a sequence (si)i∈[n]\{k} of elements of Sn−1 satisfying
∂i(sj) = ∂j−1(si) for all i < j and i, j 6= k. A filler of an (n, k)-horn (si)i is an element s
of Sn satisfying ∂i(s) = si for all i 6= k. A simplicial set S is Kan when every horn in S
has a filler.
This Kan property can be generalised to simplicial objects in a regular category as
follows (see [CKP1993]).
2.1.10 Definition (Internal Kan property): Let A be a simplicial object in a regular
category A and n ≥ 1, k ∈ [n] natural numbers. An (n, k)-horn in A is a family of maps
(bi : B −→ An−1)i∈[n]\{k} satisfying ∂ibj = ∂j−1bi for all i < j and i, j 6= k; we can view
this as a map b : B −→ (An−1)n. A filler of an (n, k)-horn b : B −→ (An−1)n is given by
a surjection p : Z −→ B and a generalised element z : Z −→ An satisfying ∂iz = bip for
i 6= k. This can be viewed as a filler “up to enlargement of domain”.
Carboni, Kelly and Pedicchio show in [CKP1993] that every simplicial object of a
regular category A is Kan if and only if A is Mal’tsev. Thus when A is regular Mal’tsev, for
example semi-abelian, we can apply the internal Kan property for every simplicial object
in A. This property is very powerful and can be used in many situations. To demonstrate
this, we present a small new result involving the maps used in the Moore complex. Recall
that the object NnA is the kernel of the map (∂0, . . . , ∂n−1) : An −→ (An−1)n.
2.1.11 Proposition: Let A be a simplicial object in a semi-abelian category A. Let In
denote the image of the map (∂0, . . . , ∂n−1) : An −→ (An−1)n. Then the following is an
equaliser diagram:
In ,2 (An−1)n
(∂0pi1,∂0pi2,...,∂0pin−1,∂1pi2,...,∂n−2pin−1) ,2
(∂0pi0,∂0pi1,...,∂0pin−2,∂1pi1,...,∂n−2pin−2)
,2 (An−2)k
where the two parallel maps are constructed to give all the horn conditions ∂ipij = ∂j−1pii
for i < j and j ≤ n− 1; so k = 12(n− 1)(n− 2).
Proof. The equaliser of these two maps is an (n, n)-horn through which all other (n, n)-
horns factor. So we have to show that all (n, n)-horns factor through In.
34
2.1 Simplicial objects
Let b : B −→ (An−1)n be an (n, n)-horn. As in a regular Mal’tsev category every
simplicial object is Kan [CKP1993, Theorem 4.2], this horn must have a filler
(p : Z −→ B, z : Z −→ An).
This gives the following diagram:
Z
z

p  ,2
z′
$>
>>
>>
>>
>>
B

b
%,SSS
SSS
(An−1)n ,2 ,2 (An−2)k
An
 ,2 In
2:
i
2:lllll
Since p is the coequaliser of some pair of maps, and iz′ = bp has equal composite with this
pair and i is monic, z′ factors through p and we get a factorisation of the horn B through
In as desired.
This fact can be used to prove that the normalisation functor is exact, a result which
Tim Van der Linden and Tomas Everaert prove in a different way in [EVdL2004b] (see
Lemma 2.1.4).
2.1.12 Lemma: Given a short exact sequence of simplicial objects
0 ,2 A
f ,2 B
g ,2 C ,2 0
in a semi-abelian category A, the induced sequence of chain complexes
0 ,2 NA ,2 NB ,2 NC ,2 0
is also exact.
Proof. As mentioned above, this is the same as Lemma 2.1.4, but we give an alternate
proof here using Proposition 2.1.11.
Given a short exact sequence of simplicial objects as above, we must show that
0 ,2 NnA ,2 NnB ,2 NnC ,2 0
is exact in A for each n ≥ 0. For n = 0 we have N0A = A0, so the result is clear. For
n ≥ 1 we use that NnA is the kernel of
(∂i)i∈[n−1] : An −→ Ann−1
35
Chapter 2. Comonadic Homology
where Ann−1 is the n-fold product of An−1. Then Proposition 2.1.11 tells us that the image
IAn of this map is an equaliser. This implies that the image sequence
0 ,2 IAn ,2 I
B
n
,2 ICn
,2 0
is also exact: we have
0 ,2 An
 ,2 fn ,2
_
Bn
gn  ,2
_
Cn ,2
_
0
0 ,2 IAn ,2
f ,2


'G
GG
GG
G IBn
g  ,2


ICn
,2


0
K[g]
7 7A
7Awwwww
v v
vv
v
]g
0 ,2 Ann−1
 ,2
fnn−1 ,2

Bnn−1
gnn−1 ,2

Cnn−1

0 ,2 Akn−2
 ,2
fkn−2 ,2 Bkn−2 ,2 Ckn−2
where the first row is exact and the columns are image factorisations. Also, as kernels
commute with products, fnn−1 is the kernel of gnn−1. Clearly f factors over the kernel K[g]
of g. The kernel property of Ann−1 also induces a map K[g] −→ Ann−1, which has equal
composite with the two maps to Akn−2, as fkn−2 is a monomorphism. So this map factors
over the equaliser IAn and by the universal properties we see that I
A
n
∼= K[g].
Now we can use the 3× 3 Lemma:
0

0

0

0 ,2 NnA ,2_

NnB ,2_

NnC_

,2 0
0 ,2 An
 ,2 ,2
_
Bn
 ,2
_
Cn ,2
_
0
0 ,2 IAn
 ,2 ,2

IBn
 ,2

ICn
,2

0
0 0 0
Here all columns and the last two rows are exact, so the first row is also exact, as desired.
36
2.2 Comonadic Homology
2.2 Comonadic Homology
When A is a semi-abelian monadic category (e.g., a semi-abelian variety; see [GR2004]
for a precise characterisation), there is a canonical forgetful/free comonad G = (G, , δ)
on A, which gives rise to a functorial simplicial resolution GA of any object A, that is, an
augmented simplicial object over A with face maps ∂i = GiGn−iA : Gn+1A −→ GnA and
degeneracies σi = GiδGn−iA : Gn+1A −→ Gn+2A.
· · ·
,2,2,2,2 G
3A GGA ,2
G2A ,2
G2A
,2 G
2A
GA ,2
GA
,2 GA
A ,2 A
This gives rise to the following Barr-Beck style [BB1969] notion of homology:
2.2.1 Definition: [EVdL2004b] LetB be a Birkhoff subcategory of a semi-abelian monadic
category A with reflector I : A −→ B and canonical comonad G. For any object A of A
and any n ≥ 0, we define
Hn+1(A, I)G = HnNIGA. (A)
In fact, comonadic homology can be defined in a more general context than that of a
semi-abelian category with a Birkhoff subcategory.
2.2.2 Definition (Comonadic homology): [EVdL2004b] Let C be any category with a
comonad
G = (G : C −→ C, δ : G =⇒ G2,  : G =⇒ 1C)
and let E : C −→ A be a functor to a semi-abelian category A. For n ≥ 1, the object
Hn(A,E)G = Hn−1NEGA
is called the nth homology object of A (with coefficients in E) relative to the
comonad G. This defines a functor Hn(−, E)G : C −→ A, for every n ≥ 1.
The dimension shift here is not present in Barr and Beck’s original definition, but
was introduced in [EVdL2004b] to make it better adjusted to the non-abelian examples
(homology of groups, Lie algebras, crossed modules) which traditionally have a shifted
numbering.
When E is a contravariant functor, we get comonadic cohomology in a similar way.
2.2.3 Example (Comonads and functors of coefficients): The most common exam-
ple of a comonad used for comonadic homology is that of a forgetful/free comonad on
a variety of algebras, such as the free group comonad on the category of groups. Using
37
Chapter 2. Comonadic Homology
this comonad, the abelianisation functor E = ab: Gp −→ Ab gives rise to integral group
homology. The forgetful/free comonad on the category R-Mod of R-modules gives rise to
two well-known homology theories. When E = N⊗R− : R-Mod −→ Ab for a fixed module
N , we get the Tor groups as homology, that is
Hn(M,N ⊗R −)G = TorRn (M,N).
The contravariant functor E = HomR(−, N) : R-Mod −→ Ab gives the Ext groups:
Hn(M,HomR(−, N))G = ExtnR(M,N)
However, if we take the covariant Hom-functor HomR(N,−), we obtain the Eckmann-
Hilton homotopy groups (see [BB1969, Example 1.1]). Notice that when taking comonadic
homology of the covariant Hom-functor, we are still using projective (in fact free) reso-
lutions, and not injective resolutions, which is why we don’t get the Ext groups in this
case.
There are also other comonads on the category of R-modules: given a ring homomor-
phism φ : S −→ R, we can view any R-module as an S-module, and any S-module can
be turned into an R-module by tensoring it with R over S. This adjunction gives rise to
another comonad on R-Mod.
R-Mod
U !)LL
LLL
LLL
LL
Gφ ,2 R-Mod
S-Mod
R⊗S(−)
5=rrrrrrrrrr
Using this so-called relative comonad, the functors given by tensoring and homing as
above give rise to Hochschild’s relative Tor and Ext groups. This comonad will be used
in Example 3.3.14.
Many more examples of comonads and functors of coefficients exist, see for exam-
ple [BB1969].
2.3 Hn(−, E)G as a functor in the variable E
Let G = (G, δ, ) be a comonad on the category C, and E : C −→ A a functor into a
semi-abelian category A. We can view the homology functor Hn(−, )G as a functor
[C,A] −→ [C,A], taking E to Hn(−, E)G. As such it has certain universal properties, which
we discuss in this section. The results in this section are straightforward generalisations
from the abelian case discussed in [BB1969], and in most parts the proofs carry over
without much change. We still give them here in our own notation for completeness.
38
2.3 Hn(−, E)G as a functor in the variable E
2.3.1 Proposition (G-acyclicity): For n ≥ 1, we have
Hn(−, EG)G = 0,
and for n = 0 the map
λ : H0(−, EG)G
∼= ,2 EG
is an isomorphism.
Proof. For any object X ∈ C, the augmented simplicial object EGGX is contractible, as
we have EGnδX : EGGnX −→ EGGn+1X for n ≥ 0, which satisfies EGGnX◦EGnδX =
1EGGnX and EGGiGn−iX◦EGnδX = EGn−1δX◦EGGiGn−i−1X . Thus using Lemma 2.1.8
the result follows.
We define a G-exact sequence to be a sequence of functors
0 −→ E1 −→ E2 −→ E3 −→ 0
such that
0 −→ E1G −→ E2G −→ E3G −→ 0
is exact. We then get
2.3.2 Proposition (G-connectedness): Any G-exact sequence
0 −→ E1 −→ E2 −→ E3 −→ 0
gives rise to a long exact sequence on homology:
· · · ,2 Hn(−, E1)G ,2 Hn(−, E2)G ,2 Hn(−, E3)G
∂
ovfffff
fffff
fffff
fffff
fff
Hn−1(−, E1)G ,2 · · · ,2 H0(−, E3)G ,2 0
Proof. For any object X ∈ C the G-exact sequence gives rise to a short exact sequence of
simplicial objects
0 −→ E1GX −→ E2GX −→ E3GX −→ 0
which in turn gives rise to the desired long exact sequence on homology using Lemma 2.1.5.
Analogously to [BB1969], we can define a theory of G-left derived functors by the
above properties:
39
Chapter 2. Comonadic Homology
2.3.3 Definition: L = (Ln, λ, ∂) is a theory of G-left derived functors if the following
are satisfied:
(1) Each Ln is a functor Ln : [C,A] −→ [C,A].
(2) λ : L0 −→ 1[C,A] is a natural transformation from L0 to the identity functor on the
functor category [C,A].
(3) (G-acyclicity) For a functor of the form EG we have
λ : L0(EG)
∼=−→ EG is an isomorphism,
Ln(EG) = 0 for n ≥ 1.
(4) (G-connectedness) Every G-exact sequence 0 −→ E1 −→ E2 −→ E3 −→ 0 gives rise
to a long exact sequence:
· · · ,2 Ln(E1) ,2 Ln(E2) ,2 Ln(E3)
∂
qxiiii
iiii
iiii
iiii
iii
Ln−1(E1) ,2 · · · ,2 L0(E3) ,2 0
where ∂ depends of the given sequence, and
LnE3
∂ ,2

Ln−1E1

LnF3 ∂
,2 Ln−1F1
commutes for any map of sequences
0 ,2 E1 ,2

E2 ,2

E3 ,2

0
0 ,2 F1 ,2 F2 ,2 F3 ,2 0.
We will show that the homology functor above is special amongst these theories of
G-left derived functors. To do this, we first prove a result which is needed in the proof of
the next theorem.
2.3.4 Lemma: The following is a coequaliser diagram:
L0(EG2)
L0(EG) ,2
L0(EG)
,2 L0(EG)
L0(E) ,2 L0E
40
2.3 Hn(−, E)G as a functor in the variable E
Proof. Notice that this diagram is the image under L0 of the lowest part of the augmented
simplicial object EG. Similarly to Lemma 2.1.7, it is enough to show that L0(E) is the
cokernel of L0(d1) : L0(N1(EG)) −→ L0(EG), which is the composite
L0(N1(EG))
L0(Ker (EG)) ,2 L0(EG2)
L0(EG) ,2 L0(EG).
For this let M be the kernel of E : EG −→ E, and form the following diagram, where the
bottom row is G-exact (as EG : EG2 −→ EG is split epic and so regular epic):
N1(EG)
ν
v
d1

0 ,2M
i
,2 EG
E
,2 E ,2 0
The morphism ν is induced by the kernel property of i. We now apply L0 to this diagram
and get
L0(N1(EG))
L0(ν)
t|ppp
ppp
ppp
pp
L0(d1)

L0(M)
L0(i)
,2 L0(EG)
L0(E)
,2 L0(E) ,2 0.
Now the bottom row is exact, as the bottom row of the previous diagram was G-exact.
Note that we do not require L0 to preserve 0, but we still have L0(E)L0(d1) = 0, as the
exact sequence
0 ,2 N1EG ,2 EG2
EG ,2 EG ,2 0
gives rise to the diagram
0 ,2 L0(N1EG) ,2
L0(d1) %,SSS
SSS
SSS
SSS
SS
L0(EG2)
L0(EG) ,2
L0(EG)

L0(EG) ,2
L0(E)

0
L0(EG)
L0(E)
,2 L0(E)
where the top row is short exact by G-connectedness and G-acyclicity.
Thus if we denote the images of L0(i) and L0(d1) by I and I ′ respectively, we get a
factorisation I ′ −→ I, which is of course monic. If we can show it is regular epic as well,
we have shown that L0(d1) followed by L0(E) is also exact. For this it is sufficient to
show that L0(ν) is a regular epi.
Let K = K[ν]; then it is sufficient to show that
0 ,2 KG ,2 N1(EGG)
νG ,2MG ,2 0
41
Chapter 2. Comonadic Homology
is exact, as then G-connectedness implies that
L0(K) ,2 L0(N1(EG))
L0(ν) ,2 L0(M) ,2 0
is exact. So we only need to prove that νG is regular epic. In the following diagram both
rows are exact, and the left downwards square and right upwards square commute:
0 ,2 N1(EGG)
νG

j ,2 EG3
EGG

EG2 ,2 EG2 ,2 0
0 ,2MG
h
LR
iG ,2 EG2
EGδ
LR
EG ,2 EG ,2
Eδ
LR
0
The dotted arrow h is induced by EG2◦EGδ◦iG = 0. Then
iGνGh = EGG◦j◦h = EGG◦EGδ◦iG = iG
so as iG is monic, νG is split epic, so in particular regular epic. Thus L0(E) is the cokernel
of L0(d1), as asserted.
This implies that L0(E) is the coequaliser of L0(EG) = h0 and L0(EG) = h1 (we
rename them for convenience), in an analogous way to Lemma 2.1.7. Let k = L0(KerEG).
In the following diagram, the outer rectangle is a pushout, and both squares commute.
L0(N1EG)

L0(KerEG)=k,2 L0(EG2)
L0(EG)=h0

L0(EG)=h1 ,2 L0(EG)
L0(E)

f

0 ,2 L0(EG)
L0(E) ,2
f
'.
L0(E)
D
To show that L0(E) is the required coequaliser, it is enough to show that if fh0 = fh1,
then fh1k = 0. But this is clear as h0k = 0. Thus f factors through L0(E).
We can now prove the following Uniqueness Theorem:
2.3.5 Theorem: Let L be a theory of G-left derived functors. Then there exists a unique
family of natural isomorphisms
Ln
σn
∼=
,2 Hn(−, )G
42
2.3 Hn(−, E)G as a functor in the variable E
for n ≥ 0, (i.e. natural isomorphisms LnE −→ Hn(−, E)G which are also natural in E),
which are compatible with the augmentation λ and the connecting homomorphism ∂:
L0E
σ0 ,2
λ &C
CC
CC
CC
C H0(−, E)G
λ
uttt
ttt
ttt
t
E
LnE3
σn

∂ ,2 Ln−1E1
σn−1

Hn(−, E3)G
∂
,2 Hn−1(−, E1)G
Proof. The proof from [BB1969] for the abelian case carries over almost completely to the
semi-abelian case, but we give it in full in our own notation for convenience.
We start by constructing σ0.
L0(EG2)
L0(EG) ,2
L0(EG)
,2
λ∼=

L0(EG)
λ∼=

L0(E) ,2 L0(E)
σ0

EG2
EG ,2
EG
,2 EG ,2 H0(−, E)G
From Lemma 2.3.4 we know that the top row of this diagram is a coequaliser, as is the
bottom row (by definition of homology and Lemma 2.1.7). Thus we get an induced map
σ0, which is an isomorphism, since both λ occurring in the diagram are isomorphisms (by
the acyclicity property). It is clear that σ0 also commutes with the augmentations λ to
E, since L0(E) is a regular epi.
Now we construct the other σn inductively. From the construction of σ0 it is clear that
it is also natural in E. We take M = K[E : EG −→ E], so that
0 ,2M
i ,2 EG
E ,2 E ,2 0
is G-exact.
For n = 1 we then have
L1(EG) = 0 ,2 L1E
σ1

∂L ,2 L0M
σ0∼=

L0i ,2 L0(EG)
σ0∼=

H1(−, EG)G = 0 ,2 H1(−, E)G ∂H ,2 H0(−,M)G H0(−,i)G ,2 H0(−, EG)G
where both rows are exact, by G-connectedness. This induces the map σ1, which is also
an isomorphism (by uniqueness of limits, or the Five Lemma).
43
Chapter 2. Comonadic Homology
For n > 1 we use the diagram
Ln(EG) = 0 ,2 LnE
σn

∂L
∼=
,2 Ln−1M
σn−1∼=

,2 0 = Ln−1(EG)
Hn(−, EG)G = 0 ,2 Hn(−, E)G ∂H∼= ,2 Hn−1(−,M)G ,2 0 = Hn−1(−, EG)G
Again both rows are exact, so σn = ∂−1H σn−1∂L is also an isomorphism. This defines the
natural isomorphisms σn for all n. Now we must verify compatibility with the connecting
homomorphisms.
Let 0 ,2E1
α ,2E2
β ,2E3 ,20 be G-exact. Let K be the kernel of the composite
γ : E2G
E2 ,2 E2
β ,2 E3 , and M3 the kernel of E3 as before. Then we get a map of
G-exact sequences
0 ,2 K
j ,2
κ

E2G
γ ,2
βG

E3 ,2 0
0 ,2M3 i3
,2 E3G E3
,2 E3 ,2 0
(B)
where κ is induced by the kernel property of i3.
This gives rise to a diagram
0 ,2 L1E3

∂′L ,2
∂L
&-UUU
UUUU
UUUU
U L0M3
σ0∼=

,2 L0(E3G)
σ0∼=

L0K
L0κ
18iiiiiiiiiiii
σ0∼=

0 ,2 H1(−, E3)G
∂′H ,2
∂H
&-TTT
TTTT
H0(−,M3)G ,2 H0(−, E3G)G
H0(−,K)G H0(−,κ)G
18jjjjjjj
Here both rows are exact, and the triangles commute by naturality of the connecting
homomorphism. The rightmost square and the front right square commute by naturality
of σ0. Now since ∂H is the kernel of H0(−, j)G (as H1(−, E2G)G = 0), we get a factorisation
L1E3 −→ H1(−, E3)G making the front left square commute. This morphism also makes
the back rectangle commute. But that rectangle defines σ1 uniquely, so the front left
square must commute with σ1 substituted in.
Now letKβ be the kernel of β, then we also have the following map ofG-exact sequences
0 ,2 K
j ,2
κ2

E2G
γ ,2
E2

E3 ,2 0
0 ,2 Kβ ,2 E2 β
,2 E3 ,2 0
(C)
44
2.3 Hn(−, E)G as a functor in the variable E
with the obvious induced κ2. We also have an induced morphism of G-exact sequences
0 ,2 E1 ,2

E2 ,2 E3 ,2 0
0 ,2 Kβ ,2 E2 ,2 E3 ,2 0
which implies that Ln(Kβ) ∼= Ln(E1) and Hn(−,Kβ) ∼= Hn(−, E1) for n ≥ 0, using G-
connectedness and the Five Lemma.
The map ofG-exact sequences (C) induces the following prism, where we can substitute
E1 for Kβ using the above isomorphisms:
L1E3
σ1

∂L ,2
∂′L %,
SSSS
SSSS
S L0K
σ0

L0(κ2)rykk
kkkk
kkk
L0E1
σ0

H1(−, E3)G ∂H ,2
∂′H
%,SS
SSS
S
H0(−,K)G
H0(−,κ2)Grzlll
lll
H0(−, E1)G
The triangles again commute by naturality of the connecting homomorphisms, the right
front square commutes by naturality of σ0, and the back square is the square whose
commutativity we have shown above. Thus the front left square also commutes.
The case n ≥ 2 works similarly, again using diagram (B) to get
0 ,2 LnE3
σn

∂′L
∼=
,2
∂L
∼= &-UUUU
UUUU
UUUU
Ln−1M3
σn−1∼=

,2 0
Ln−1K
07hhhhhhhhhhhh
σn−1∼=

0 ,2 Hn(−, E3)G
∂′H
∼=
,2
∂H
∼=
&-UUU
UUUU
U
Hn−1(−,M3)G ,2 0
Hn−1(−,K)G
07hhhhhhhh
This time it is clear that the front left square commutes with σn substituted in, as ev-
erything in sight is an isomorphism (by G-connectedness and G-acyclicity). Then we can
again use (C) to get a similar prism to the above, which proves that
LnE3
σn

∂L ,2 Ln−1E1
σn−1

Hn(−, E3)G
∂H
,2 Hn−1(−, E1)G
commutes.
45
Chapter 2. Comonadic Homology
Thus we have shown that the comonadic homology theories are “the only” such theories
of G-left derived functors, up to isomorphism.
46
Chapter 3
A Comparison Theorem for
Simplicial Resolutions
Introduction
In Section 2.3 we studied comonadic homology as a functor in the second variable. This
chapter addresses a different question: how does the comonadic homology theory depend
on the given comonad G? Barr and Beck showed in [BB1969] that, in the abelian setting,
two comonads G and K give rise to the same comonadic homology theory when they
generate the same class of projective objects. In this chapter, we prove a suitable extension
of their result to the semi-abelian case. That is, we show that given any category C with
two comonads G and K, and any functor E : C −→ A to a semi-abelian category A, the
functors Hn(−, E)G and Hn(−, E)K : C −→ A are isomorphic for n ∈ N when G and K
generate the same Kan projective class. The condition on the projective class ensures that
homming from a (G- or K-)projective object into a (G- or K-)simplicial resolution gives a
Kan simplicial set. It is exactly the condition needed to prove a comparison theorem for
these simplicial resolutions.
The examples reveal that this condition is not too strong. In an additive category,
homming from any object into any simplicial object results in a Kan simplicial set, and
when C is a regular Mal’tsev category, which includes semi-abelian ones, then the condition
is fulfilled as soon as the G-projective objects are also regular projectives.
For their proof in the abelian case, Barr and Beck make heavy use of additive structure
via a free additive completion of the category C. As a semi-abelian category is only additive
when it is abelian, we have to use a different approach to extend their result to the semi-
abelian setting. We prove a comparison theorem which shows that, given a projective
class P, any two P-resolutions are homotopy equivalent and consequently have the same
homology. The advantage of our method is that it shows that any P-resolution of a given
object will give the same homology, so that it is also possible to use resolutions not coming
from a comonad, should this turn out to be more convenient. The only subtlety lies in
the definition of a P-resolution of an object A: this is an augmented simplicial object
A = (An)n≥−1 where A−1 = A, all other An ∈ P, and for any object P ∈ P the augmented
simplicial set Hom(P,A) is Kan and contractible. This is the reason for our condition
47
Chapter 3. A Comparison Theorem for Simplicial Resolutions
on the projective class stated above: we must make sure that a G-resolution is also a
P-resolution.
Section 3.1 sets the scene by explaining the definition of a P-resolution in detail. Sec-
tion 3.2 is devoted to the Comparison Theorem 3.2.3: if P is a simplicial object over B
with each Pi ∈ P, and A is a simplicial object over A such that all augmented simplicial
sets Hom(Pi,A) are contractible and Kan, then any map f : B −→ A extends to a semi-
simplicial map f : P −→ A, and any two such extensions are simplicially homotopic. In
this section we also relate our comparison theorem to that of Tierney and Vogel [TV1969],
which uses a different definition of resolution, in a category with finite limits.
The Comparison Theorem is used in Section 3.3 to prove the main result of this chapter,
Theorem 3.3.11: under the condition on C mentioned above, any two comonads G and
K that generate the same class of projectives induce isomorphic homology theories. We
obtain it as an immediate consequence of Corollary 3.3.10 which states that, in a semi-
abelian category, simplicially homotopic maps have the same homology: if f ' g then, for
any n ∈ N, Hnf = Hng.
All results in this chapter are joint work with Tim Van der Linden and also appear in
our paper [GVdL2007].
3.1 Simplicial resolutions
To obtain the comonadic homology of a given object, we need to consider simplicial reso-
lutions relative to a chosen class of projectives. Here we recall the definition of a projective
class and give some examples.
3.1.1 Definition (Projective class): Let C be a category, P an object and e : B −→ A
a morphism of C. Then P is called e-projective, and e is called P -epic, if the induced
map
Hom(P, e) = e◦(·) : Hom(P,B) −→ Hom(P,A)
is a surjection. That is, for every map P −→ A, there is a (not necessarily unique) map
making the following diagram commute:
P
y
B
e ,2 A
Let P be a class of objects of C. A morphism e is called P-epic if it is P -epic for every
P ∈ P; the class of all P-epimorphisms is denoted by P-epi. Let E be a class of morphisms
in C. An object P is called E-projective if it is e-projective for every e in E; the class of
48
3.1 Simplicial resolutions
E-projective objects is denoted E-proj. C is said to have enough E-projectives if for
every object Y there is a morphism P −→ Y in E with P in E-proj.
A projective class on C is a pair (P,E), P a class of objects of C, E a class of
morphisms of C, such that P = E-proj, P-epi = E and C has enough E-projectives. Since,
given a projective class (P,E), P and E determine each other, we will sometimes abusively
write the projective class P or the projective class E.
It is easy to see that any retract of a projective object is also projective, as is any
coproduct of projectives.
3.1.2 Example: If E is the class of regular epimorphisms, P is called the class of regular
projectives. In a variety, the class of regular projectives is generated by the free objects,
hence there are enough projectives.
The regular projectives in a variety C are also generated by the values of the canonical
comonad C, induced by the forgetful functor to Set. More generally, any comonad on a
category C generates a projective class:
3.1.3 Definition (Projective class generated by a comonad): Let G = (G, , δ)
be a comonad on a category C. An object P in C is called G-projective if it is in the
projective class (PG,EG) generated by the objects of the form GA. A map in EG is called
a G-epimorphism.
The A-component A : GA −→ A of the counit  is always a G-epimorphism. Indeed,
any map f : GB −→ A factors over A as Gf◦δB, because A◦Gf◦δB = f◦GB◦δB = f .
It is now clear that C has enough projectives of this class, since for any A we have
A : GA −→ A.
This definition coincides with the definition of G-projectives in [BB1969]. There a
G-projective object is an object P which admits a map s : P −→ GP such that P s = 1P .
Indeed, if P ∈ P, then the identity on P factors over the P-epimorphism P , which gives
the splitting s.
A simplicial resolution or a simplicial object in a semi-abelian category A gives rise to
simplicial sets, for example by homing into the simplicial object from a fixed object of the
category. As we saw in Section 2.1, an important classical property simplicial sets may
satisfy is the Kan property defined in 2.1.9. We need the simplicial objects in the category
C to satisfy a similar property, but relative to a chosen projective class P on C. For the
purposes of this chapter, we will call this the relative Kan property.
3.1.4 Definition (relative Kan property): A simplicial object A is Kan (relative to
P) when for every object P ∈ P the simplicial set Hom(P,A) is Kan.
49
Chapter 3. A Comparison Theorem for Simplicial Resolutions
3.1.5 Example: If C is regular with enough regular projectives and P the induced pro-
jective class, saying that A is Kan relative to P is the same as saying that the simplicial
object A is Kan, in the internal sense of Definition 2.1.10. Every simplicial object of C has
this Kan property if and only if C is a Mal’tsev category [CKP1993, Theorem 4.2]. Thus
when C is semi-abelian, every simplicial object is Kan with respect to the class of regular
projectives.
Note, however, that C need not have enough projectives for the internal Kan condition
to make sense. The projective objects in the definition of the relative Kan property given
here may be replaced by an enlargement of domain as in Definition 2.1.10. In case there
are enough projectives, of course the two notions do coincide.
3.1.6 Example: It is well known that the underlying simplicial set of a simplicial group
is always Kan. This may be seen as a consequence of the previous example, because the
category Gp is a Mal’tsev variety and the forgetful functor U : Gp −→ Set is represented
by the group of integers Z.
Since C is an arbitrary category (without any extra structure) and A is just semi-
abelian (rather than abelian), we have to be careful when considering simplicial resolutions
of objects of C. Definition 3.1.7 seems to suit our purposes.
3.1.7 Definition (Simplicial resolution): Let P be a projective class. A P-resolution
of A is an augmented simplicial object A = (An)n≥−1 with A−1 = A, where An ∈ P for
n ≥ 0, and for every object P ∈ P the augmented simplicial set Hom(P,A) is Kan and
contractible.
In this chapter, we focus on simplicial resolutions in a category C which are generated
by a comonad G on C. For any G-projective object P , the simplicial set Hom(P,GA) is
contractible: choose a splitting s for P : GP −→ P ; given a map f : P −→ Gn+1A, define
hn(f) = Gf◦s : P −→ Gn+2A. The morphisms hn : Hom(P,Gn+1A) −→ Hom(P,Gn+2A)
then satisfy ∂0hn = 1Hom(P,Gn+1A) and ∂ihn = hn−1∂i−1 for i > 0. Thus they give a
contraction of the simplicial set Hom(P,GA). Later we assume that the category C and
the projective class P generated by G are such that GA is Kan relative to P for any object
A, so that GA is a P-resolution of A.
In the case when C is a category with finite limits, there exists another definition of
simplicial resolution, using simplicial kernels. We give the definition of simplicial kernels
here so that we can relate our Comparison Theorem of the next section to that of Tierney
and Vogel [TV1969].
50
3.1 Simplicial resolutions
3.1.8 Definition (Simplicial kernels): [TV1969] Let
(fi : B −→ A)0≤i≤n
be a sequence of n+ 1 morphisms in the category C. A simplicial kernel of (f0, . . . , fn)
is a sequence
(ki : K −→ B)0≤i≤n+1
of n+ 2 morphisms in C satisfying fikj = fj−1ki for 0 ≤ i < j ≤ n+ 1, which is universal
with respect to this property. In other words, it is the limit for a certain diagram in C.
For example, the simplicial kernel of one map is just its kernel pair. If C has finite
limits, simplicial kernels always exist. We can then factor any augmented simplicial object
through its simplicial kernels as follows:
· · ·
,2,2,2,2
%B
BB
BB
BB
B A2 ,2
,2
,2
%B
BB
BB
BB
B A1
,2 ,2
%B
BB
BB
BB
B A0
,2 A−1
K3
9C||||||||
9C||||||||
9C||||||||
9C|||||||| K2
9C||||||||
9C||||||||
9C||||||||
K1
9C||||||||
9C||||||||
Here the Kn+1 are the simplicial kernels of the maps (∂i)i : An −→ An−1. This gives a
definition of P-exact simplicial objects:
3.1.9 Definition: [TV1969] Let P be a projective class. An augmented simplicial object
A = (An)n≥−1 is called P-exact when the comparison maps to the simplicial kernels and
the map ∂0 : A0 −→ A−1 are P-epimorphisms.
3.1.10 Remark: It can be shown that for any P-exact simplicial object A, the simplicial
set Hom(P,A) is contractible for any P ∈ P. We will call this property of A relative
contractibility.
A resolution in the Tierney-Vogel sense is then a P-exact augmented simplicial object
A in which all objects An for n ≥ 0 are in the projective class P. For their definition
they need the presence of simplicial kernels, so they have to assume for example that the
category C has finite limits. In our definition all assumptions are on the comonad G or
rather the induced projective class P, and not on the category C. In the next section we
will make clear the connections between our definition and theirs.
51
Chapter 3. A Comparison Theorem for Simplicial Resolutions
3.2 The Comparison Theorem
Let P be a projective class on C.
3.2.1 Lemma: Let P ∈ P, and let A be an augmented simplicial object for which the aug-
mented simplicial set Hom(P,A) is contractible and Kan. Let n ≥ 0. Given a sequence of
maps (ai : P −→ An−1)i∈[n] satisfying ∂iaj = ∂j−1ai for i < j, there is a map a : P −→ An
with ∂ia = ai.
Proof. Define the maps bi+1 = hn−1(ai), where (hn)n≥−1 is the contraction of the simplicial
set Hom(P,A). These maps satisfy ∂0bi+1 = ai, and also ∂jbi+1 = ∂i+1bj+1 for i < j ≤ n,
since (∂jhn−1)(ai) = hn−2(∂j−1ai), and ∂j−1ai = ∂iaj for i < j.
a2
a0
a1
Thus they form an (n + 1, 0)-horn in the simplicial set Hom(P,A), and since we are
assuming that this simplicial set is Kan, this horn has a filler b : P −→ An+1. This gives
the required map a = ∂0b.
3.2.2 Remark: This lemma shows that in the presence of finite limits our P-resolutions
are also simplicial resolutions in the sense of Tierney and Vogel [TV1969]; that is, the com-
parison maps to the simplicial kernels are P-epimorphisms. Together with Remark 3.1.10
we see that P-exactness and relative contractibility are equivalent in the situation when
we have finite limits and any simplicial object is Kan relative to P. So if C has finite limits,
the Comparison Theorem 2.4 from [TV1969] is more general than the one following in this
section, but in the absence of finite limits our Comparison Theorem still works.
We now prove our Comparison Theorem using the above lemma.
3.2.3 Theorem (Comparison Theorem): Let P be a simplicial object over B with each
Pi ∈ P, and let A be a simplicial object over A, for which all the augmented simplicial sets
Hom(Pi,A) are contractible and Kan. Then any map f : B −→ A can be extended to a
semi-simplicial map f : P −→ A, and any two such extensions are simplicially homotopic.
52
3.3 Comonads generating the same projective class
Proof. We construct this semi-simplicial map inductively, using Lemma 3.2.1.
· · ·
,2,2,2,2 P2
f2

,2,2,2 P1
f1

,2,2 P0
f0

,2 B
f−1=f
· · ·
,2,2,2,2 A2
,2,2,2 A1
,2,2 A0 ,2 A
Suppose the maps fj : Pj −→ Aj are given for −1 ≤ j < n, and commute appropriately
with the ∂i. This gives us n + 1 maps ai : Pn −→ An−1, where i ∈ [n], by composing the
∂i : Pn −→ Pn−1 with fn−1. These maps satisfy ∂iaj = ∂j−1ai for i < j, since ∂ifn−1 =
fn−2∂i, and the ∂j in P satisfy the simplicial identities. Thus we can use Lemma 3.2.1 to
obtain the map fn : Pn −→ An such that ∂ifn = ai = fn−1∂i.
Now suppose f : P −→ A and g : P −→ A are two semi-simplicial maps commuting with
f : B −→ A. We construct a homotopy hni : Pn −→ An+1 for n ≥ 0 and 0 ≤ i ≤ n, which
satisfies ∂0hn0 = fn, ∂n+1h
n
n = gn and
∂ih
n
j =

hn−1j−1 ∂i for i < j
∂ih
n
i−1 for i = j 6= 0
hn−1j ∂i−1 for i > j + 1.
h00 can be constructed using Lemma 3.2.1. Suppose the h
k
j exist for k < n and commute
appropriately with the ∂i. Then hn0 must satisfy ∂0h
n
0 = fn, ∂1h
n
0 = ∂1h
n
1 and ∂ih
n
0 =
hn−10 ∂i−1 for i > 1. Of these maps, all are known except for ∂1h
n
1 . Setting a
0
0 = fn and
a0i = h
n−1
0 ∂i−1 for i > 1, we form an (n + 1, 1)-horn in Hom(Pn, A). A filler for this
horn gives hn0 , and also a
0
1 = ∂1h
n
1 , which is needed for the next step. Now suppose h
n
j
are given for j < l, and we have al−1l = ∂lh
n
l = ∂lh
n
l−1. Then a
l
i = h
n−1
l−1 ∂i for i < l,
all = a
l−1
l and a
l
i = h
n−1
l ∂i−1 for i > l+1 form an (n+1, l+1)-horn. A filler for this gives
hnl and a
l
l+1 = a
l+1
l+1 for the next step. In the last step we have a
n
i = h
n−1
n−1∂i for i < n,
ann = a
n−1
n = ∂nh
n
n−1 and ann+1 = gn. Then we use Lemma 3.2.1 again to get hnn.
3.3 Comonads generating the same projective class
In this section we will need an assumption on the category C and the projective class P
generated by the comonad G.
3.3.1 Definition (Kan projective class): Let G be a comonad on a category C and let
P be the projective class generated G. The projective class P is called a Kan projective
class on C when any augmented simplicial object A which is relatively contractible is also
Kan relative to P.
53
Chapter 3. A Comparison Theorem for Simplicial Resolutions
In particular, when G generates a Kan projective class, the simplicial object GA is
Kan relative to P for any object A. Thus if K is a second comonad which generates the
same projective class, the simplicial object KA is automatically also Kan relative to P.
3.3.2 Example (Additive categories): If C is an additive category, any projective class
P in C is a Kan projective class, since then for any simplicial object A and any object
P , the simplicial set Hom(P,A) is actually a simplicial group and thus Kan (cf. Exam-
ple 3.1.6). For example, any comonad on the category R-Mod of (left) R-modules generates
a Kan projective class, including both the “absolute” comonad generated by the forget-
ful/free adjunction to Set and the “relative” comonad induced by a ring homomorphism
φ : R −→ S as in Example 2.2.3.
3.3.3 Example (Regular projectives): When C is a regular category and the projective
class P is the class of regular projectives, as remarked in Example 3.1.5, saying that a
simplicial object A is Kan relative to P is the same as saying A is internally Kan in C.
Thus when C is also Mal’tsev, every simplicial object is Kan [CKP1993, Theorem 4.2], and
GA is a P-resolution. This includes the forgetful/free comonads on the categories Gp of
groups, Rng of non-unital rings, XMod of crossed modules, Comm of commutative rings,
K-Alg of associative K-algebras, etc. In fact it includes any variety C with the comonad
generated by the forgetful functor to Set, as the following argument shows.
Given a comonad G on a category C which comes from an adjunction
C
U $?
??
??
??
G ,2 C
D
F
:D
we can determine the class of morphisms of the projective class (P,E) generated by G in
the following way:
Given an object A and a morphism e : B −→ C in C, the diagram
GA
f

B
e ,2 C
corresponds via the adjunction to
UA

UB
Ue ,2 UC
54
3.3 Comonads generating the same projective class
If e is in E, by choosing A = C and f = C , we see that Ue must be split in D, since
C corresponds under the adjunction to 1UC . Conversely if Ue is a split epimorphism in
D, we can factor any map UA −→ UC over Ue, which implies that we can factor any
map f : GA −→ C over e in C, thus e ∈ E. Therefore the class E is exactly the class of
morphisms whose images under U are split in D. Thus when C is a variety and U is the
forgetful functor to Set, we will always get the class of regular projectives on C.
The first step towards our goal is to show that the two simplicial resolutions GA and
KA are homotopy equivalent.
3.3.4 Lemma: Let G and K be two comonads on C which generate the same Kan projective
class P. Then the simplicial objects GA and KA are homotopy equivalent for any object A.
Proof. Our assumptions on C and P imply that for any object A, the simplicial objects
GA and KA are both P-resolutions of A. Thus we can use the Comparison Theorem 3.2.3
to get semi-simplicial maps f : GA −→ KA and g : KA −→ GA which commute with the
identity on A.
· · ·
,2 ,2,2,2 G
3A
f2

,2,2,2 G2A
f1

,2,2 GA
f0

,2 A
1A
· · ·
,2,2,2,2 K
3A
g2

,2,2,2 K2A
g1

,2,2 KA
g0

,2 A
1A
· · ·
,2,2,2,2 G
3A ,2
,2
,2 G2A
,2,2 GA ,2 A
Using the second part of the Comparison Theorem we see that both fg and gf are ho-
motopic to the identity on KA and GA respectively. Thus GA and KA are homotopy
equivalent.
3.3.5 Remark: In this case we don’t actually need the full strength of the second half
of Theorem 3.2.3. For any semi-simplicial map f : GA −→ GA which commutes with the
identity on A, we can use the homotopy hni = (G
i+1fn−i)σi to see that it is homotopic to
the identity on GA.
Given a functor E : C −→ A, the simplicial objects EGA and EKA are still homotopy
equivalent. We now show that, when A is a semi-abelian category, two simplicially ho-
motopic semi-simplicial maps induce the same map on homology (see also [VdL2006]).
For this we need to define a special simplicial object, so that all the maps that form a
simplicial homotopy are taken together to form a single semi-simplicial map. We do this
by defining the following limit objects AIn.
55
Chapter 3. A Comparison Theorem for Simplicial Resolutions
3.3.6 Notation: Suppose thatA has finite limits and let A be a simplicial object inA. Put
AI0 = A1 and, for n > 0, let A
I
n be the limit (with projections pr1,. . . , prn+1 : A
I
n −→ An+1)
of the zigzag
An+1
∂1 $?
??
??
??
??
An+1
∂1z




∂2 $?
??
??
??
??
· · ·
z $
An+1
∂nz




An An An
in A.
Let 0(A)n, 1(A)n : AIn −→ An and s(A)n : An −→ AIn denote the morphisms respec-
tively defined by
0(A)0 = ∂0
0(A)n = ∂0pr1,
1(A)0 = ∂1
1(A)n = ∂n+1prn+1,
and s(A)n = (σ0, . . . , σn).
3.3.7 Proposition: Let A be a simplicial object in a finitely complete category A. Then
the faces ∂Ii : A
I
n −→ AIn−1 and degeneracies σIi : AIn −→ AIn+1 given by
∂I0 = ∂0pr2 : A
I
1 −→ AI0
∂I1 = ∂2pr1 : A
I
1 −→ AI0
σI0 = (σ1, σ0) : A
I
0 −→ AI1
prj∂
I
i =
∂i+1prj if j ≤ i∂iprj+1 if j > i : AIn −→ An
prkσ
I
i =
σi+1prk if k ≤ i+ 1σiprk−1 if k > i+ 1 : AIn −→ An+2,
for i ∈ [n], 1 ≤ j ≤ n and 1 ≤ k ≤ n+2, determine a simplicial object AI . The morphisms
mentioned in Notation 3.3.6 above form simplicial morphisms
0(A), 1(A) : AI −→ A and s(A) : A −→ AI
such that 0(A)◦s(A) = 1A = 1(A)◦s(A). In other words, (AI , 0(A), 1(A), s(A)) forms a
cocylinder on A.
Two semi-simplicial maps f, g : B −→ A are simplicially homotopic if and only they
are homotopic with respect to the cocylinder (AI , 0(A), 1(A), s(A)): there exists a semi-
simplicial map h : B −→ AI satisfying 0(A)◦h = f and 1(A)◦h = g.
Using a Kan property argument, we now give a direct proof that homotopic semi-
simplicial maps have the same homology.
56
3.3 Comonads generating the same projective class
3.3.8 Proposition: Let A be a simplicial object in a semi-abelian category A; consider
0(A) : AI −→ A.
For every n ∈ N, Hn0(A) is an isomorphism.
Proof. Recall that in a semi-abelian category every simplicial object is Kan, relative to the
class of regular epimorphisms. Using the Kan property, we show that the commutative
diagram
Nn+1AI
Nn+10(A)  ,2
d′n+1

Nn+1A
d′n+1

ZnAI Zn0(A)
 ,2 ZnA
is a generalised regular pushout (see Definition 1.1.19); then it is also a pushout, and
Lemma 1.1.17 implies that the induced map HnAI −→ HnA is an isomorphism. Consider
morphisms z : Y0 −→ ZnAI and a : Y0 −→ Nn+1A that satisfy d′n+1◦a = Zn0(A)◦z. It
is enough to show that there exist a regular epimorphism y : Y −→ Y0 and a morphism
h : Y −→ Nn+1AI satisfying d′n+1◦h = z◦y and Nn+10(A)◦h = a◦y: this implies that the
comparison map to the pullback is a regular epimorphism, by Lemma 1.1.10 and the fact
that the morphisms of a limit cone form a jointly monic family.
We first sketch the geometric idea of this in the case n = 0. Consider a = a0 and
z = z0 as in Figure 3.1; then (up to enlargement of domain) using the Kan property twice
yields the needed (h0, h1) in N1AI .
a
z
0
Z00(A) ◦ z = d′1 ◦ a
0
h1
h0
Figure 3.1: Using the Kan property twice to obtain (h0, h1) in N1AI .
For arbitrary n, write
a0 =
⋂
j
ker∂j◦a : Y0 −→ An+1,
and (z0, . . . , zn) =
⋂
j ker∂j◦z. Note that as z : Y0 −→ ZnAI , we have ∂Ii z = 0 for i ∈ [n],
which implies ∂izj−1 = 0 for i < j − 1 and i > j, where 1 ≤ j ≤ n + 1. We also have
∂jzj−1 = ∂jzj for 1 ≤ j ≤ n from the definition of the objects AIn. The map a0 in
turn satisfies ∂ia0 = 0 for i ∈ [n], and ∂n+1a0 = ∂0z0. This last equality follows from
d′n+1◦a = Zn0(A)◦z.
57
Chapter 3. A Comparison Theorem for Simplicial Resolutions
Suppose we have regular epimorphisms yk : Yk −→ Yk−1 for 1 ≤ k ≤ n + 2, and mor-
phisms hk−1 : Yk −→ An+2 satisfying ∂i◦hk−1 = 0 for 1 ≤ k ≤ n+2 and i /∈ {k−1, k, n+2},
and ∂n+2hk−1 = zk−1y1 · · · yk for 1 ≤ k ≤ n + 1, and also ∂0h0 = a0y1. We set
y = y1◦ · · · ◦yn+2. This gives us the required map
h = (h0◦y2◦ · · · ◦yn+2, . . . , hn+1) : Yn+2 −→ Nn+1AI
which satisfies d′n+1◦h = z◦y and Nn+10(A)◦h = a◦y.
We construct these maps yk and hk inductively. To get h0 we form an (n+ 2, 1)-horn
(bi : Y0 −→ An+1)i of A by setting b0 = a0, bn+2 = z0 and bi = 0 for 1 < i < n + 2. A
filler for this horn gives y1 : Y1 −→ Y0 and h0 : Y1 −→ An+2 satisfying ∂0h0 = a0y1. Now
suppose for 1 ≤ k ≤ n+ 1 we have ak = ∂khk−1 : Yk −→ An+1 with ∂i◦ak = 0 for i ∈ [n],
and ∂n+1ak = ∂kzk−1y1y2 · · · yk. Then we can form an (n + 2, k + 1)-horn by setting
bk = ak, bn+2 = zky1 · · · yk and bi = 0 for i < k and k + 1 < i < n+ 2, which, when filled,
induces yk+1 and hk with the desired properties.
3.3.9 Remark: A homology functor Hn involves an implicit choice of colimits: the coker-
nels involved in the construction of the HnA. We may, and from now on we will, assume
that these colimits are chosen in such a way that Hn0(A) is an identity instead of just an
isomorphism. This gives us the equality in the next corollary.
3.3.10 Corollary: If f ' g then, for any n ∈ N, Hnf = Hng.
Proof. Proposition 3.3.8 states that Hn0(A) is an isomorphism; by a careful choice of
colimits in the definition of Hn, we may assume that Hn0(A) = 1HnA = Hn1(A).
A
B h ,2
g
.4
f *0
AI
1(A)
4=qqqqqqqqqqqqq
0(A)
!*MM
MMM
MMM
MMM
MM As(A)
lr
A
If now h is a homotopy f ' g, then Hnf = Hn0(A)◦Hnh = Hn1(A)◦Hnh = Hng.
Using the above, we can now prove our Main Theorem.
3.3.11 Theorem: Let G and K be two comonads on C which generate the same Kan
projective class P. Let E : C −→ A be a functor into a semi-abelian category. Then the
functors Hn(−, E)G and Hn(−, E)K from C to A are isomorphic for all n ≥ 1.
58
3.3 Comonads generating the same projective class
Proof. It follows from Lemma 3.3.4 that the simplicial objects EGA and EKA are ho-
motopy equivalent. Thus Corollary 3.3.10 implies that Hn(A,E)G ∼= Hn(A,E)K. Given a
map f : B −→ A, the two semi-simplicial maps
GB ,2 KB
Kf ,2 KA and GB
Gf ,2 GA ,2 KA
are both semi-simplicial extensions of f , so they are homotopic by the Comparison Theo-
rem 3.2.3. Again using Corollary 3.3.10, we see that the square
HnGB
HnGf ,2
∼=

HnGA
∼=

HnKB HnKf
,2 HnKA
commutes, which proves that the isomorphisms are natural.
3.3.12 Remark: In fact, the above isomorphism is also natural in the second variable. If
α : E =⇒ F is a natural transformation, then the square
Hn(A,E)G
∼= ,2
Hn(A,α)G

Hn(A,E)K
Hn(A,α)G

Hn(A,F )G ∼=
,2 Hn(A,F )K
also commutes, since
EGA ,2
αGA

EKA
αKA

FGA ,2 FKA
already commutes.
3.3.13 Remark: We could define homology just using a projective class instead of a
comonad, since the Comparison Theorem and Corollary 3.3.10 imply that any P-resolution
of A will give the same homology. Consider for example the following (Tierney-Vogel)
resolution in a category with finite limits:
Given an object A, we can find a P-projective object A0 with a P-epimorphism
∂0 : P0 −→ A, since there are enough P-projectives. We call this a presentation of A.
Take the kernel pair of ∂0, and take the presentation of the resulting object to get P1.
Composition gives two maps ∂0 and ∂1 to P0, and we can take the simplicial kernel of
59
Chapter 3. A Comparison Theorem for Simplicial Resolutions
these and the presentation of the resulting object to get P3 and so on. This gives a reso-
lution in the Tierney-Vogel sense [TV1969]. When P is a Kan projective class, it is also a
P-resolution in our sense and thus gives the same homology. This resolution is often easier
to work with than the functorial GA.
3.3.14 Example (Two comonads on R-Mod): Given a ring homomorphism φ : S −→ R,
every R-module can also be considered as an S-module by restricting the R-action to S
via φ. This gives rise to an adjunction R⊗S (−) a HomR(R,−) between the categories of
modules, where R is viewed as an S-module.
Now consider two rings S1 and S2 with a surjective ring homomorphism ψ : S1 −→ S2.
Let R be another ring, with ring homomorphisms as below which make the diagram
commute:
S1
φ1
!*MM
MMM
MMM
ψ
_
R
S2
φ2
4=qqqqqqqq
For each i = 1, 2 this gives us a comonad on R-Mod using the adjunction above:
R-Mod
Ui !)LL
LLL
LLL
LL
Gi ,2 R-Mod
Si-Mod
R⊗Si (−)
5=rrrrrrrrrr
We write Ui for the forgetful functor HomR(R,−) from R-modules to Si-modules.
As seen in Example 3.3.3, the projective class generated by Gi is given by the class of
maps in R-Mod which are split as Si-module maps. An R-module map e : B −→ C is split
as an Si-module map if and only if there exists a function f : C −→ B with f(sc+ s′c′) =
sf(c)+s′f(c′) for s ∈ I[φi]. Since ψ is a surjection, we have I[φ1] = I[φ2], so any R-module
map e has the property that U1(e) is split if and only if U2(e) is split. Thus the comonads
G1 and G2 induce the same projective class, and thus give rise to the same homology on
R-Mod, for any functor E : R-Mod −→ A to a semi-abelian category A. As mentioned in
[BB1969], when using the functors E = N ⊗R − or E = HomR(N,−) for an R-module
N , this homology is Hochschild’s S-relative Tor or Ext respectively; so we see we get the
same relative Tor or Ext functor for a ring S and any quotient of S. This means we can
replace S by the image of φ as a submodule of R.
60
Chapter 4
Homology via Hopf Formulae
In the early 1940s, Hopf proved the well-known formula for the second integral homology
of a special kind of topological space, which calculates the homology purely in terms of the
fundamental group of the space. This led to the definition of group homology independent
of a topological space. So today we can write his famous formula as
H2A =
[P, P ] ∩K[p]
[K[p], P ]
where p : P −→ A is a projective presentation of a group A. The group commutators used
here can be generalised so that this formula makes sense in any semi-abelian category.
Furthermore, it can be extended to higher dimensional Hopf formulae which give the
higher homology objects HnA. This course was taken by Everaert, Gran and Van der
Linden in [EGVdL2008] and further pursued by Everaert in his thesis [Eve2007]. We give
the main background and results here, as they will be needed later on in the thesis.
Though we will work in semi-abelian categories, most of the results in this section
do not need coproducts. In particular, all constructions borrowed from [EGVdL2008]
and [Eve2007] which take place in a semi-abelian category still work in pointed exact
protomodular ones: though these categories need not have coproducts, they still have
cokernels of kernels (see [BB2004, Corollary 4.1.3]). This allows us to apply our results
to examples where the category being considered is pointed exact protomodular but lacks
coproducts, such as the category of finite groups.
In Section 4.1 we introduce the concept of an axiomatically defined class of extensions
and the associated higher extensions which are the starting point for the whole theory.
Section 4.2 defines strongly E-Birkhoff subcategories, which are a generalisation of the
ordinary Birkhoff subcategories we met in Section 1.3. The Galois structures and the
induced centralisation and trivialisation functors on which the Hopf formulae crucially
build are discussed in Section 4.3. Finally in Section 4.4 we define homology via the
Hopf formulae and exhibit the Everaert sequence: a long exact homology sequence which
generalises and extends the Stallings-Stammbach sequence known in the case of groups.
This sequence and its universal properties play a crucial role in Chapter 5.
The content of this chapter is known material, and mainly taken from the work of
Everaert, Gran and Van der Linden [EGVdL2008] and Everaert’s thesis [Eve2007]. We
will need the results and concepts introduced here in Chapter 5.
61
Chapter 4. Homology via Hopf Formulae
4.1 Extensions and higher extensions
The main ingredient for higher Hopf formulae is the concept of higher-dimensional exten-
sions. To arrive at this notion, we will first introduce higher-dimensional arrows. Here A
always denotes a semi-abelian category, unless stated otherwise.
4.1.1 Definition (Higher-dimensional arrows): The category ArrkA consists of k-
dimensional arrows in A: Arr0A = A, Arr1A = ArrA is the category of arrows Fun(2,A)
where 2 is generated by a single map ∅ −→ {∅}, and Arrk+1A = ArrArrkA. Thus a dou-
ble arrow is a commutative square in A, a 3-arrow is a commutative cube, and a
k-arrow is a commutative k-cube. Clearly, ArrkA is also semi-abelian. The functor
ker : Arrk+1A −→ ArrkA maps a (k + 1)-arrow a to its kernel K[a], and a morphism (f ′, f)
between (k + 1)-arrows b and a to the induced morphism between their kernels.
K[b]
ker(f ′,f)

 ,2Ker b ,2 B′
f ′

b ,2
⇓
B
f

K[a]  ,2
Ker a
,2 A′ a ,2 A
Repeating it n times gives a functor kern : Arrk+nA −→ ArrkA which sends a (k+n)-arrow
a to the object Kn[a] of ArrkA.
We now axiomatically define a class of extensions as in [Eve2007]. The definitions
and most of the results and proofs in this section are taken from Tomas Everaert’s thesis
[Eve2007].
Given a class of morphisms E in a semi-abelian category A, we write obE for the class
of objects A ∈ |A| that occur as domains or codomains of the arrows in E: A ∈ obE if and
only if there is at least one f ∈ E with f : A −→ B or f : C −→ A. We also write AE for
the full subcategory of A determined by the objects of obE.
4.1.2 Definition (Extensions): [Eve2007] Let E be a class of regular epimorphisms in
A with 0 ∈ obE. Then E is called a class of extensions when it satisfies the following
properties:
(1) E contains all split epimorphisms f : B −→ A with A and B in obE;
(2) (a) if f : B −→ A and g : C −→ B are in E, then so is their composite f◦g;
(b) if f◦g is in E and B is in obE, then g is in E;
62
4.1 Extensions and higher extensions
(3) morphisms f ∈ E are stable under pullback along arrows h : D −→ A in A with
D ∈ obE, i.e. in the diagram below, h∗f is again in E whenever D ∈ obE;
D ×A B ,2
h∗f

B
f

D
h ,2 A
(4) given a short exact sequence in A as below with B ∈ obE, we have f ∈ E whenever
K ∈ obE;
0 ,2K ,2B
f ,2A ,20
(5) given a commutative diagram as below with short exact rows in A, whenever both
g and k are in E and B ∈ obE, then we also have b ∈ E.
0 ,2 K ,2
k

C
g ,2
b

A ,2 0
0 ,2 L ,2 B
f ,2 A ,2 0
We call an arrow f ∈ E an E-extension, or just an extension, and write B  ,2A .
4.1.3 Remark: As we have 0 ∈ obE, condition (3) implies that the kernel K[f ] of any
extension f is in obE. This gives the converse to (4), so that we have: for any short exact
sequence in A as in (4), K ∈ obE if and only if f ∈ E.
4.1.4 Example: The leading example of a class of extensions is the class of all regular
epimorphisms in A. In this case we have obE = |A|, all objects of the category A. In fact,
it follows from (4) that the class of all regular epimorphisms is the only class E with this
property.
We also introduce a concept of higher extensions.
4.1.5 Definition (Higher extensions): Given a class of extensions E in a semi-abelian
category A, we define the class of n-fold (E-)extensions (called n-extensions when E
is understood) inductively as follows:
a 0-extension is an object in obE, a 1-extension is an arrow in E, and for n ≥ 1,
an (n+ 1)-extension is a morphism (f ′, f) in ArrnA such that all arrows in the induced
63
Chapter 4. Homology via Hopf Formulae
diagram
B′
f ′
 '
b
z"
r
 %
P
 ,2
_
A′
a
_
B
f
 ,2 A
(D)
are n-extensions. Here P is the pullback of a and f . We will say double (E-)extension
for a 2-fold extension. We denote the class of (n+1)-fold E-extensions by En, thus E0 = E
and E1 denotes the double E-extensions.
To justify the name of (n+ 1)-extension, we will have to show that the class En really
is a class of extensions in the sense of 4.1.2. Clearly it is enough to show that E1 is a class
of extensions, and the rest follows by induction, as we can then view En as (En−1)1. So
we will now concentrate on double extensions.
4.1.6 Remark: Notice that a morphism (f ′, f) : b −→ a in ArrA is a double extension if
and only if (b, a) : f ′ −→ f is a double extension.
B′
f ′  ,2
b
_
A′
a
_
B
f
 ,2 A
We can see that, as any extension is a regular epimorphism, a double extension gives a
square in A which is a regular pushout (see Section 1.1). In particular, a double extension
is a regular epimorphism in ArrA, both viewed as (f ′, f) : a −→ b and as (a, b) : f ′ −→ f .
We will now prove a property relating double extensions to extensions, reminiscent of
condition (4) in Definition 4.1.2. This will be a key ingredient in proving that E1 is a class
of extensions.
4.1.7 Lemma: Given a commutative diagram in A with short exact rows as below, where
f ′, f , a and b are extensions,
0 ,2 K ′
k

 ,2 ,2 B′
b
_
f ′  ,2 A′ ,2
a
_
0
0 ,2 K  ,2 ,2 B
f
 ,2 A ,2 0
the right hand square is a double extension if and only if k is an extension.
64
4.1 Extensions and higher extensions
Proof. We can decompose the above diagram as follows, with pullback squares as indicated.
0 ,2 K ′  ,2 ,2
k

B′
f ′  ,2
r

A′ ,2 0
0 ,2 K  ,2 ,2 P  ,2
_
A
a
_
,2 0
0 ,2 K  ,2 ,2 B
f ,2 A ,2 0
By definition, the right hand square of the original diagram is a double extension if and
only if the factorisation r to the pullback is an extension. Notice that by Condition (3)
of Definition 4.1.2 the pullback P −→ A of f along a is again an extension and so P is in
obE, and by Remark 4.1.3 both K and K ′ are also in obE. If r is an extension, then k is
an extension by 4.1.2 (3). Conversely, if k is an extension, then so is r by 4.1.2 (5).
4.1.8 Proposition: Given a class of extensions E in A, the induced class E1 is a class of
extensions in ArrA.
Proof. We remarked earlier that a double extension is indeed a regular epimorphism in
ArrA. Notice that obE1 = E, and clearly 10 ∈ E. We have to show that E1 satisfies
conditions (1) to (5) of Definition 4.1.2. To distinguish between these conditions applied
to E and to E1, we will denote the conditions referring to E1 by (1)1 to (5)1.
A split epimorphism in ArrA is a morphism (f ′, f) : b −→ a such that both f ′ and f
are split in A by s′ and s respectively, and (s′, s) : a −→ b is also a morphism in ArrA.
Using Remark 4.1.6, to show that (1)1 holds it is enough to show that (b, a) : f ′ −→ f is
a double extension whenever a and b are extensions and (f ′, f) is a split epimorphism in
ArrA as above. This follows from (1) and Lemma 4.1.7, as the map between the kernels
K[b] and K[a] is also a split epimorphism.
For the next two conditions we again use Remark 4.1.6, which makes it easy to verify
that (2a)1 and (2b)1 follow from Lemma 4.1.7 and (2a) or (2b) respectively.
Condition (3)1 follows from (3) and (2a). The key point here is that when pulling
back (f ′, f) : b −→ a to say (g′, g) : e −→ d, the square formed by the comparison to the
pullbacks
E′ ,2

Q

B′  ,2 P
is also a pullback square, and this implies that (g′, g) is also an extension.
Condition (4)1 follows easily from (4), (2) and Lemma 4.1.7, and finally Condition (5)1
follows from (5) and (3). Again the squares induced by comparison maps to the pullbacks
play a crucial role.
65
Chapter 4. Homology via Hopf Formulae
Thus the classes En−1 are indeed classes of extensions for any n ≥ 1. We have obEn =
En−1. The n-extensions En−1 determine a full subcategory ExtnA of ArrnA. Notice that
ExtnA = (ArrnA)En . We sometimes analogously write Ext0A = AE. When we say that a
sequence
0 ,2 K[f ]  ,2 ,2 B
f  ,2 A ,2 0
is exact in ExtnA, we mean that it is an exact sequence in ArrnA, and the objects are
n-extensions. Recall from Remark 4.1.3 that f is an n+1-extension if and only if all three
objects are n-extensions.
Roughly, the idea behind this definition of k-extensions is the following: suppose we
are given a double extension (f ′, f) of an object A of A as in Diagram (D), and let α be
any element of A. Then in addition to the existence of elements β of B and α′ of A′ such
that f(β) = α and a(α′) = α, there is also an element β′ ∈ B′ such that b(β′) = β and
f ′(β′) = α′, whichever β and α′ were chosen.
4.2 Strongly (E-)Birkhoff subcategories
Given a class of extensions as in Section 4.1, we will now generalise the notion of Birkhoff
subcategory given in Definition 1.3.2. Again this concept is taken from [EGVdL2008] and
[Eve2007]. This more general definition will allow us to handle higher extensions at the
same time as ordinary Birkhoff subcategories, which makes for clearer statements and
proofs later on.
4.2.1 Definition: Given a class of extensions E in a semi-abelian category A, and a
reflective subcategory B of AE, we write I : AE −→ B for the reflector and denote the unit
of the adjunction by η. We call B a strongly E-Birkhoff subcategory of A if for every
(E-)extension f : B −→ A the induced square
B
ηB

f ,2 A
ηA

IB
If ,2 IA
is a double (E-)extension.
4.2.2 Remark: Notice that this immediately implies that the reflector I takes extensions
to extensions, and also that ηA is an extension (and so a regular epimorphism) for each
A. This in turn implies that B is closed under subobjects: let A ,2 m ,2B be a mono in A
66
4.2 Strongly (E-)Birkhoff subcategories
with B ∈ B. Then as η is a natural transformation we have
A ,2
m ,2
ηA
_
B
ηB
IA
Im ,2 IB
and so ηA is both a mono and a regular epi, and thus an isomorphism. In fact it is well
known that a reflective subcategory is closed under subobjects if and only if each ηA is a
regular epimorphism, but we only need this one direction.
Note that B is also closed under any limits that exist in AE, as it is a reflective
subcategory (see for example Proposition 3.5.3 in [Bor1994]).
4.2.3 Example (strongly E-Birkhoff subcategories): When E is the class of regular
epimorphisms in A, strongly E-Birkhoff subcategories of A coincide with the usual Birkhoff
subcategories (see Lemma 1.3.4), and AE = A. Thus the category Ab of abelian groups is
a strongly (regular epi)-Birkhoff subcategory of Gp. Later we will meet special classes of
higher extensions, the central n-extensions, which form a strongly E-Birkhoff subcategory
of ArrnA when E is the class of n-extensions, so AE = ExtnA. This allows us to state
results that work at the same time for a semi-abelian category A with a usual Birkhoff
subcategory and for a category of higher extensions ExtnA in A.
There is another criterion for a reflective subcategory B to be strongly E-Birkhoff,
which uses the kernel of the unit ηA for a given object A. We first introduce some notation.
We can view the reflector I as a functor I : AE −→ AE. Then we have another functor
J : AE −→ AE, given by JA = K[ηA], which fits into the following short exact sequence of
functors.
0 ,2J  ,2
µ ,21AE
η  ,2I ,20
4.2.4 Proposition: Let B be a reflective subcategory of AE such that the unit ηA : A −→ IA
is a regular epimorphism for any object A ∈ A. Then B is a strongly E-Birkhoff subcategory
of A if and only if the functor J : AE −→ AE preserves extensions.
Proof. Consider the following diagram with short exact rows:
JB
Jf

 ,2 ,2 B
f
_
ηB  ,2 IB
If
_
JA
 ,2 ,2 A ηA
 ,2 IA
By 4.1.7 it follows that if the right hand square is a double extension, then Jf is an
extension. Conversely, if Jf is an extension, then 4.1.2(4) implies that ηB and ηA are
67
Chapter 4. Homology via Hopf Formulae
extensions, and so by 4.1.2(2) If is as well. Then again by 4.1.7 the right hand square is
a double extension for any extension f , and so B is strongly E-Birkhoff.
4.3 The Galois structures Γn, centralisation and trivialisa-
tion
The strongly E-Birkhoff subcategories give rise to Galois structures Γ as defined in 1.4.1.
In fact, we will see that a strongly E-Birkhoff subcategory B induces a whole sequence Γn
of Galois structures, each one giving rise to the next.
4.3.1 Proposition: Let E be a class of extensions (in the sense of 4.1.2) in a semi-abelian
category A, and B a strongly E-Birkhoff subcategory of A. Let Z be the class of arrows f
in B such that f ∈ E, and let I : AE −→ B be the reflector and ⊆ : B −→ AE the inclusion
functor. Then (AE,B,E,Z, I,⊆) is a Galois structure.
Proof. This is an immediate consequence of the definitions of a class of extensions 4.1.2
and a strongly E-Birkhoff subcategory 4.2.1.
4.3.2 Remark: Notice that we take the extensions E of the Galois structure to be a class
of extensions in the sense of 4.1.2, so there is no clash of terminology.
This Galois structure Γ = (AE,B,E,Z, I,⊆) gives rise to central extensions as defined
in 1.4.3. We denote the full subcategory of ArrA determined by these central extensions
by CExtBA. This gives a subcategory of the category ExtA determined by the class of
extensions E, which is dependant on the strongly E-Birkhoff subcategory B. When B
is clear from the context, we might also write CExtA. We will show that this category
of central extensions forms a strongly E1-Birkhoff subcategory of ArrA, thus giving us
another Galois structure as above. To show this, we must first construct a reflector
I1 : ExtA −→ CExtBA.
Recall the short exact sequence of functors AE −→ AE induced by the reflector I = I0:
0 ,2J  ,2
µ ,21A
η  ,2I ,20
From this, we build a similar short exact sequence of functors ExtA −→ ExtA as follows.
(The construction is made pointwise in ArrA, which has good categorical properties, but
the result turns out to be an extension.)
68
4.3 The Galois structures Γn, centralisation and trivialisation
4.3.3 Definition (Centralisation functor): Consider an extension f : B −→ A and its
kernel pair (R[f ], pi1, pi2). Write J1[f ] = K[Jpi1] and J1f : J1[f ] −→ 0.
J1[f ] = K[Jpi1]_

 ,2Ker Jpi1,2 JR[f ]
_
µR[f ]

Jpi1 ,2
Jpi2
,2 JB_
µB

K[f ] = K[pi1]
 ,2
Kerpi1
,2 R[f ]
pi1 ,2
pi2
,2 B
This clearly determines a functor J1 : ExtA −→ ExtA. Note that pi2◦Kerpi1 = Ker f , and
the left hand square is a pullback. We define the map µ1f : J1f −→ f as in the left hand
square below.
J1[f ] ,2
µB◦Jpi2◦Ker Jpi1
J1f_
µ1f
=⇒
B
f
_
0 ,2 A
B
ρ1f  ,2
f
_
η1f
=⇒
I1[f ]
I1f_
A A
The composition µB◦Jpi2◦Ker Jpi1 is a normal monomorphism, so we can take cokernels,
yielding the right hand square. Since µ1f is the kernel of its cokernel, we obtain the short
exact sequence
0 ,2J1
 ,2 µ
1
,21ExtA
η1  ,2I1 ,20
of functors ExtA −→ ExtA.
4.3.4 Proposition: The functor I1 : ExtA −→ ExtA corestricts to I1 : ExtA −→ CExtBA.
Proof. For a proof see for example [Eve2007, Lemma 1.4.2].
This justifies the name of centralisation functor for I1.
4.3.5 Theorem: Given a strongly E-Birkhoff subcategory B of A, the category CExtBA is
a strongly E1-Birkhoff subcategory of ArrA. The reflector is given by I1 : ExtA −→ CExtBA.
Proof. For a proof see for example [Eve2007, Theorem 1.4.3].
Now Proposition 4.3.1 implies that Γ1 = (ExtA,CExtBA,E1,Z1, I1,⊆) forms another
Galois structure, where Z1 is defined analogously to Z above as those maps in CExtBA that
lie in E1. This process may be repeated inductively to obtain Galois structures Γn and
functors Jn : ExtnA −→ ExtnA and In : ExtnA −→ CExtnBA. For n ≥ 1 and an n-extension
f , we often call the extension Inf the centralisation of f . We now give some examples
of centralisation functors I1 for the different Birkhoff subcategories we have met.
69
Chapter 4. Homology via Hopf Formulae
4.3.6 Remark (Property of group commutators): In the next example, we will be
using a property of group commutators which is easy to check: for normal subgroups M
and N of a group B with N ⊆M , we have[
M
N
,
B
N
]
=
[M,B]
N
.
4.3.7 Example (centralisation functors): When A = Gp and B = Ab, the category of
abelian groups, we saw in Example 1.2.6 that I = ab takes a group G to G/[G,G]. We also
saw in Example 1.4.4 that an extension f : B −→ A is central if and only if K[f ] ⊆ ZB,
or equivalently if and only if [K[f ], B] = 0. It easily follows, using Remark 4.3.6, that the
functor I1 = centr takes an extension f : B −→ A to
I1f = centr f :
B
[K[f ], B]
−→ A,
that is, J1[f ] = [K[f ], B].
When A = Gp and B = Nilm, the category of nilpotent groups of class at most m, we
saw in Example 1.3.3 that I = nilm takes a group G to G/LCmG, for example nil2 takes
G to G/[[G,G], G]. Here the centralisation functor I1 takes an extension f : B −→ A to
I1f :
B
lm(K[f ], B, . . . , B)
−→ A
with lm as in Example 1.4.4. Thus, when m = 2, we have I1f : B/[[K[f ], B], B] −→ A.
Again this follows from Example 1.4.4 and Remark 4.3.6.
Similarly when A = Gp and B = Solm, the category of solvable groups of class at
most m, the reflector I = solm takes a group G to G/DmG, for example sol2 takes G to
G/[[G,G], [G,G]], and I1 takes an extension f : B −→ A to
I1f :
B
dm(K[f ], B,B, . . . , B)
−→ A.
For example, when m = 2, we have I1f : B/[[K[f ], B], [B,B]] −→ A.
When A = LeibK and B = LieK for a field K (of characteristic 6= 2), we saw in
Example 1.3.3 the reflector lie : g 7−→ g/gAnn, where gAnn is the two-sided ideal generated
by elements of the form [x, x]. Given an extension f : b −→ a, let [K[f ], b]lie be the ideal
generated by elements of the form ([k, b] + [b, k]) for k ∈ K[f ] and b ∈ b. Notice that
[k, b] + [b, k] = [b+ k, b+ k]− [b, b]− [k, k] is an element of bAnn. Then the centralisation
functor sends f : b −→ a to
I1f :
b
[K[f ], b]lie
−→ a.
We see that ([b, b], [b+k, b+k])−([b, b], [b, b])−([0, 0], [k, k]) is an element of R[f ]Ann which
is sent to 0 in b by the first projection, so it (or isomorphically, its second projection
70
4.3 The Galois structures Γn, centralisation and trivialisation
[k, b] + [b, k]) is an element of [K[f ], b]lie = J1[f ] = K[Jpi1]. Clearly, for any k ∈ K[f ],
we have [k, k] = 12([k, k] + [k, k]) ∈ [K[f ], b]lie. Conversely, any element of R[f ]Ann which
maps to 0 under the first projection must be made up of elements of the following sort:
either the first entry is of the form [b, b] − [b, b], which must have in the second entry
[b + k, b + k] − [b + k′, b + k′] = [b, k] + [k, b] + [k, k] − [b, k′] − [k′, b] − [k′, k′] for some
k, k′ ∈ K[f ], which is an element of [K[f ], b]lie; or the first entry is of the form [a, a] = 0,
accompanied by [b, b] in the second entry, such that f(a) = f(b). But then b − a ∈ K[f ]
and we have [b− a, b+ a] + [b+ a, b− a] = 2[b, b] so [b, b] ∈ [K[f ], b]lie. Compare this with
the Lie-centre
ZLie(b) = {z ∈ b | [b, z] = −[z, b] ∀ b ∈ b}
from Example 1.4.4.
As mentioned earlier, Theorem 4.3.5 allows us to apply results about strongly E-
Birkhoff subcategories at the same time to an ordinary Birkhoff subcategory of a semi-
abelian category A and to the higher central extensions viewed as strongly E-Birkhoff
subcategories of ArrnA, where E is the class of n-extensions. Thus when we say “B is a
strongly E-Birkhoff subcategory of A”, we have one of these two cases in mind.
4.3.8 Remark: Given an n-extension A, for n ≥ 0, the centralisation of the (n + 1)-
extension !A : A −→ 0 turns out to be In+1!A : InA −→ 0.
The following is also often useful, and quite easy to show using the 3× 3-Lemma and
the fact that CExtnBA is strongly E
n-Birkhoff.
4.3.9 Lemma: For an (n+ 1)-extension f : B −→ A, we have
InIn+1f = Inf : InB −→ InA,
i.e., In(In+1[f ]) = InB.
Proof. This proof is taken from [EGVdL2008, Lemma 6.2]. Consider the following dia-
gram, in which the rows and the middle column are exact sequences (here pi1 and pi2 are
71
Chapter 4. Homology via Hopf Formulae
the projections of the kernel pair R[f ] to B, as in Definition 4.3.3):
0

0

0 ,2 J1[f ]_
Jpi2◦kerJpi1

J1[f ] ,2_
µ1f

0

0 ,2 JB  ,2
µB ,2
Jρf_
B
ηB  ,2
ρf
_
IB ,2
Iρf_
0
0 ,2 JI1[f ]
 ,2
µI1[f ] ,2

I1[f ]
ηI1[f ] ,2

II1[f ] ,2

0
0 0 0
The top left square commutes by definition of µ1f . Note that this square is a pullback, as
µB is a monomorphism. Thus Jpi2◦kerJpi1 is the kernel of Jρf , and the first column is
also an exact sequence. Thus by the 3 × 3 Lemma, the last column is also exact, which
makes Iρf an isomorphism. This gives
II1[f ]
II1f  ,2
∼=

IA
IB
If  ,2 IA
4.3.10 Remark: Given an n-extension f , the only object of Jnf which is non-zero is
domn Jnf , the “initial” object of the n-cube Jnf . This follows easily from the inductive
construction of Jnf . Thus we have domn Jnf = Kn[Jnf ] for any n-extension f . This also
implies that the only object of the n-cube Inf which differs from f is the initial object
domn Inf .
The Galois structures Γn also give rise to trivial extensions as defined in 1.4.3. Similarly
to the central extensions, the trivial (n+ 1)-extensions of A form a reflective subcategory
TExtn+1A of Extn+1A; the reflector
Tn+1 : Extn+1A −→ TExtn+1A
72
4.4 Hopf formulae
maps an extension f to the pullback Tn+1f : Tn+1[f ] −→ A of Inf along ηnA, the triviali-
sation of f .
A
ηnA
$
??
??
??
B
f  *0
ηnB
# .4
ρnf
 ,2 In+1[f ]
In+1f
$ .5
ηn
In+1[f ]
 )0
 ,2 Tn+1[f ]
????


Tn+1f
?:D

$
??
??
??
InA
InB
Inf
?:D

Thus we obtain a comparison map rn+1f : In+1[f ] −→ Tn+1[f ], which is a (n+1)-extension
by the strong En-Birkhoff property of the reflector In (see 4.2.1) and 4.1.2(2b). This gives
an (n+ 2)-extension In+1f −→ Tn+1f .
4.3.11 Remark: Recalling Remark 4.3.10, we see that again the only object of the (n+1)-
cube Tn+1f which differs from f is the initial object domn+1 Tn+1f . This implies that the
comparison map rn+1f is the identity everywhere except for on this initial object.
4.4 Hopf formulae
As in the groups case, the Hopf formulae in the general categorical context use projective
presentations, so we must define what exactly we mean by a higher projective presentation.
4.4.1 Definition (Projective presentations): An object of ArrkA is called extension-
projective if it is projective with respect to the class of (k + 1)-extensions. A k-
extension f : B −→ A is called a (projective) presentation of A when the object B is
extension-projective. A k-extension f : B −→ A is called a k-fold presentation, or just
k-presentation, when the object B is extension-projective and A is a (k−1)-presentation.
(A 0-presentation is an object of AE.) Given an object A of AE, a k-fold presentation
p of A is a k-fold presentation with codk p = A, i.e. the “terminal object” of the k-cube
p in A is A. We will often denote the “initial object” of a k-presentation p by Pk.
We can now state the theorem connecting comonadic homology to the Hopf formulae.
4.4.2 Theorem (Hopf formula): [EGVdL2008, Eve2007] Let E be a class of extensions
in a semi-abelian monadic category A, and let B be a strongly E-Birkhoff subcategory of
A with reflector I. Given an n-presentation p of an object A of AE with initial object Pn,
we have
Hn+1(A, I)G ∼= JPn ∩K
n[p]
Kn[Jnp]
.
73
Chapter 4. Homology via Hopf Formulae
Proof. The case where B is a straightforward Birkhoff category of A is proved in Theorem
8.1 of [EGVdL2008]. For the axiomatic extensions version see Theorem 3.6.10 in [Eve2007].
Notice that different notation is used there. The functor J is denoted by [·], and Kn[Jn·]
by [·]n. The n-fold kernel Kn[p] =
⋂n
i=0K[pi] is the intersection of all maps pi with domain
Pn in p.
In their paper [EG2007], Everaert and Gran define homology in a semi-abelian category
with enough projectives via these higher Hopf formulae, which has been shown to give
the same result as comonadic homology in a monadic setting. But for this theory no
monadicity conditions are required. We will follow this strategy here as well.
4.4.3 Definition (Hopf homology): Let E be a class of extensions in a semi-abelian
category A with enough projectives, and let B be a strongly E-Birkhoff subcategory of A
with reflector I. We define
Hn+1(A, I)E =
JPn ∩Kn[p]
Kn[Jnp]
with notation and p : P −→ A as in Theorem 4.4.2 above. We also write
H1(A, I)E = IA.
Notice we have changed the subscript G of the comonadic homology to E to distinguish
between these two different definitions of homology. As Theorem 4.4.2 shows, the two
definitions coincide as soon as they are both defined, namely in a semi-abelian monadic
category.
4.4.4 Example: In the category Gp of groups, the functor J becomes the commutator
subgroup functor, that is JA = [A,A]. So in the case n = 1 we recover the well-known
formula
H2(A,Z) =
[P, P ] ∩K[p]
[K[p], P ]
for integral group homology. We could write the functor J as a commutator in a general
semi-abelian category, as it really is a commutator in many examples, but this would give
too many different sorts of square brackets in our notation, so we prefer to call it J in
most cases.
When A = LieK is the category of Lie algebras over a field K, with the Birkhoff
subcategory B = AbLie of abelian Lie algebras, the homology defined as in Definition 4.4.3
is the Chevalley-Eilenberg homology of Lie algebras.
We can give an alternative formulation of the Hopf formulae, involving centralisation
and trivialisation of the n-fold presentation p (cf. [Eve2008, Remark 5.12]).
74
4.4 Hopf formulae
4.4.5 Proposition: Let E be a class of extensions in a semi-abelian category A with
enough projectives, and let B be a strongly E-Birkhoff subcategory of A with reflector I.
Given an n-presentation p of an object A of AE with initial object Pn, we have
JPn ∩Kn[p]
Kn[Jnp]
∼= Kn+1[Inp −→ Tnp]
and thus
Hn+1(A, I)E ∼= Kn+1[Inp −→ Tnp].
Proof. Notice that as Inp and Tnp coincide in all but their initial object, this (n+ 1)-fold
kernel is really just the initial object of the n-cube K[Inp −→ Tnp].
We first prove the case n = 1, then we use the strongly (extension)-Birkhoff prop-
erties to apply this case to higher n and use induction to prove the whole statement.
For ease of notation let p : P1 = P −→ A have kernel K. Writing I1p : I1[p] −→ A and
T1p : T1[p] −→ A, we see that
K2[I1p −→ T1p] = K[I1[p] −→ T1[p]]
as remarked above.
K2[I1p −→ T1p]
K[I1[p] −→ T1[p]]  ,2 ,2
_
I1[p]
 ,2
I1p
_
T1[p]
T1p
_
0  ,2 ,2 A A
So if we manage to prove
I1[p] =
P
K[J1p]
and T1[p] =
P
K ∩ JP ,
then Noether’s Isomorphism Theorem 1.1.15 will give the required result.
By definition
I1[p] =
P
J1[p]
75
Chapter 4. Homology via Hopf Formulae
(see 4.3.3), and we have J1[p] = K[J1p] as J1p : J1[p] −→ 0. For T1[p], consider the following
diagram, where we are taking kernels to the left:
D
 ,2 ,2

JP
 ,2
_
µP

JA_
µA

K
 ,2 ,2 P
p  ,2
_
A
_
IP
Ip  ,2 IA
As µA is a mono, the upper left square is a pullback, so D = K ∩ JP . Now consider
K ∩ JP  ,2 ,2 JP  ,2_

JA_

JA_

K ∩ JP  ,2 ,2 P
τ1p  ,2
_
T1[p]
T1p  ,2
_
A
_
IP IP
Ip  ,2 IA
where the three columns are short exact. As the bottom right square is a pullback, the
kernels of its two vertical maps coincide. Now the middle square in the top line is also a
pullback, since the map IP −→ IP at the bottom is a monomorphism. Thus K ∩ JP is
also the kernel of τ1p and thus T1[p] = P/K ∩ JP as required.
For higher n we make use of the fact that CExtn−1A is a strongly En−1-Birkhoff sub-
category of Arrn−1A. If we write p : P −→ Q with P and Q in Extn−1A, we can apply the
above to get
K2[Inp −→ Tnp] = Jn−1P ∩K[p]K[Jnp] .
Thus taking more kernels gives
Kn+1[Inp −→ Tnp] = K
n−1[Jn−1P ] ∩Kn[p]
Kn[Jnp]
(E)
as K[B/C] = K[B]/K[C] by the 3 × 3-Lemma, and kernels commute with intersections
as both are limits. So we only have to prove that Kn−1[Jn−1P ] ∩ Kn[p] = JPn ∩ Kn[p].
This holds because Kn−1[Jn−1P ] is a subobject of JPn and JPn ∩Kn[p] is a subobject of
Kn−1[Jn−1P ], which we will now show.
Notice that the (n− 1)-extension P has projective codomain and hence is split. This
means that its centralisation and trivialisation coincide (see Lemma 1.4.5). Using (E) for
P we see that
Kn[In−1P −→ Tn−1P ] = K
n−2[Jn−2(domP )] ∩Kn−1[P ]
Kn−1[Jn−1P ]
= 0
76
4.4 Hopf formulae
and so Kn−2[Jn−2(domP )] ∩ Kn[p] ⊆ Kn−2[Jn−2(domP )] ∩ Kn−1[P ] = Kn−1[Jn−1P ] as
clearly Kn[p] ⊆ Kn−1[P ]. For n = 2 this immediately gives
JP2 ∩K2[p] ⊆ K[J1P ].
Now we use induction. By the induction hypothesis we have
JPn ∩Kn−1[P ] ⊆ Kn−2[Jn−2(domP )]
and so
JPn ∩Kn[p] ⊆ Kn−2[Jn−2(domP )].
We also have
Kn−2[Jn−2(domP )] ∩Kn[p] ⊆ Kn−1[Jn−1P ]
from above. Thus we can fit all these together to give
JPn ∩Kn[p] ⊆ Kn−2[Jn−2(domP )] ∩Kn[p] ⊆ Kn−1[Jn−1P ]
as desired.
The crucial point here is that the information in the higher homology objects is
entirely contained in higher-dimensional versions Ik : ExtkA −→ CExtkBA of the reflector
I : A −→ B. One could say that homology measures the difference between the centralisa-
tion and the trivialisation of an n-presentation p of A.
We now give a long exact homology sequence, which plays a crucial role in our subse-
quent theory in Chapter 5. For this we use the homology defined via the Hopf formulae
as in 4.4.3 above, so no monadicity conditions are needed. Even so we here only give the
proof for the monadic case and refer the reader to [Eve2007] for a proof in full generality.
4.4.6 Theorem (Everaert sequence): [Eve2007, Theorem 2.4.2] Let E be a class of
extensions in a semi-abelian category A, and B a strongly E-Birkhoff subcategory of A
with reflector I : AE −→ B. Then any short exact sequence
0 ,2K[f ]  ,2
Ker f ,2B
f  ,2A ,20
in AE induces a long exact homology sequence
· · · ,2 Hn+1(A, I)E
δn+1f ,2 K[Hn(f, I1)E1 ]
γnf ,2 Hn(B, I)E
Hn(f,I)E,2 Hn(A, I)E ,2 · · ·
· · · ,2 H2(A, I)E
δ2f
,2 K[H1(f, I1)E1 ]
γ1f
,2 H1(B, I)E
H1(f,I)E
,2 H1(A, I)E ,2 0
(F)
in B. This sequence is natural in f .
77
Chapter 4. Homology via Hopf Formulae
Proof. A proof of this theorem in its full generality is given in [Eve2007]. However, when
we restrict ourselves to the monadic case it becomes relatively easy to understand why
the sequence takes this shape. So suppose that we are in a semi-abelian monadic setting,
AE = ExtkA with B = CExtkBA, and G is the induced comonad on Ext
kA. This comonad
produces canonical simplicial resolutions GA and GB of A and B and, by functoriality, also
a simplicial resolution Gf of f . The Everaert sequence (F) is the long exact homology
sequence (see [EVdL2004b, Corollary 5.7]) obtained from the short exact sequence of
simplicial objects
0 ,2K[IkGf ]  ,2 ,2IkGB
IkGf ,2IkGA ,20;
it remains to be shown that Hn−1K[IkGf ] = K[Hn(f, Ik+1)G] for all n ≥ 1. (Remember
the dimension shift in Equation (A).) Now degree-wise, the (k + 1)-extension
Ik+1Gf : Ik+1[Gf ] −→ GA
is a split epimorphic central extension: it is a centralisation, and GA is degree-wise projec-
tive. Via [EGVdL2008, Proposition 4.5], this implies that, degree-wise, it is a trivial exten-
sion. This means that Ik+1Gf is the pullback of IkGf along the unit ηkGA : GA −→ IkGA,
which in turn implies that K[IkGf ] is the kernel K[Ik+1Gf ] of Ik+1Gf . Since HnGA = 0
for all n ≥ 1, GA being a simplicial resolution, the long exact homology sequence induced
by the short exact sequence of simplicial objects
0 ,2K[Ik+1Gf ]  ,2 ,2Ik+1[Gf ]
Ik+1Gf ,2GA ,20
gives the needed isomorphism Hn−1K[Ik+1Gf ] ∼= K[Hn(f, Ik+1)G].
Note that in [Eve2007], this sequence has a slightly different appearance: there it
contains the objects domHn(f, I1)E1 instead of K[Hn(f, I1)E1 ] for n ≥ 2. But the codomain
of Hn(f, I1)E1 is zero (because J1f has zero codomain, hence I1 only changes the domain of
an extension), so its domain coincides with its kernel. For us, the sequence in its present,
more uniform, shape will be easier to work with.
4.4.7 Corollary: For any n ≥ 2 and any projective presentation p : P −→ A of an object
A ∈ |AE|,
Hn(p, I1)E1 ∼= (Hn+1(A, I)E −→ 0).
Proof. It suffices to note that in the Everaert sequence (F), all Hn+1(P, I)E are zero,
because P is projective.
This shows how the degree of the homology may be lowered from n+1 to n by raising
the degree of the reflector.
78
4.4 Hopf formulae
It is well known that the integral group homology objects are abelian groups. The
analogous result holds for the homology objects defined via Hopf formulae.
4.4.8 Lemma: Let n ≥ 1. For any object A ∈ AE, the homology object Hn+1(A, I)E is an
abelian object.
Proof. For a proof see [Eve2007, Proposition 2.3.16]. This result uses Lemma 1.2.7.
When the Birkhoff subcategory B is the subcategory of abelian objects AbA, it is
easy to see that all objects occurring in the Everaert sequence (F) are abelian objects.
However, when B is not made up of abelian objects, then H1(A, I)E = IA will not in
general be abelian. In the Everaert sequence, there is a nice bridge between the abelian
and the non-abelian objects: the map from the last abelian object H2(A, I)E to the first
non-abelian object K[I1f ] is central in the sense of Huq. This is here shown for the first
time.
4.4.9 Lemma: Let f : B −→ A be an extension in A. Then
δ2f : H2(A, I)E −→ K[H1(f, I1)E1 ]
in the Everaert sequence (F) is central in the sense of Huq.
Proof. We use some steps leading to the proof of Theorem 2.1 in [Bou2005], which is here
quoted as Lemma 1.2.7 (without proof). First of all consider the image
Im δ2f : I[δ
2
f ] −→ K[I1f ] = K[H1(f, I1)E1 ].
By Lemma 1.5.8 it is enough to show that Im δ2f is central. As the Everaert sequence
is exact, this image is the kernel of γ1f : K[I1f ] −→ IB or equivalently the kernel of the
corestriction of γ1f to its image, K[I1f ] −→ K[If ]. Now recalling the definition of I1 and
J1 from 4.3.3, we see that
K[I1f ] =
K[f ]
J1[f ]
=
K[f ]
pi2(JR[f ] ∩K[f ])
since J1[f ] = JR[f ] ∩ K[f ] as a normal subobject of R[f ], and its direct image under
pi2 gives us a normal subobject of B (note that pi2(J1[f ]) = J1[f ] as µ1f is a normal
monomorphism). Similarly
K[If ] =
K[f ]
JB ∩K[f ] =
K[f ]
pi2(JR[f ]) ∩ pi2(K[f ]) .
79
Chapter 4. Homology via Hopf Formulae
Thus, by Noether’s Isomorphism Theorem, we have
I[δ2f ] =
JB ∩K[f ]
J1[f ]
=
pi2(JR[f ]) ∩ pi2(K[f ])
pi2(JR[f ] ∩K[f ])
which is an abelian object by Lemma 1.2.7. But in fact, following the proof of this result,
we see that
JB ∩K[f ]
J1[f ]
=
JB
J1[f ]
∩ K[f ]
J1[f ]
=
pi2(JR[f ])
pi2(JR[f ] ∩K[f ]) ∩
pi2(K[f ])
pi2(JR[f ] ∩K[f ])
as quotienting out by J1[f ] is a regular epimorphism. Now clearly
JR[f ]
JR[f ] ∩K[f ] ∩
K[f ]
JR[f ] ∩K[f ] = 0
so by [Bou2005, Proposition 2.1] these two subobjects of R[f ]/(JR[f ] ∩ K[f ]) cooperate.
Now we take the images under pi2, and [Bou2005, Proposition 1.1] implies that these
images JB/J1[f ] and K[f ]/J1[f ] also cooperate, as subobjects of B/J1[f ]. Thus, using
Lemma 1.5.7, we see that (JB ∩ K[f ])/J1[f ] and K[f ]/J1[f ] cooperate as subobjects of
K[f ]/J1[f ] = K[I1f ], which says exactly that Im δ2f is central.
4.4.10 Remark: Notice that γ1f coincides with the composite in the diagram below, show-
ing that the corestriction to the image is in fact K[(I1f, If)] : K[I1f ] −→ K[If ].
K[I1f ]
_
γ1f
HH
H
(H
HHH
 ,2 ,2 I1[f ]
ηI1[f ]
_
I1f  ,2 A
ηA
_
K[If ]  ,2 ,2 IB
If
 ,2 IA
Here the map (I1f, If) : ηI1[f ] −→ ηA is a double extension, so its kernel is an extension
by Lemma 4.1.7. Thus we have established that
K[γ1f ] =
JB ∩K[f ]
J1[f ]
which is reminiscent of the Hopf formula; the only difference is that here f is not a
projective presentation and so this expression is not independent of the choice of f . The
form of this kernel is not surprising, as when f is a projective presentation, the kernel of
γ1f is exactly H2(A, I)E.
80
Chapter 5
Homology via Satellites
Introduction
Having defined homology via Hopf formulae and introduced the Everaert sequence in the
previous chapter, we now use the universal properties of the Everaert sequence to define
homology via pointwise Kan extensions or limits. Recall that any short exact sequence
0 ,2K[f ]  ,2
Ker f ,2B
f  ,2A ,20
in A gives rise to a long exact homology sequence
· · · ,2 Hn+1(A, I)E
δn+1f ,2 K[Hn(f, I1)E1 ]
γnf ,2 Hn(B, I)E
Hn(f,I)E,2 Hn(A, I)E ,2 · · ·
· · · ,2 H2(A, I)E
δ2f
,2 K[H1(f, I1)E1 ]
γ1f
,2 H1(B, I)E
H1(f,I)E
,2 H1(A, I)E ,2 0
which is natural in f . Janelidze’s theory of generalised satellites now helps us to compute
homology objects step by step: the (n + 1)st homology functor Hn+1(−, I) is obtained
from Hn(−, I1) as a pointwise right Kan extension, and the connecting homomorphism
δ in the Everaert sequence is exactly what makes this work. This approach removes the
dependence on projective objects from the definition of homology. In a further step it also
cements the connection between homology and central extensions: gluing all the step by
step Kan extensions together we see that homology is the limit of the diagram of kernels of
all central extensions of a given object. More precisely, given an object A and the category
CExtnAA of central n-extensions of A, we have
Hn+1(A, I) = lim
f∈CExtnAA
Kn[f ]
for any n ≥ 1.
When the category A does have enough projective objects, we can use projective
presentations to cut down the size of the diagram of which the homology object is the
limit. Given a projective presentation p of A, the (n + 1)st homology Hn+1(A, I) forms
the limit of the diagram consisting of the n-fold kernel of p and all maps induced by
endomorphisms of p over A. This can be interpreted to say that calculating homology
81
Chapter 5. Homology via Satellites
amounts to calculating common fixed points of these maps induced by endomorphisms of
a projective presentation of A.
The first four sections of this chapter give an analysis of homology in terms of satellites.
We start by stating the main definitions in Section 5.1. Then, in Section 5.2, we interpret
Hn+1(−, I)E (together with the connecting map δn+1) as a satellite of Hn(−, I1)E1 . In
Section 5.3 we prove one of the main results of this chapter: a formula which gives Hn+1
in terms of In. Finally in Section 5.4 we explain how the situation is entirely symmetric,
in that the connecting map γn also arises as a pointwise satellite.
In the last two sections we discuss the theory obtained by defining homology via satel-
lites. Section 5.5 gives this definition of homology without projectives and the result that
homology is the limit of the diagram of kernels of central extensions, in Corollary 5.5.10.
It also establishes that the homology objects are both objects of the Birkhoff subcategory
B and abelian objects of A. In Section 5.6 we investigate the consequences of the new
definition when enough projective objects are available. This leads to the interpretation
of homology as calculating fixed points of endomorphisms of a projective presentation.
Most material in this chapter is based on joint work with Tim Van der Linden and can
also be found in our paper [GVdL2008a], though I use some different concepts and proof
techniques here that make many statements and proofs easier.
5.1 Satellites and pointwise satellites
Modulo a minor terminological change, the following definition is due to Janelidze.
5.1.1 Definition (Satellites): [Jan1976, Definition 2] Let I ′ : A′ −→ B′ be a functor.
A left satellite (H, δ) of I ′ (relative to F : A′ −→ A and G : B′ −→ B) is a functor
H : A −→ B together with a natural transformation δ : HF =⇒ GI ′
A′
F
z



I′
$?
??
??
??
A
H $
B′
Gz



B
δ +3
universal amongst such, i.e., if there is another functor L : A −→ B with a natural trans-
formation λ : LF =⇒ GI ′, then there is a unique natural transformation µ : L =⇒ H satis-
fying δ◦µF = λ. This means that (H, δ) is the right Kan extension RanFGI ′ of the functor
GI ′ along F : A′ −→ A.
82
5.1 Satellites and pointwise satellites
This makes it possible to compute derived functors in quite diverse situations. The fol-
lowing example, borrowed from [Jan1976], explains how satellites may be used to capture
homology in the classical abelian case.
5.1.2 Example: In the abelian context, the (n + 1)st homology functor Hn+1 may be
seen as a left satellite of Hn. For instance, let A = B′ and B be categories of modules
and G : A −→ B an additive functor. Then G = H0(−, G). Let SESeqA be the category
of short exact sequences
0 ,2K  ,2 k ,2B
f  ,2A ,20
in A, the functor I ′ : SESeqA −→ A the projection pr1 that maps a sequence (k, f) to the
objectK, and F : SESeqA −→ A the projection pr3 that maps (k, f) to A. LetH : A −→ B
be the first homology functor H1(−, G). We obtain a satellite diagram
SESeqA
pr3
z


 pr1
$?
??
??
??
A
H1(−,G) $
A
H0(−,G)z



B
δ +3
where the natural transformation δ = (δ(k,f))(k,f)∈|SESeqA| consists of the connecting maps
from the (classical) long exact homology sequence
· · · ,2 H1K H1k ,2 H1B H1f ,2 H1A
δ(k,f) ,2 H0K
H0k ,2 H0B
H0f ,2 H0A ,2 0.
The universality of the Kan extension follows from the universality of the long exact
homology sequence amongst similar sequences and may for instance be shown as follows.
Given any functor L : A −→ B and any natural transformation
λ : L◦pr3 =⇒ H0(−, G)◦pr1,
we will construct the component at an object A ∈ |A| of the required natural transforma-
tion
L =⇒ H1(−, G)
by using a projective presentation p : P −→ A of A. Let k : K −→ P be the kernel of this
projective presentation of A. Since H1P is zero (as P is projective), the exactness of the
long homology sequence induced by (k, p) means that δ(k,p) : H1A −→ H0K is the kernel
of H0k. Then the string of equalities
H0k◦λ(k,p)
(1)
= λ(1P ,!P )◦L!A
(2)
= H0(¡P )◦λ(10,10)◦L!A
(3)
= 0
83
Chapter 5. Homology via Satellites
yields the required factorisation LA −→ H1A: (1) expresses the naturality of λ at the
upper, downward-pointing morphism of the diagram
0 ,2 K  ,2
k ,2
k

⇓
P
1P
p  ,2
⇓
A
!A

,2 0
0 ,2 P
1P
⇑
P
!P
 ,2
⇑
0 ,2 0
0 ,2 0
10
¡P
LR
0
¡P
LR
10
0
10
LR
,2 0
(G)
in SESeqA, while (2) follows from λ(1P ,!P ) = λ(1P ,!P )◦L10 = H0(¡P )◦λ(10,10), which is the
naturality of λ at the lower, upward-pointing morphism; the last equality (3) holds because
H00 = 0.
Note that, as such, this example does not follow the terminology of Definition 5.1.1.
From its point of view one is tempted to call H a left satellite of G (rather than a satellite
of I ′), and actually this is how the definition appears in the paper [Jan1976]. But the
situation we shall be considering in this thesis demands the change in terminology, and
the present example may easily be modified to comply with Definition 5.1.1.
Indeed, the functor G may be lifted to a functor
SSeqH0(−, G) : SESeqA −→ SSeqB
where the latter category consists of short (not necessarily exact) sequences in B. Together
with the obvious projection pr1 : SSeqB −→ B (s.t. H0(−, G)◦pr1 = pr1◦SSeqH0(−, G)),
this gives us the satellite diagram
SESeqA
pr3
z


 SSeqH0(−,G)
$?
??
??
??
A
H1(−,G) $
SSeqB.
pr1z



B
δ +3
Whereas such a viewpoint may seem rather far-fetched in the abelian case, it is the only
one still available when the context is widened to semi-abelian categories. In fact, even in
the abelian setting, this formulation is slightly reminiscent of the universal property for a
derived functor (see for example [Wei1997, Section 10.5]), so it is not all that far-fetched
after all.
In practice, satellites may almost always be computed explicitly using limits—namely,
as pointwise Kan extensions. Then the definition given above is strengthened as follows.
84
5.2 Hn+1(−, I)E as a satellite of Hn(−, I1)E1
5.1.3 Notation: Let A be an object of A. We denote by (A ↓ F ) the category of elements
of the functor Hom(A,F−) : A′ −→ Set: its objects are pairs (A′, α : A −→ FA′), where
A′ is an object of A′ and α is a morphism in A, and its morphisms are defined in the
obvious way (cf. [Bor1994, Theorem 3.7.2]). The forgetful functor U : (A ↓ F ) −→ A′
maps a pair (A′, α) to A′. The natural transformation (H, δ) now induces a cone δ on
GI ′U : (A ↓ F ) −→ B with vertex HA defined by
δ(A′,α : A−→FA′) = δA′◦Hα : HA
Hα ,2HFA′
δA′ ,2GI ′A′ = GI ′U(A′, α).
5.1.4 Definition (Pointwise satellites): A left satellite (H, δ) of I ′ relative to the
functors F : A′ −→ A and G : B′ −→ B is called pointwise when it is pointwise as a Kan
extension, i.e., for every object A of A, the cone (HA, δ) on GI ′U : (A ↓ F ) −→ B is a
limit cone.
To check that a pair (H, δ) is a pointwise satellite it is not necessary to prove its univer-
sality as in Definition 5.1.1, but it suffices to check the limit condition from Definition 5.1.4;
see, for example, MacLane [Mac1998, Theorem X.3.1].
5.2 Hn+1(−, I)E as a satellite of Hn(−, I1)E1
We are now ready to prove the first main result of this chapter: we focus on the universal
properties of the Everaert sequence (F), and prove that they allow us to interpret the
(n+1)st homology with coefficients in I as a satellite of the nth homology with coefficients
in I1.
For the whole of this section, let E be a class of extensions in a semi-abelian category
A with enough projectives, and let B be a strongly E-Birkhoff subcategory of A with
reflector I : AE −→ B.
5.2.1 Lemma: For n ≥ 1 and A ∈ |AE|,
K[Hn(!A : A −→ 0, I1)E1 ] = Hn(A, I)E.
Proof. This follows from the exactness of the Everaert sequence (F) and the fact that all
Hn(0, I)E are zero.
85
Chapter 5. Homology via Satellites
5.2.2 Lemma: For all n ≥ 1 and f : B −→ A ∈ |ExtA|,
γnf = ker
Hn

B
f

⇒
B
!B

A
!A
,2 0
, I1

E1
 : K[Hn(f, I1)E1 ] −→ Hn(B, I)E.
Proof. This follows from the previous lemma and the naturality of γn. Indeed, its natu-
rality square at the map (1B, !A) is nothing but
K[Hn(f, I1)E1 ]
kerHn((1B ,!A),I1)E1 ,2
γnf

K[Hn(!B, I1)E1 ]
γn!B

Hn(B, I)E Hn(B, I)E;
and all kernels may be chosen in such a way that γn!B is an identity.
5.2.3 Proposition: Let E be a class of extensions in a semi-abelian category A with
enough projectives, and let B be a strongly E-Birkhoff subcategory of A with reflector
I : AE −→ B. Let n ≥ 1. Then Hn+1(−, I)E : AE −→ AE with the connecting natural
transformation
ExtA
cod
z


 Hn(−,I1)E1
$?
??
??
??
?
AE
δn+1 +3
Hn+1(−,I)E $
ExtA
kerz



AE
(H)
is the pointwise left satellite of Hn(−, I1)E1. That is, for any object A of AE,
Hn+1(A, I)E = Rancod(ker◦Hn(−, I1)E1)(A) = lim
(f,g)∈|(A↓cod)|
K[Hn(f, I1)E1 ].
Proof. Let A be an object of AE. Let p : P −→ A be a projective presentation of A. We
have to show that (Hn+1(A, I)E, δn+1) is the limit of
(A ↓ cod) U ,2ExtA Hn(−,I1)E1 ,2ExtA ker ,2AE.
To do so, let (L, λ) be another cone on ker◦Hn(−, I1)E1◦U ; we use the presentation p of A
to construct a map of cones l : L −→ Hn+1(A, I)E.
First we consider the case n = 1. Recall from Definition 4.4.3 that H1(−, I)E = I
and H1(−, I1)E1 = I1. Since p : P −→ A is a projective presentation of A, and thus
H2(P, I)E = 0, the lower end of the Everaert sequence (F) of p becomes
86
5.2 Hn+1(−, I)E as a satellite of Hn(−, I1)E1
0 ,2H2(A, I)E
 ,2
δ2p ,2K[I1p]
γ1p ,2IP.
In other words, δ2p is the kernel of γ
1
p . Recalling Diagram (G), consider the following two
morphisms in (A ↓ cod):
P
p  ,2
⇓
A
!A

1A
⇓
A
P
!P  ,2
⇑
0
⇑
A
!Alr
0
¡P
LR
10
0
10
LR
A
!A
lr
(I)
By Lemma 5.2.2, the naturality of λ at the downward-pointing morphism in Diagram (I)
means γ1p◦λ(p,1A) = λ(!P ,!A). This latter morphism is zero, since the naturality of λ at the
upward-pointing morphism in (I) means λ(!P ,!A) = I(¡P )◦λ(10,!A), and I0 = 0. Hence there
exists a unique morphism l : L −→ H2(A, I)E satisfying λ(p,1A) = δ2p◦l.
Higher up in the Everaert sequence (F) of p, for n ≥ 2, Corollary 4.4.7 gives us the
isomorphism
δn+1p : Hn+1(A, I)E
∼=−→ K[Hn(p, I1)E1 ].
Here we may simply put l = (δn+1p )
−1◦λ(p,1A).
It remains to be shown that, in both cases, the constructed map l is a map of cones.
Given any object (f : B −→ C, g : A −→ C) of (A ↓ cod), there is a map
P
p  ,2

⇓
A
g

⇓
A
B
f
 ,2 C Ag
lr
as P is projective. Writing h for the image of this morphism under ker◦Hn(−, I1)E1◦U , we
see that the diagram
L
l ,2
λ(p,1A)

λ(f,g)
OOOO
#+OO
OOO
OOO
OO
Hn+1(A, I)E
δp
oo
ooo
s{ooo
ooo
o δ(f,g)

K[Hn(p, I1)E1 ] h
,2 K[Hn(f, I1)E1 ]
commutes: λ(f,g) = h◦λ(p,1A) = h◦δp◦l = δ(f,g)◦l. Thus l is indeed a map of cones, and
Hn+1(A, I)E is the limit of the given diagram.
5.2.4 Remark: This gives a way to derive Hn+1(−, I)E from Hn(−, I1)E1 for n ≥ 2 in
exactly the same way as H2(−, I)E is derived from H1(−, I1)E1 = I1. In other approaches
such as [Eve2007, EGVdL2008] the two cases are formally different.
87
Chapter 5. Homology via Satellites
5.2.5 Remark: Notice that we can apply this proposition for higher extensions as well,
making use of the fact that CExtkBA is a strongly E
k-Birkhoff subcategory of ArrkA for
k ≥ 1. This allows us to use induction, as we will see in the next subsection.
5.3 Hn+1(−, I)E as a satellite of In
Proposition 5.2.3 gives a way to construct Hn+1(−, I)E from Hn(−, I1)E1 . Here, with The-
orem 5.3.2, we obtain a one-step construction of Hn+1(−, I)E out of In. To be able to apply
Proposition 5.2.3 repeatedly, we have to show that satellite diagrams like Diagram (H)
may be composed in a suitable way (cf. [Jan1976, Theorem 9]).
The kernel functor
ker : Extk+1A −→ ExtkA
that maps an extension f : B −→ A to its kernel K[f ] has a left adjoint, namely the functor
ExtkA −→ Extk+1A that sends an object C of ExtkA to the extension !C : C −→ 0. This
allows us to use the following result.
5.3.1 Proposition: Suppose that (I ′, δ′) = RanF ′G′I ′′ and (H, δ) = RanFGI ′ as in the
diagrams
A′
F
z


 I′
$?
??
??
??
?
A
H $
δ +3 B′
Gz



B
and
A′′
F ′
z


 I′′
$?
??
??
??
?
A′
I′ $
δ′ +3 B′′.
G′z



B′
If G is a right adjoint then (H,Gδ′◦δF ′) = RanFF ′GG′I ′′: the two diagrams may be
composed to form a single Kan extension diagram
A′′
FF ′
z


 I′′
$?
??
??
??
?
A
H $
Gδ′◦δF ′ +3 B′′.
GG′z



B
If G preserves limits and (I ′, δ′) and (H, δ) are pointwise satellites then (H,Gδ′◦δF ′) is
also a pointwise satellite.
Proof. We prove the pointwise case. Let A be an object of A, and (C, σ) a cone on the
diagram GG′I ′′U : (A ↓ FF ′) −→ B.
88
5.3 Hn+1(−, I)E as a satellite of In
For any A′ in A′, the pair (I ′A′, δ′) is the limit of the diagram G′I ′′U ′ : (A′ ↓ F ′) −→ B′.
Since G preserves limits, (GI ′A′, Gδ′) is the limit of GG′I ′′U ′ : (A′ ↓ F ′) −→ B. Now for
every α : A −→ FA′ the collection
(σ(A′′,Fα′◦α))(A′′,α′)∈|(A′↓F ′)|
also forms a cone on GG′I ′′U ′; hence there is a unique map µ(A′,α) : C −→ GI ′A′ such that
Gδ′A′′◦GI
′α′◦µ(A′,α) = σ(A′′,Fα′◦α).
The collection (µ(A′,α))(A′,α)∈|(A↓F )| in turn forms a cone on GI ′U : (A ↓ FF ′) −→ B.
Indeed, if (B′, β) is an object of (A ↓ F ) and f ′ : B′ −→ A′ is a map in A′ such that
Ff ′◦β = α, then GI ′f ′◦µ(B′,β) = µ(A′,α), because for every (A′′, α′) ∈ |(A′ ↓ F ′)|,
Gδ′A′′◦GI
′α′◦GI ′f ′◦µ(B′,β) = σ(A′′,F (α′◦f ′)◦β)
= σ(A′′,Fα′◦α)
= Gδ′A′′◦GI
′α′◦µ(A′,α),
and the Gδ′A′′◦GI
′α′ are jointly monic.
This cone gives rise to the required unique map c : C −→ HA. Since it satisfies
µ(A′,α) = δA′◦Hα◦c for all (A′, α) ∈ |(A ↓ F )|, we have that
Gδ′A′′◦δF ′A′′◦Hα
′′◦c = Gδ′A′′◦δF ′A′′◦HFα
′◦Hα◦c
= Gδ′A′′◦GI
′α′◦δA′◦Hα◦c
= Gδ′A′′◦GI
′α′◦µ(A′,α)
= σ(A′′,Fα′◦α) = σ(A′′,α′′)
for all α′′ = Fα′◦α : A −→ FA′ −→ FF ′A′′ in |(A ↓ FF ′)|—and any α′′ allows such a
decomposition.
5.3.2 Theorem: Let E be a class of extensions in a semi-abelian category A with enough
projectives, and let B be a strongly E-Birkhoff subcategory of A with reflector I : AE −→ B.
Let n ≥ 1. Then
Hn+1(−, I)E : AE −→ AE
with the connecting natural transformation
∂n+1 = kern−1δ2◦ · · · ◦kerδn◦δn+1 : Hn+1(−, I)E◦ codn =⇒ kern◦In
89
Chapter 5. Homology via Satellites
is the pointwise left satellite of In.
ExtnA
codn
z



In
$?
??
??
??
?
AE
Hn+1(−,I)E $
∂n+1 +3 ExtnA
kernz



AE
This means, for any object A of A,
Hn+1(A, I)E = Rancodn(kern◦In)(A) = lim
(f,g)∈|(A↓codn)|
Kn[Inf ].
Proof. This follows from gluing diagrams as in Proposition 5.2.3 together using Proposi-
tion 5.3.1 and Remark 5.2.5.
5.4 Hn(−, I)E as a right satellite of Hn(−, I1)E1
Proposition 5.2.3 gives an interpretation of the connecting morphisms δnf in the Everaert
sequence as left satellites. The connecting morphisms γnf have a dual interpretation:
(Hn(−, I)E, γn) is a right satellite (left Kan extension) of Hn(−, I1)E1 .
5.4.1 Proposition: Let E be a class of extensions in a semi-abelian category A with
enough projectives, and let B be a strongly E-Birkhoff subcategory of A with reflector
I : AE −→ B. Consider n ≥ 1. Then (Hn(−, I)E, γn), i.e., Hn(−, I)E : AE −→ AE with the
connecting natural transformation
ExtA
Hn(−,I1)E1
z


 dom
$?
??
??
??
ExtA
ker $?
??
??
??
γn +3 AE
Hn(−,I)Ez
AE
is the pointwise right satellite of Hn(−, I1)E1.
Proof. For any A, the category (dom↓A) has a terminal object (!A : A −→ 0, 1A), so the
colimit object of the diagram
(dom↓A) U ,2 ExtA
Hn(−,I1)E1 ,2 ExtA ker ,2 AE
90
5.5 Homology without projectives
is K[Hn(!A, I1)E1 ] = Hn(A, I)E. The component of the colimit cocone at
(g : B −→ C, f : B −→ A) ∈ |(dom↓A)|
is
ker
Hn

B
f ,2
g

⇒
A
!A

C
!C
,2 0
, I1

E1
 = ker
Hn
 B
f ,2
!B

⇒
A
!A

0 0
, I1

E1
 ◦ker
Hn

B
g

⇒
B
!B

C
!C
,2 0
, I1

E1

= Hn(f, I)E◦γng
= γn(g,f)
by Lemma 5.2.2 and Lemma 5.2.1.
5.5 Homology without projectives
In this section we set up a homology theory without projectives by defining homology via
pointwise satellites as they appear in Proposition 5.2.3.
5.5.1 Proposition: Let E be a class of extensions in a semi-abelian category A, and let
B be a strongly E-Birkhoff subcategory of A with reflector I : AE −→ B. Let k ≥ 0, and
consider an object A ∈ |AE|. If it exists, write
H(2,k) = Rancod(ker◦Ik+1)
for the pointwise left satellite of Ik+1 relative to the functors cod and ker. Now suppose
H(n,k+1) exists for n ≥ 2, and write
H(n+1,k) = Rancod(ker◦H(n,k+1))
for the pointwise left satellite of H(n,k+1) relative to cod and ker, if this exists. Then
H(n+1,k) is also the left satellite of In relative to the functors codn and kern.
Proof. The proof is the same as the proof of Theorem 5.3.2.
5.5.2 Definition (Homology): Let E be a class of extensions in a semi-abelian category
A, and let B be a strongly E-Birkhoff subcategory of A with reflector I : AE −→ B. Con-
sider an object A ∈ |AE|, and let n ≥ 1. If the functor H(n+1,0) from Proposition 5.5.1
exists, we call it the (n+ 1)st homology functor
Hn+1(−, I) : AE −→ AE
91
Chapter 5. Homology via Satellites
(with coefficients in I).
ExtA
cod
z


 Hn(−,I1)
$?
??
??
??
AE
δn+1 +3
Hn+1(−,I) $
ExtA
kerz



AE
We also write H1(−, I) = I.
5.5.3 Remark: As CExtkBA is strongly E
k-Birkhoff in ArrkA, Definition 5.5.2 also gives
us functors Hn+1(−, Ik) : ExtkA −→ ExtkA. These are sometimes needed directly, but
whenever possible we will include these higher-dimensional cases in the setting of strongly
(extension)-Birkhoff subcategories, to make statements and proofs easier.
5.5.4 Remark: For any object A ∈ |AE|, if H2(A, I) exists, it is the limit object of the
diagram
(A ↓ cod) U ,2 ExtA I1 ,2 ExtA ker ,2 AE.
Similarly, if Hn+1(A, I) exists, it is the limit object of the diagram
(A ↓ cod) U ,2 ExtA
Hn(−,I1)E1 ,2 ExtA ker ,2 AE
or equivalently of
(A ↓ codn) U ,2ExtnA In ,2ExtnA kern ,2AE. (J)
Potentially, these limits may exist for a given object A even if the homology functors
Hn+1(−, I) do not exist in full.
5.5.5 Example (When the reflection is the identity): If B = A then all In are iden-
tity functors, and the Hn are zero for n ≥ 2. To see this, we have to prove that the functor
0: AE −→ AE is a pointwise Kan extension of ker : ExtA −→ AE along cod: ExtA −→ AE,
for all k ≥ 0. This shows that H2 is zero, which immediately implies that the higher
homologies are also zero, being satellites of the zero functor.
92
5.5 Homology without projectives
Let A be an object of AE and (L, λ) a cone on ker◦U : (A ↓ cod) −→ AE. Then any map
λ(f,g), where (f : B −→ C, g : A −→ C) ∈ |(A ↓ cod)|, fits into the commutative diagram
L
λ(f,g) ,2
λ(!0,!A)

λ(!B,!A)
??
??
$?
??
?
K[f ]
_
Ker f

0 ,2 B,
which means that λ(f,g) is the zero map. If now (L, λ) is a limit cone, this implies that L
is zero.
The category (A ↓ cod) is rather large, and in a given situation it may be very hard
to decide whether the needed limits do indeed exist. Even if they do, they may still be
hard to compute. But we may replace the above diagrams with simpler ones, for example
using the concept of an initial subcategory. Recall its definition as it occurs in [Mac1998,
Section IX.3]:
5.5.6 Definition: An initial functor is a functor F : D −→ C such that for every object
C of C, the slice category (F ↓ C) is non-empty and connected. A subcategory D of a
category C is called initial when the inclusion of D into C is an initial functor, i.e., for
every object C ∈ |C|, the full subcategory (D ↓ C) of (C ↓ C) determined by the maps
D −→ C with domain D in D is non-empty and connected.
If D is initial in C then limits of diagrams over C may be computed as the limit of
their restriction to D. More generally, if F : D −→ C is initial then a diagram G : C −→ E
has a limit if and only if GF does, in which case it may be computed as the limit of GF .
For any object A of AE, let ExtAA denote the category of extensions of A, the
preimage in ExtA of the arrow 1A under the functor cod: ExtA −→ AE. Then the functor
U ′ : ExtAA −→ (A ↓ cod) that sends an extension f : B −→ A of A to the pair (f, 1A) is
easily seen to be initial: for every object (f : B −→ C, g : A −→ C) of (A ↓ cod) there is
the natural morphism U ′f −→ (f, g)
B

f  ,2
⇓
A
g

1A
⇓
A
B
f
 ,2 C A,g
lr
where f is the pullback of f along g; this f is an extension by Definition 4.1.2(3).
93
Chapter 5. Homology via Satellites
Also, any other morphism
D

h  ,2
⇓
A
g

1A
⇓
A
B
f
 ,2 C A,g
lr
factors over this morphism U ′f −→ (f, g), by the universal property of a pullback. This
means that the limit of ker◦Hn(−, I1)E1◦U may also be computed as the limit of the dia-
gram ker◦Hn(−, I1)E1◦UU ′ and moreover, since UU ′ is just the inclusion of the subcategory
ExtAA into ExtA, as the limit of
ker◦Hn(−, I1)E1 : ExtAA −→ AE.
But even now the diagram of shape ExtAA over which the limit is computed may be
too large, in the sense that even if A is small-complete, it is still unclear whether the
limit of ker◦Hn(−, I1)E1 exists. In the case where A has enough projectives, however, it
is possible to further cut down the size of this diagram. In this case Proposition 5.2.3
shows that the limit of this diagram exists and is equal to the homology object defined
via the Hopf formulae. But making the diagram smaller gives a new way to calculate this
homology. This situation is discussed in Section 5.6.
5.5.7 Notation: Let A ∈ |AE|. Denote by ExtnAA the category of n-extensions of A,
defined as the preimage of the arrow 1A under the functor codn : ExtnA −→ AE. This
generalises the category ExtAA of extensions of A defined above. Thus the objects are
n-extensions with “terminal object” A, when viewed as diagrams in the category A, and
the maps are those maps in ExtnA which restrict to the identity on A under codn. Sim-
ilarly the category CExtnAA denotes the full subcategory of Ext
n
AA determined by those
n-extensions which are central. The strongly E-Birkhoff subcategory B is understood, and
not mentioned in the notation.
5.5.8 Remark: The functor U ′ : ExtnAA −→ (A ↓ codn) which sends an n-extension f of A
to (f, 1A) is still initial. This may be shown by induction, using the fact that in a category
of n-fold extensions, the (n+ 1)-extensions are pullback-stable (see Definition 4.1.2(3)).
5.5.9 Proposition: Let E be a class of extensions in a semi-abelian category A, and let
B be a strongly E-Birkhoff subcategory of A with reflector I : AE −→ B. Consider n ≥ 1
and A ∈ |AE|. If it exists, Hn+1(A, I) is also the limit of the diagram
kern◦In : ExtnAA −→ AE.
Proof. This uses Diagram (J) and the fact that U ′ : ExtnAA −→ (A ↓ codn) is initial.
94
5.5 Homology without projectives
5.5.10 Corollary: For n ≥ 1 and A ∈ |AE|, if it exists, Hn+1(A, I) is the limit of the
diagram
kern : CExtnAA −→ AE.
Proof. The functor In : ExtnAA −→ CExtnAA is initial because, for any central extension
f ∈ |CExtnAA|, we have Inf = f , so the slice category (In ↓ f) is non-empty and connected.
Since limits commute with kernels, Corollary 5.5.10 also says that Hn+1(A, I) may be
computed as the n-fold kernel of a certain n-fold arrow in A, namely, the limit in ArrnA
of the inclusion of CExtnAA into Arr
nA. Sometimes this n-fold arrow in A itself happens to
be an n-fold central extension of A. We say that an n-fold central extension of an object
A ∈ |AE| is universal when it is an initial object of CExtnAA. We will see in Lemma 6.2.3
that, when A is a semi-abelian category and I = ab: A −→ AbA is the abelianisation
functor, then an object A of A admits a universal central extension p if and only if it is
perfect: its abelianisation is zero. In fact, this result holds for any reflector I to a Birkhoff
subcategory B of A, not just the abelianisation functor (see [CVdL2009, Theorem 2.9]).
In this case, H2(A, I) is the kernel of p. This latter property holds in general, also for
higher extensions:
5.5.11 Corollary: Consider n ≥ 1 and A ∈ |AE|. If A has a universal n-fold central
extension p then Hn+1(A, I) = Kn[p]. In particular, if A ∈ |A| has a universal central
extension p : P −→ A then H2(A, I) = K[p].
Proof. The limit of a functor from a category that has an initial object is the value of the
functor at this object.
5.5.12 Example (The homology of zero is zero): If A = 0 then, for any n ≥ 1,
the category CExtnAA has an initial object, the zero n-cube. Taking kernels as in Corol-
lary 5.5.11 gives Hn+1(0, I) = 0.
5.5.13 Remark: Note that in certain special cases a weakly universal extension can also
determine the homology of an object A. When 1A is a weakly universal extension of
A, i.e., if every extension f : B −→ A of A is split, we have H2(A, I) = 0. This is be-
cause K[I1A] = 0 for any object A, so if 1A is weakly initial, every leg of a cone over
ker◦I : ExtAA −→ AE factors over K[I1A] and thus is zero. In particular, we get:
5.5.14 Example (The homology of a projective object is zero): For any projective
object P and any n ≥ 1 we have Hn+1(P, I) = 0, since 1P (and also the n-extension only
consisting of the maps 1P ) is always weakly initial when P is projective.
95
Chapter 5. Homology via Satellites
5.5.15 Example (Homology of finite groups): For a finite group, we compare its
second homology groups with respect to two different adjunctions. On the one hand we
have the abelianisation functor ab: Gp −→ AbGp, where Gp is the category of groups, AbGp
is the Birkhoff subcategory of abelian groups, and abG = G/[G,G]. This example has been
studied in the classical setting in [EVdL2004b] (for lower dimensions) and in [Eve2007,
EGVdL2008] (higher dimensions). Here the centralisation functor ab1 takes an extension
f : B −→ A to centr f : B/[K[f ], B] −→ A. As mentioned in Example 1.3.6, in this case
Definition 5.5.2 gives the classical integral homology of groups.
On the other hand, we could focus on finite groups and let A = FinGp be the category
of finite groups and B = FinAb = AbFinGp its Birkhoff subcategory of finite abelian
groups. Note that FinGp is not semi-abelian and doesn’t have enough projectives, but
nevertheless it is pointed, Barr exact and Bourn protomodular. All the results that we
apply in this example do not use coproducts, so they are still valid in this context. Here
I : A −→ B again sends a group G to finabG = G/[G,G] and I1 : ExtFinGp −→ ExtFinGp
sends an extension f : B −→ A to fincentr f : B/[K[f ], B] −→ A. We show that, for any
finite group, its second homology groups with respect to the two theories coincide.
For perfect groups this is clear. Recall from Corollary 5.5.11 that if a group G has
a universal central extension p : P −→ G, then the homology is H2(G, ab) = K[p]; this
is the case when G is perfect: abG = 0. So given a finite perfect group G, we know
that it has a universal central extension p : P −→ G in the category Gp of all groups, and
that H2(G,Z) = H2(G, ab) = K[p]. But we also know that the integral homology of a
finite group is a finite group, therefore the group P must also be finite, and the universal
central extension p : P −→ G lies in the category FinGp of finite groups. Thus we also
have H2(G,finab) = K[p]. So for a finite perfect group G we have
H2(G,finab) = H2(G, ab) = H2(G,Z).
For a general group, we need a few more steps to prove this equality.
Step 1: First we want to show that, for any finite group G, there is a central extension
G∗ −→ G with kernel H2(G,Z), such that in the diagram
ker : CExtGGp −→ Gp, (K)
the leg from the limit H2(G, ab) to this object is an isomorphism. We consider stem
extensions: central extensions g : H −→ G with K[g] ≤ [H,H]. This condition implies
that abH −→ abG is an isomorphism, or equivalently that the map K[g] −→ abH is zero.
So it follows from exactness in (F) that the leg H2(G, ab) −→ K[g] is a surjection when g is
a stem extension. To find a stem extension with H2(G,Z) as its kernel, we use the Schur
multiplier M(G) of a finite group G introduced in [Sch1904]. Schur proved in [Sch1907]
96
5.5 Homology without projectives
that for a finite group G, this multiplier M(G) may be expressed in terms of what is now
called the Hopf formula (which, in the infinite case, was only introduced in [Hop1942]),
and so we have M(G) ∼= H2(G,Z) (see also, e.g., [Kar1987, Theorem 2.4.6]). In [Sch1904]
he showed that, for any finite group G, there is a stem extension f : G∗ −→ G of G with
kernel M(G) (see also [Kar1987, Theorem 2.1.4]).
Putting these two facts together, we see that H2(G,Z) occurs in the diagram (K) as
the kernel of this stem extension f , and that the leg from H2(G, ab) to it must be an
isomorphism, being a surjection between finite groups of the same size. From now on we
shall assume that this isomorphism is an identity.
Step 2: We now consider the diagram of kernels of finite central extensions of G,
ker : CExtGFinGp −→ FinGp, (L)
which is a small diagram and so has a limit in Gp which we denote by L. We shall show
in Step 3 that L ∼= H2(G, ab) and so is actually the limit of (L) in the category FinGp as
well, as H2(G, ab) is a finite group.
H2(G, ab) forms a cone on (L), using the legs from (K). The induced map of cones to
L gives a splitting for the leg p : L −→ H2(G,Z) = K[f ]. As these are all abelian groups,
we have L ∼= H2(G,Z) ⊕ E for some abelian group E, and p = pi1 : L −→ H2(G,Z), the
first projection. We consider the following central extensions and maps between them:
H2(G,Z)  ,2 ,2_

G∗
f  ,2
_
(1G∗ ,0)

G
H2(G,Z)⊕ E
p
_LR
 ,2 ,2
pi2
_
G∗ × E
pi1
_LR
f◦pi1 ,2
f×1E
_
G
E
 ,2 ,2 G× E pi1
 ,2G
Since the extension pi1 : G× E −→ G is split, the leg from L to E = K[pi1] must be the zero
map. So the leg from L to K[f◦pi1] is 1H2(G,Z) ⊕ 0: L ∼= H2(G,Z)⊕ E −→ H2(G,Z)⊕ E,
as H2(G,Z)⊕ E is a product.
Step 3: Finally we consider a third, even smaller diagram. Let C be the full subcategory
of CExtFinGp containing those extensions g of G for which there exists a map f −→ g in
CExtGFinGp. We consider the subdiagram
ker : C −→ FinGp, (M)
the limit of which is H2(G, ab). For any cone D over the diagram (M), the two legs
d : D −→ H2(G,Z) = K[f ] and 0: D −→ E = K[pi1] again determine the leg to the prod-
97
Chapter 5. Homology via Satellites
uct, (d, 0) : D −→ H2(G,Z)⊕ E = K[f◦pi1]. The leg d also forms the unique cone map
D −→ H2(G, ab). Notice that in (M) we also have maps from H2(G,Z)⊕E to any other ob-
ject, as p : H2(G,Z)⊕ E −→ H2(G,Z) is part of the diagram. So as (1H2(G,Z)⊕ 0)◦(d, 0) =
(d, 0), the map (d, 0) : D −→ L is a cone map and makes L into a limit of (M). So
L ∼= H2(G, ab) as promised, and we have H2(G,finab) = H2(G, ab) = H2(G,Z) for any
finite group G.
It is well known that all integral homology groups of a group are abelian. More
generally, both approaches to homology discussed in Chapter 1 are such that the homology
objects are abelian objects of the Birkhoff subcategoryB. We now prove that our homology
objects Hn+1(A, I) also satisfy these properties.
5.5.16 Lemma: Consider an object A ∈ |AE|. The kernel K[f ] of a central extension
f : B −→ A of A is an object of the strongly E-Birkhoff subcategory B.
Proof. Let A ∈ |AE|. First consider a trivial extension f : B −→ A. This means f is
the pullback of If : IB −→ IA along ηA, so K[f ] is isomorphic to K[If ]. This kernel of
the extension If : IB −→ IA is an object of B because B is closed under subobjects (see
Remark 4.2.2). Now for a central extension f : B −→ A, recall from Definition 1.4.3 that
there exists an extension g such that the pullback f of f along g is trivial.
K[f ]  ,2 ,2 B
f  ,2
_
A
g
_
K[f ]  ,2 ,2 B
f
 ,2 A
But then K[f ] = K[f ], which is an object of B as f is trivial.
5.5.17 Remark: The converse implication does not hold, as for example in the category
of groups not every extension with abelian kernel is central.
5.5.18 Proposition: Let A be an object of AE and n ≥ 0. Then Hn+1(A, I) is an object
of B.
Proof. If n = 0 the result is clear as H1(A, I) = IA. For n ≥ 1, we use Lemma 5.5.16
repeatedly to see that the diagram from Corollary 5.5.10 factors over B and becomes the
functor kern : CExtnAA −→ B. Since B is closed under limits in A, the limit Hn+1(A, I) of
this diagram is still an object of B.
98
5.5 Homology without projectives
5.5.19 Example (When the reflection is zero): If B = 0, the zero subcategory in A,
then all homology objects are zero, because they are in B by Proposition 5.5.18.
The proofs of the next result Proposition 5.5.22 and its lemma were offered to us by
Tomas Everaert. Recall that an object A of a homological category A is abelian if it
carries an internal abelian group structure. Such a structure is necessarily unique, and is
given by a morphism m : A×A −→ A satisfying m◦(1A, 0) = 1A = m◦(0, 1A), called its
addition (see Definition 1.2.1). The abelian objects form a Birkhoff subcategory AbA of
A.
5.5.20 Lemma: For any extension f : B −→ A in A, the image of the connecting mor-
phism
δ2f : H2(A, I) −→ K[H1(f, I1)] = K[I1f ]
is an abelian object of A.
Proof. We show that I[δ2f ] is a subobject of an abelian object in A, namely the kernel of
γ1f . Recall from Remark 4.4.10 that
K[γ1f ] =
JB ∩K[f ]
J1[f ]
=
pi2(JR[f ]) ∩ pi2(K[f ])
pi2(JR[f ] ∩K[f ])
and thus is abelian by Lemma 1.2.7.
Now the composite
H2(A, I)
δ2f ,2 K[I1f ]
γ1f ,2 IB = H1(B, I)
is zero, as it is the leg from the limit H2(A, I) to the kernel IB = K[I1!B] of a split central
extension. Thus the image of δ2f factors over the kernel of γ
1
f , and I[δ
2
f ] is abelian, as a
subobject of the abelian object K[γ1f ] = (JB ∩K[f ])/J1[f ].
5.5.21 Remark: Notice that we can not assume any more that I[δ2f ] = K[I1f ], as we did
in Lemma 4.4.9. As we are using a different definition of homology, we can not assume
that the Everaert sequence is exact. It is however still a complex, and the map γ1f does not
change, so K[γ1f ] is still the same abelian object. Note that Lemma 1.5.8 still implies that
δ2f is central in the sense of Huq, as it factors over the central morphism Ker γ
1
f . Of course,
as soon as we are working in a category with enough projectives, the two definitions of
homology coincide and the Everaert sequence is exact even when using the definition via
limits.
99
Chapter 5. Homology via Satellites
5.5.22 Proposition: Let A be an object of A and n ≥ 1. Then Hn+1(A, I) is an abelian
object of A.
Proof. It suffices to show that, for any A ∈ |AE|, the object H2(A, I) is abelian in A. We
can then use this fact also in higher dimensions when B = CExtkBA, so then the higher
homology objects are limits of a diagram of abelian objects, and thus abelian by induction.
To show H2(A, I) is abelian, consider the functor
H2(−, I)×H2(−, I) : AE −→ AE
that sends an object A to the product H2(A, I) × H2(A, I). The previous lemma gives
rise to a natural transformation (H2(−, I)×H2(−, I)) ◦ cod =⇒ ker ◦ I1 of functors from
ExtA to AE; the component of this natural transformation at an extension f : B −→ A is
the composition
H2(A, I)×H2(A, I) −→ I[δ2f ]× I[δ2f ] −→ I[δ2f ] −→ K[I1f ].
Here the first arrow is the corestriction of δ2f × δ2f , the second arrow is the addition on the
abelian object I[δ2f ], and the last arrow is the inclusion of the image into the codomain
of δ2f . The universal property of the Kan extension (H2(−, I), δ2) now yields a natural
transformation H2(−, I)×H2(−, I) =⇒ H2(−, I) which is easily seen to define an abelian
group structure on all H2(A, I).
5.5.23 Example (Heyting semi-lattices): As mentioned in Example 1.1.4, Heyting
semi-lattices form a semi-abelian category (see [Joh2004]). It would be interesting to study
homology there, as traditionally this setting is not connected to any homology theories.
However, the previous proposition implies that any semi-abelian homology theory in the
category of Heyting semi-lattices is necessarily trivial, as the only abelian object is the zero
object. We will prove this fact. For the axioms of Heyting semi-lattices see for example
[Joh2004] or [Joh2002, A 1.5.11].
Given a morphism f : A −→ B in the category of Heyting semi-lattices which has equal
composite with (1A, 0) : A×A −→ A and (0, 1A) : A×A −→ A, we show that f has to be
the zero map. This implies that any abelian object is zero. The condition on f can be
written as
f(>, a) = f(a,>) ∀a ∈ A.
We have
f(>, a) = f((a⇒ a), (> ⇒ a)) = f((a,>)⇒ (a, a)) = (f(a,>)⇒ f(a, a))
100
5.6 Homology with projectives
so writing b = f(>, a) = f(a,>) and c = f(a, a), we have
b = (b⇒ c).
But then b = b ∧ (b⇒ c) = b ∧ c and c ∧ b = c ∧ (b⇒ c) = c, so b = c and we have
b = (b⇒ b) = >.
Therefore, translating back, we have f(>, a) = f(a,>) = > and f(a, a) = > for all a ∈ A.
We then see by
f(b, a) = (> ⇒ f(b, a)) = (f(b,>)⇒ f(b, a)) = f(>, a) = >
that f is the zero map, as claimed.
5.6 Homology with projectives
In this section we investigate our new definition of homology in the situation when A
does have enough projectives. In this case we know that homology exists, for example
via Everaert’s definition using the Hopf formulae, and Proposition 5.2.3 shows that it
coincides with the notion introduced in Definition 5.5.2. But by reducing the size of the
diagram which defines the homology objects, we obtain a new way to calculate homology.
Our main aim is to show Theorem 5.6.6 which states that the (n + 1)st homology of an
object A may be computed as a limit over the category Endp of all endomorphisms of an
n-presentation p of A.
5.6.1 Notation: For any n-extension f of an object A ∈ |AE|, let Endf , the category
of endomorphisms of f over A, be the full subcategory of ExtnAA determined by the
object f . Thus maps in Endf are maps from f to itself which restrict to the identity on
A under the functor codn.
When A has enough projectives we can view Proposition 5.2.3 the other way round:
5.6.2 Theorem (Hopf formula): Let E be a class of extensions in a semi-abelian cate-
gory A with enough projectives, and let B be a strongly E-Birkhoff subcategory of A with
reflector I : AE −→ B. Let n ≥ 1. Given an n-fold presentation p of an object A ∈ |AE|
with initial object Pn, we have
Hn+1(A, I) ∼= JPn ∩K
n[p]
Kn[Jnp]
.
101
Chapter 5. Homology via Satellites
Proof. This is just Proposition 5.2.3 viewed from the perspective of Definition 5.5.2.
5.6.3 Remark: In [Eve2007, Eve2008] Everaert gives a direct proof that the right hand
side of the Hopf formula is a Baer invariant of A: an expression independent of the
chosen n-fold presentation p of A (see also [EVdL2004a, Fro¨1963]). More precisely, any
morphism p −→ p over A induces the identity on (JPn ∩Kn[p])/Kn[Jnp].
Of course we can still calculate homology as a limit, as defined in Section 5.5. It turns
out that in this case, homology may also be computed as a limit over the small subdiagram
of shape Êndp, which is a subcategory of (A ↓ codn).
5.6.4 Notation: Let p be an n-presentation of a k-extension A, with initial object Pn.
Let ιnPn be the n-cube which has initial object Pn and all other objects zero. The
category Êndp is inspired by a higher-dimensional variation of Diagram (I): it is the
subcategory of (A ↓ codn) that is generated by the objects (p, 1A), (ιnPn, !A) and (0, !A),
all endomorphisms of p over A, and the three maps
p
f

A
!A

1A
⇓
A
1A
ιnPn

0

⇑⇓
A
!Alr
0
LR
0
LR
A
!A
lr
in (A ↓ codn). Here f is the identity on Pn and obvious everywhere else, and the right side
of the diagram displays the maps from A to the “terminal” object of the n-cube depicted
on the left. Note that there is an obvious inclusion Endp −→ Êndp sending p to (p, 1A).
5.6.5 Proposition: Consider n ≥ 1 and A ∈ |AE|, and let p be an n-fold presentation
of A with initial object Pn. Then
Jp ∩Kn[p]
Kn[Jnp]
= lim
(
kern◦In◦U : Êndp −→ AE
)
.
Proof. The diagram we are considering is
Kn[p]
Kn[Jnp]
= Kn[Inp]
kern(Inf)=f¯ ,2Kn[InιnPn] = IPn
,20lr
Notice that the initial object of the n-cube Inp is Pn/Kn[Jnp]. We will show that the
object (Jp∩Kn[p])/Kn[Jnp] is the kernel of the map f¯ , which will in turn imply that it is
indeed the limit of our diagram.
102
5.6 Homology with projectives
Forming the cokernel Q of Kn[p] −→ Pn, we construct the following maps:
JPn ∩Kn[p]  ,2 ,2_

JPn_

 ,2 JQ_

Kn[p]
_
 ,2 ,2 Pn
 ,2
_
Q
_
Kn[p]
JPn∩Kn[p]
 ,2 ,2 IPn
 ,2 IQ
Here the columns and the first two rows are short exact sequences, and the top left square
is a pullback because JQ −→ Q is a mono. The last row clearly composes to give the
zero-map, so using the 3×3-Lemma we see that the last row is also a short exact sequence.
To avoid fractions in the text we will name the map g : IPn −→ IQ and so refer to its
kernel as K[g], and may use Kn[Inp] instead of its explicit description as a fraction.
We now wish to show that f¯ factors over K[g]. For this we first see that taking domn
of the commutative square
p
ηnp  ,2
f

Inp
Inf

ιnPn
ηn
(ιnPn) ,2 In(ιnPn)
gives
Pn
 ,2 Pn
Kn[Jnp]
_
Pn
ηPn  ,2 IPn.
Also, since domn In(ιnPn) = Kn[In(ιnPn)] = IPn, we see that f¯ factors as
Kn[Inp] =
Kn[p]
Kn[Jnp]
 ,2 ,2
f¯

Pn
Kn[Jnp]
= domn Inp
_
IPn IPn
.
So in the following diagram, all possible squares and triangles commute.
Kn[p]  ,2 ,2
_
 '
GG
GG
GG
GG
Pn
 ,2
_
 '
FF
FF
FF
FF
Q
_
Kn[p]
Kn[Jnp]
 ,2 ,2
f¯
'F
FF
FF
FF
F
hw
 w
w
w
w
Pn
Kn[Jnp]
8w
 x
xx
xx
xx
x
K[g]  ,2 ,2 IPn
g  ,2 IQ
103
Chapter 5. Homology via Satellites
By considering the composite Kn[p]  ,2Kn[Inp]
f¯ ,2IPn
g  ,2IQ in this diagram, we see
that g◦f¯ = 0 and so f¯ factors over K[g] by the regular epimorphism
Kn[p]
Kn[Jnp]
h  ,2 K
n[p]
JPn∩Kn[p] = K[g] .
By Noether’s Isomorphism Theorem its kernel is
K[f¯ ] = K[h] =
JPn ∩Kn[p]
Kn[Jnp]
.
Any cone (C, σ) on kern◦In◦U : Êndp −→ AE consists of three maps as shown below.
C
σ(0,!A)

σ(ιnPn,!A)
OOO
OOO
OO
#+OO
OOO
OOO
σ(p,1A) ,2 Kn[Inp]
f¯

0 ,2 IPn
Thus σ(p,1A) factors over K[f¯ ], which we claim to be the limit of ker
n◦In◦U : Êndp −→ AE.
It remains to show that K[f¯ ] is itself a cone over this diagram. Given any endomor-
phism e of p over A, we write e¯ for the induced endomorphism of Kn[Inp] and en : Pn −→ Pn
for its “top” component. To show K[f¯ ] forms a cone over the diagram, we have to prove
that e¯◦Ker f¯ = Ker f¯ . But this follows from the fact that (JPn ∩Kn[p])/Kn[Jnp] is a Baer
invariant of A (see Remark 5.6.3). Indeed, in the diagram
JPn∩Kn[p]
Kn[Jnp]
 ,2 Ker f¯ ,2
e¯′

Kn[Inp]
f¯ ,2
e¯

IPn
Ien

JPn∩Kn[p]
Kn[Jnp]
 ,2
Ker f¯
,2 Kn[Inp]
f¯
,2 IPn
the induced map e¯′ is the identity, and the needed equality follows. Thus
JPn ∩Kn[p]
Kn[Jnp]
forms a cone on kern◦In◦U : Êndp −→ AE and is indeed the limit, as claimed.
5.6.6 Theorem: Consider n ≥ 1 and A ∈ |AE|. If AE has enough projectives and p is an
n-fold presentation of A then
Hn+1(A, I) = lim
(
kern◦In : Endp −→ AE
)
.
Proof. By Theorem 5.6.2 the (n + 1)st homology of A is (JPn ∩ Kn[p])/Kn[Jnp]. Hence
104
5.6 Homology with projectives
by Proposition 5.6.5 it suffices to show that Endp is initial in Êndnp. We must check that
the slice categories (Endp ↓ (p, 1A)), (Endp ↓ (ιP , !A)) and (Endp ↓ (10, !A)) are non-empty
and connected (here we view Endp as the full subcategory of Êndp determined by (p, 1A)).
There is only one possible map from (p, 1A) to (10, !A), and the other two categories fulfil
the needed conditions essentially because ((1P , 1Q), (1A, 1A)) is a terminal object of the
slice category (Endp ↓ (p, 1A)), and the only maps in Êndp from (p, 1A) to (ιP , !A) are
compositions of an endomorphism of (p, 1A) with f .
5.6.7 Remark: This means that computing the homology of an object essentially amounts
to finding fixed points of endomorphisms of a projective presentation of this object. The
use of this technique will be illustrated in Examples 5.6.9 and 5.6.10.
5.6.8 Remark: We now come back to Remark 5.6.3 and interpret Definition 5.5.2 in terms
of Baer invariants. It provides an alternative answer to the following question: “Given
a functor I : A −→ A and an object A of A, how can we construct an object Hn+1(A, I)
out of the n-extensions of A in a manner which is independent of any particular chosen
extension of A?” The classical example is the Hopf formula
H2(A, I)G ∼= JP ∩K[p]K[J1p]
which expresses H2(A, I)G in terms of a projective presentation p : P −→ A of A. Of
course, the very existence of the isomorphism implies that the expression on its right hand
side cannot depend on the choice of p. The idea behind Definition 5.5.2 is different but
straightforward: simply take the limit of all extensions of A. The independence might
now be understood as follows. If p is an n-presentation of A then Hn+1(A, I) is the
limit of kern◦In : Endp −→ A, which means that Hn+1(A, I) is the universal object with
the property that all endomorphisms of p are mapped to the same automorphism of this
object, its identity.
Finally we show, as worked out examples, that we can retrieve well-known results in
group homology using our new definition.
5.6.9 Example (Finite cyclic groups): We use the methods of our theory to calculate
H2(Cn, ab) for any n ∈ N, where Cn is the cyclic group of order n. As Z is projec-
tive and abelian, the map p : Z −→ Cn which sends 1 ∈ Z to a generator c ∈ Cn is a
projective presentation of Cn, and central. Thus H2(Cn, ab) is the limit of the diagram
105
Chapter 5. Homology via Satellites
ker : Endp −→ Gp. Now any endomorphism of p must be
Z
·(nk+1)

p  ,2
⇓
Cn
Z p
 ,2 Cn
i.e., multiplication by (nk + 1) for some k ∈ Z. So H2(Cn, ab) is the limit of the diagram
which has nZ as the only object , and maps ·(nk+1): nZ −→ nZ. If λ : H2(Cn, ab) −→ nZ
is the leg of the limit cone, we must have λ(x) · (nk + 1) = λ(x) for every element
x ∈ H2(Cn, ab) and every k. So we are looking for fixed points of the map ·(nk + 1). But
as, in nZ, 0 is the only fixed point of multiplication by (nk + 1) for all k 6= 1, we have
λ(x) = 0 for all x ∈ H2(Cn, ab). Thus, as λ is a limit cone and so a monomorphism,
H2(Cn, ab) = 0.
Notice that as Cn and Z are abelian, they are also nilpotent of any class m ≥ 1 and
solvable of any derived length m ≥ 1. Therefore p : Z −→ Cn is also central with respect
to Nilm and Solm for any m ≥ 1, and so we have H2(Cn,nilm) = 0 and H2(Cn, solm) = 0
for any m ≥ 1.
5.6.10 Example (Generators and relations): Given a presentation of a group in terms
of generators and relations, for example
A = 〈a1, . . . , an | ri = 1〉
for some relations ri, the kernel of the free presentation
p : Fn −→ A
is generated by the relations ri as a normal subgroup of Fn. Here Fn is the free group on
n generators. But when we go to the centralisation
centr p :
Fn
[K[p], Fn]
−→ A,
every element of the kernel commutes with every other element, so now K[centr p] is
generated by the relations ri as a subgroup of Fn/[K[p], Fn]. Every endomorphism of p
over A must send a generator ai to aiki for some ki ∈ K[f ], and any choice of ki gives such
an endomorphism. Thus on centr p we get endomorphisms that send ai ∈ Fn/[K[p], Fn]
to ai
∏
j r
αij
j , for some αij ∈ Z, and again any choice of αij gives an endomorphism. Note
that K[centr p] is an abelian group, since it is in the centre of Fn/[K[p], Fn]. From here it
is relatively easy to find the fixed points of the induced endomorphism of K[centr p], given
106
5.6 Homology with projectives
a specific group in terms of generators and relations. We give as an example
Cn × Cn = 〈a, b | an = 1 = bn, aba−1b−1 = 1〉.
Here p : F2 −→ Cn × Cn, and K[centr p] is generated by x = an, y = bn and z = aba−1b−1.
Note that as aba−1b−1 commutes with everything, we get (aba−1b−1)n = abna−1b−n,
and as bn also commutes with everything, we have zn = 1. As described above, any
endomorphism of centr p induced by one on p sends a ∈ F2/[K[p], F2] to axα1yα2zα3 and
b to bxβ1yβ2zβ3 . On K[centr p] this gives
x 7−→ xnα1+1ynα2
y 7−→ xnβ1ynβ2+1
z 7−→ z
as the x, y and z commute with everything, and zn = 1. For xl1yl2zl3 to be a fixed point
for any of these endomorphisms, we need
l1α1 + l2β1 = 0
l1α2 + l2β2 = 0
for any choice of αi and βi, or in other words we need
l1α+ l2β = 0
for any choice of α and β. Hence l1 = l2 = 0, and we have fixed points zl3 . Since zn = 1,
we get
H2(Cn × Cn, ab) = Cn.
Note that we can use the diagram over Êndp instead of Endp to see that any fixed
point must be of the form aba−1b−1 for some a and b (or a product of such), since the
fixed point must be sent to the identity in abFn = Fn/[Fn, Fn].
H2(A, ab) ,2
"*NN
NNN
NNN
NNN

K[centr p]

0 ,2 abFn
Comparing this to the Hopf formula
H2(A, ab) =
[Fn, Fn] ∩K[p]
[K[p], Fn]
,
we see that the calculation using our method is exactly the same as the one using the Hopf
107
Chapter 5. Homology via Satellites
formula; the only thing that is different is the interpretation of these elements as fixed
points of certain endomorphisms. Note that we of course proved in Proposition 5.6.5 that
the limit of the diagram ker◦I1 : Êndp −→ A is the expression of the Hopf formula, so this
is exactly what you would expect.
In contrast to the situation using abelianisation, we give another example which uses
the Birkhoff subcategory of nilpotent groups.
5.6.11 Example (Gp vs. Nil2): We calculate the second homology group of C2×C2 with
respect to the Birkhoff subcategory Nil2 of nilpotent groups of class at most 2, using the
Hopf formula. We again use the projective presentation
p : F2 −→ C2 × C2
with kernel K, but now centralisation gives
K
[[K,F2],F2]
 ,2 ,2 F2
[[K,F2],F2]
I1p  ,2 C2 × C2.
So the Hopf formula gives
H2(C2 × C2,nil2) = K ∩ [[F2, F2], F2][[K,F2], F2] =
K
[[K,F2], F2]
∩ [[F2, F2], F2]
[[K,F2], F2]
using Examples 1.3.3 and 4.3.7.
If F2 is generated by a and b, a long (and tedious) calculation shows that K[I1p] is
generated by the set
{x1 = a2, y1 = b2, z = [a, b], x2 = ba2b−1, y2 = ab2a−1, x3 = b−1a2b, y3 = a−1b2a}
with many relations, some of which we will list. The main fact we need to get these
identities is that any element in [K,F2] commutes with any element of F2. We get that
the element z commutes with all other elements of K[I1p], the xi commute amongst each
other and so do the yi. We have
[x1, y1] = x1y1x−11 y
−1
1 = y1x
−1
1 y
−1
1 x1 = [y1, x
−1
1 ] = [x
−1
1 , y
−1
1 ] = [y
−1
1 , x1] = z
4
that is, all cyclic permutations of x1y1x−11 y
−1
1 are the same as z
4. Similarly
[x1, y2] = [x2, y1] = [x1, y3] = [x3, y1] = [x3, y2] = [x2, y3] = [x2, y2] = [x3, y3] = z4
108
5.6 Homology with projectives
with the same cyclic permutations as for [x1, y1] above. Thus we see that z4 commutes
with every element of F2, not just elements of the kernel as z does. We also have
[a2, b]2 = (x1x−12 )
2 = [a−2, b−1]2 = (x−11 x3)
2 = z4
and the corresponding expression for y instead of x. Notice that the elements [a2, b] etc.
already commute with any element in F2.
The following Witt-Hall identities of commutators are very useful:
[g, h1h2] = [g, h1]h1[g, h2]h−11
[g1g2, h] = g1[g2, h]g−11 [g1, h]
These identities give for example that [a, ba] = [a, b], and also [[a, ab], a] = [[a, b], a] and
[[a, ab], b] = [[a, b], b] (using as always that elements of [K,F2] commute with every element
of F2). As soon as one each of g, h2 and g2, h lie in the kernel K, these identities become
[g, h1h2] = [g, h1][g, h2]
[g1g2, h] = [g2, h][g1, h].
We also see that [k, g−1] = [k, g]−1 = [k−1, g] for any k ∈ K and g ∈ F2. Thus we find
that the elements
[z, a] = [[a, b], a] and [z, b] = [[a, b], b]
generate K∩[[F2,F2],F2][[K,F2],F2] , and that
[z, a]4 = [z4, a] = 1 and [z, b]4 = [z4, b] = 1.
Thus we have H2(C2 × C2,nil2) = C4 × C4.
109
Chapter 5. Homology via Satellites
110
Chapter 6
Stem Extensions in the Context of
Abelianisation
In this chapter we look at certain special central extensions such as trivial and stem
extensions, and establish a bijection between the isomorphism classes of stem extensions
of an object A and the subobjects of H2(A, ab). This bijection is induced by the well-
known isomorphism Centr(A,K) ∼= Hom(H2A,K) which is usually achieved in two steps,
using Centr(A,K) ∼= H2(A,K) ∼= Hom(H2A,K). Here we give an explicit description of
the direct isomorphism, without going via the second cohomology group. The Stallings-
Stammbach sequence, which is the lower part of the Everaert sequence we have met before,
plays a crucial role in these results.
6.1 Stem extensions and stem covers
Let us first review some of the material from Chapters 4 and 5 and restate it appropriately
for its use in this chapter.
As we are dealing with a semi-abelian category A and its Birkhoff subcategory of
abelian objects AbA, we will denote the kernel of the unit of abelianisation of A by [A,A]
instead of JA:
0 ,2 [A,A]  ,2 ,2A
ηA  ,2 abA = H1A ,20
This extends to a functor [−,−] : A −→ A. We also write [K[p], P ] for the object J1[p] =
K[J1p] (see Section 4.3).
We use homology defined via Hopf formulae, as in Definition 4.4.3, but also use prop-
erties of homology exhibited by viewing it as a limit, as in Chapter 5. As we will only need
the second homology object, we will repeat the definition here with our slightly changed
notation.
6.1.1 Definition ([Eve2007]): Given an object A in A, let
K  ,2 ,2P
p  ,2A
111
Chapter 6. Stem Extensions in the Context of Abelianisation
be a projective presentation of A. We define the second homology object of A by the
Hopf Formula
H2A =
K ∩ [P, P ]
[K,P ]
.
The Stallings-Stammbach sequence [EVdL2004b, EG2007], which is the lowest part of
the Everaert sequence (F), plays a big role in our results, but we only use it for central
extensions, so we restate it here for this special case.
6.1.2 Proposition: For any central extension
0 ,2K  ,2 ,2B
f  ,2A ,20 (N)
in A, there is an exact sequence
H2B
H2f ,2H2A
δf ,2K
γf ,2H1B
H1f ,2H1A ,20 (O)
in AbA which is natural in f .
Proof. This is just the lowest part of (F) in Theorem 4.4.6. See also [EVdL2004b, EG2007].
Notice that as here f is already central, we get the kernel of f instead of the kernel of
I1f = H1(f, I1)E1 .
For convenience we also recall Corollary 5.5.10 for the case of H2.
6.1.3 Proposition: Given an object A in A, the second homology object H2A is the limit
of the diagram
ker : CExtAA −→ A.
The legs of the limit cone are the δf : H2A −→ K[f ] as given by the Stallings-Stammbach
sequence (O) for a central extension f .
Proof. See Proposition 5.2.3 and Corollary 5.5.10.
We now define some special central extensions that we will consider throughout the
chapter. Trivial extensions have been defined before, in 1.4.3, but we restate the definition
here with more equivalent conditions, along with diagrams which clarify the connection
to the new definitions. We have also met stem extensions before in the context of groups,
in Example 5.5.15. The la´ma´ra extensions, which I have named thus just for the moment,
are not particularly important in what follows and are only included for completeness.
6.1.4 Definition: Given a central extension f as in (N), we say it is a
112
6.1 Stem extensions and stem covers
(1) trivial extension if one of the following equivalent conditions is satisfied:
• δf = 0,
• [B,B] −→ [A,A] is an isomorphism,
• K ∩ [B,B] = 0;
0 = K ∩ [B,B]  ,2 ,2
_

[B,B]
∼= ,2
_

[A,A]
_

K
 ,2 ,2 B
f  ,2
_
A
_
H2A
0 ,2 K
 ,2 ,2 H1B
 ,2 H1A
(2) la´ma´ra extension if δf is a monomorphism;
(3) stem extension if one of the following equivalent conditions is satisfied:
• δf is a regular epimorphism,
• H1B −→ H1A is an isomorphism,
• γf = 0,
• K ≤ [B,B];
[B,B]
_

 ,2 [A,A]
_

K
 ,2 ,2
7B
7B
B
f  ,2
_
A
_
H2A
δf  ,2K
0 ,2 H1B
∼= ,2 H1A
(4) stem cover if one of the following equivalent conditions is satisfied:
• δf is an isomorphism,
• H1B −→ H1A is an isomorphism and H2B −→ H2A is the zero map.
[B,B]  ,2
_

[A,A]
_

K
 ,2 ,2
7B
7B
B
f  ,2
_
A
_
H2B
0 ,2 H2A
∼= ,2 K 0 ,2 H1B
∼= ,2 H1A
Clearly every stem cover is also a stem extension.
113
Chapter 6. Stem Extensions in the Context of Abelianisation
6.2 Perfect objects
From now on we only consider central extensions (N) where A is a perfect object. It is
well known that a group has a universal central extension if and only if it is perfect. This
holds more generally in any semi-abelian category.
6.2.1 Definition (Universal central extension): A universal central extension
(with respect to abelianisation) is a central extension g : C −→ A such that for any other
central extension h : D −→ A there is a unique map C −→ D making the following square
commute:
C
g  ,2
∃!

A
D
h  ,2 A
6.2.2 Definition: An object A of a semi-abelian category A is called perfect when its
abelianisation is zero: AbA = 0.
6.2.3 Lemma: [GVdL2008b, Proposition 4.1] Let A be a semi-abelian category with
enough projectives. An object A of A is perfect if and only if A admits a universal central
extension.
Recall from Corollary 5.5.11 that if A admits a universal central extension g, then the
second homology group is the kernel of this central extension: H2(A, ab) = K[g]. In fact,
we can say more:
6.2.4 Lemma: Any universal central extension is a stem cover.
Proof. Let K  ,2 ,2C
g  ,2A be a universal central extension of A. The limit of the dia-
gram
ker : CExtAA −→ A,
is (H2A, δf )f∈CExtAA. As g is initial in CExtAA, the limit H2A is isomorphic to the kernel
K of g, and the isomorphism is given by the leg δg. Thus, by definition, g is a stem cover.
Notice that we could always make δg into an actual identity by changing the kernel of g
to an isomorphic one.
When A is a perfect object, the special extensions defined in 6.1.4 take a more specific
form.
6.2.5 Lemma: When A is perfect,
(1) any central extension (N) has H1B ∼= K/(K ∩ [B,B]),
114
6.2 Perfect objects
(2) any trivial extension is isomorphic to a product projection K ×A pi2  ,2A,
(3) any la´ma´ra extension satisfies H2A ∼= K ∩ [B,B],
(4) any stem extension has perfect domain B.
Proof. (1) If H1A = 0, we have the following commuting diagram:
K ∩ [B,B]  ,2 ,2
_

[B,B]
_

 ,2 [A,A]
K
 ,2 ,2
_
B
f  ,2
_
A
_K
K∩[B,B] ,2 H1B ,2 0
Here all columns and the top two rows are exact sequences, so using the 3 × 3
Lemma [Bou2001], we conclude that the bottom row is also short exact and the
result follows. Notice that the map γf in the Stallings-Stammbach sequence (O)
then becomes the canonical quotient K  ,2K/(K ∩ [B,B]).
(2) If H1A = 0, the pullback in 6.1.4 becomes a product:
K
 ,2 ,2 B
f  ,2
_
A
_
H2A
0 ,2 K
∼= ,2 H1B ,2 0
Thus B = H1B ×A ∼= K ×A and f is a product projection.
(3) Combining (1) with the fact that δf is a monomorphism, the Stallings-Stammbach
sequence (O) becomes
0 ,2H2A
 ,2 ,2K  ,2 KK∩[B,B]
∼= H1B ,20
and so H2A ∼= K ∩ [B,B].
(4) If H1A = 0 and H1B ∼= H1A, clearly H1B = 0 as well.
K
 ,2 ,2 B
f  ,2
_
A
_
H2A
δf  ,2K ,2 0
∼= ,2 0
115
Chapter 6. Stem Extensions in the Context of Abelianisation
6.3 A natural isomorphism
In this section, we want to define a natural isomorphism
θ : Centr(A,−) −→ Hom(H2A,−)
of functors AbA −→ Ab. First we must define the abelian group Centr(A,K) for an abelian
object K.
6.3.1 Remark: Notice that we will be talking about two different kinds of isomorphism
classes of central extensions. Given an object A, we say two central extensions of A are
isomorphic if there is an isomorphism
K
 ,2 ,2
∼=

B
 ,2
∼=

A
K ′  ,2 ,2 B′  ,2 A
of central extensions. However, when we fix the kernel K, we say two central extensions
of A by K are isomorphic if there is a morphism
K
 ,2 ,2 B
 ,2
∼=

A
K
 ,2 ,2 B′  ,2 A
where we demand the kernel part to be an actual identity. It follows from the Short Five
Lemma that two central extensions of A by K are isomorphic if and only if there is a
morphism between them.
Let Centr(A,K) denote the set of isomorphism classes of central extensions of A by
K, as defined in the remark above. Recall from [Ger1970] how Centr(A,K) can be made
into an abelian group using the Baer sum: given two central extensions
K  ,2
k
B
f  ,2A and K  ,2 k
′
B′
f ′  ,2A,
let h be their pullback
B ×A B′  ,2
_
h
	 (I
III
III
II
B′
f ′
_
B
f
 ,2 A
or equivalently
B ×A B′ h  ,2

A
∆

B ×B′
f×f ′
 ,2 A×A
116
6.3 A natural isomorphism
where ∆ is the diagonal. Then the extension f + f ′ is formed as follows, using the
multiplication m of the abelian object K:
K[m]
_
i

K[m]
_
(k×k′)◦i

K ×K  ,2k×k
′
,2
m
_
B ×A B′ h  ,2
Coker (k×k′)◦i
_
A
K
 ,2
k×k′
,2 D
f+f ′
 ,2 A
As h is a central extension with respect to abelianisation, the subobject (k×k′)◦i is normal
(see Proposition 1.5.16). So we can form its cokernel, or equivalently the pushout of k×k′
and m. Then k × k′ is a monomorphism as the top square is a pullback. We take its
cokernel, the codomain of which is A since the bottom left square is a pushout. Note that
this square is also a pullback. This cokernel represents the isomorphism class of f + f ′; it
is central because the category of central extensions in A is closed under quotients in the
category of extensions in A (see [GVdL2008b] for more details). It is easily checked that
this gives a well-defined sum on the isomorphism classes of central extensions of A by K.
The zero for this addition is the class of split central extensions, a representative of which
is the product projection K  ,2 ,2K ×A pi2  ,2B . Note that any split central extension is
trivial [JK1994, Section 4.3], and compare this with 6.2.5(2).
Recall from [GVdL2008b] how Centr(A,−) is made into a functor AbA −→ Ab. The
main ingredient is the following result:
6.3.2 Lemma: [GVdL2008b, Corollary 3.3] Given a central extension (N) and a map
from K to an abelian object K ′, there is a central extension K ′  ,2 ,2B′  ,2A making
the diagram below commute.
K
 ,2 ,2

B
 ,2

A
K ′  ,2 ,2 B′  ,2 A
Proof. See Corollary 3.3 in [GVdL2008b]. For this construction to work it is crucial that
the extension B −→ A is central in the sense of Huq, which is the main reason why this
isomorphism only works in the setting of abelianisation.
This indeed makes Centr(A,−) into a functor AbA −→ Ab (see Propositions 6.1 and
6.2 in [GVdL2008b]).
117
Chapter 6. Stem Extensions in the Context of Abelianisation
For a given abelian object K, the set Hom(H2A,K) also forms an abelian group. Two
maps α and β are added using the multiplication m of K as follows:
α+ β : H2A
(α,β) ,2K ×K m ,2K
The zero of the addition is the zero map H2A
0 ,2K . Therefore we can view the functor
Hom(H2A,−) : A −→ Set also as a functor Hom(H2A,−) : AbA −→ Ab.
We can now finally define, for a perfect object A, the required natural transformation.
6.3.3 Definition: Given a perfect object A in A, we define a natural transformation
θ : Centr(A,−) −→ Hom(H2A,−)
as follows: for each abelian object K in A, θK takes the (isomorphism class of the)
central extension K  ,2 k ,2B
f  ,2A to the map δf : H2A −→ K defined by the Stallings-
Stammbach sequence (O). This is well-defined: if f is isomorphic to K  ,2 k
′
,2B′
f ′  ,2A ,
the Stallings-Stammbach sequence gives
H2A
δf ,2 K ,2 H1B ,2
∼=

H1A = 0
H2A
δf ′ ,2 K ,2 H1B′ ,2 H1A = 0
so δf = δf ′ . Naturality of θ follows from the definition of the functor Centr(A,−) and the
fact that δ itself is a natural transformation. For θ to be a natural transformation between
functors to Ab, we must show that each θK is a homomorphism of abelian groups. We do
this in a separate lemma.
6.3.4 Lemma: For each abelian object K in A, θK is a homomorphism of abelian groups.
Proof. Given two central extensions f and f ′ representing two isomorphism classes in
Centr(A,K), we consider their pullback B ×A B′ h  ,2A as above in the definition of
the sum in Centr(A,K). We again use the limit cone (H2A, δf )f∈CExtAA of the diagram
118
6.3 A natural isomorphism
ker : CExtAA −→ A to see that δh = (δf , δf ′) : H2A −→ K ×K:
H2A
δf
vuu
uu
uu
uu
u
δh

δf ′
(II
III
III
II
K_

K ×Kpi1lr pi2  ,2_

K_

B
f
_
B ×A B′lr  ,2
h
_
B′
f ′
_
A A A
Remembering that f + f ′ is given by
K ×K  ,2k×k
′
,2
m
_
B ×A B′
_
h  ,2 A
K
 ,2 ,2 D
f+f ′  ,2 A
the Stallings-Stammbach sequence gives us a commuting diagram
H2A
δh=(δf ,δf ′ ) ,2
δf+δf ′
!)LL
LLL
LLL
LLL
LLL
LLL
K ×K
m
_
H2A δf+f ′
,2 K
and the result follows.
6.3.5 Lemma: For each abelian object K in A, θK is injective.
Proof. If the central extension f maps to δf = 0, it is a trivial extension by definition,
and so as A is perfect, f is (isomorphic to) the product projection K  ,2 ,2K ×A pi2  ,2A
(see Lemma 6.2.5).
6.3.6 Lemma: For each abelian object K in A, θK is surjective.
Proof. Given a morphism α : H2A −→ K, we use a universal central extension of A and
Lemma 6.3.2 to construct a central extension of A by K. Let H2A
 ,2 ,2C
g  ,2A be a
universal central extension of A, which is a stem cover by Lemma 6.2.4. We can choose
g such that δg : H2A −→ H2A is the identity on H2A. Then as K is an abelian object,
119
Chapter 6. Stem Extensions in the Context of Abelianisation
Lemma 6.3.2 gives us a central extension f and a map from g to f as below:
H2A
 ,2 ,2
α

C

g  ,2 A
K
 ,2 ,2 B
f  ,2 A
Forming the Stallings-Stammbach sequence of f and g, we see that δf = α:
H2A
δg
H2A
α

H2A δf
,2 K
Thus the isomorphism class of f really is a preimage of α.
We can now put together the above results into our main theorem.
6.3.7 Theorem: Given a perfect object A in a semi-abelian category A, there is a natural
isomorphism
θ : Centr(A,−) −→ Hom(H2A,−)
of functors AbA −→ Ab. Fixing an abelian object K,
(1) trivial extensions correspond to the zero map H2A
0 ,2K (i.e. there is only one
equivalence class of trivial extensions of A by K, one representative of which is the
product projection K  ,2 ,2K ×A pi2  ,2A),
(2) la´ma´ra extensions correspond to monomorphisms H2A
 ,2 ,2K ,
(3) stem extensions correspond to regular epimorphisms H2A
 ,2K ,
(4) stem covers correspond to isomorphisms H2A
∼= ,2K .
Proof. θ is a natural isomorphism by Lemmas 6.3.4, 6.3.5 and 6.3.6. The correspondences
(1) to (4) all follow directly from Definition 6.1.4 and the definition of θ.
6.3.8 Corollary: There is only one isomorphism class of stem covers of a perfect object.
Proof. If we only look at isomorphisms of central extensions of a perfect object A by a
fixed kernel K, as in Centr(A,K), Theorem 6.3.7(4) gives us several isomorphism classes
120
6.3 A natural isomorphism
of stem extensions, one for each isomorphism H2A −→ K. But as soon as we vary K, we
see that all stem covers of A are isomorphic to each other (cf. Remark 6.3.1).
H2A
 ,2 ,2
∼=δf

B
g  ,2
∼=

A
K
 ,2 ,2 B′
f  ,2 A
6.3.9 Corollary: Any stem cover of a perfect object is a universal central extension.
Proof. As any universal central extension is a stem cover by Lemma 6.2.4 and there is
only one isomorphism class of stem covers, any stem cover is universal.
6.3.10 Remark: The isomorphism in Theorem 6.3.7 is not new, it exists as a two-step iso-
morphism Centr(A,K) ∼= H2(A,K) ∼= Hom(H2A,K) given for example in [GVdL2008b].
We have merely given a direct correspondence without going via the cohomology group.
6.3.11 Corollary: There is a bijective correspondence between the isomorphism classes
of stem extensions of a perfect object A and subobjects of H2A.
Proof. When we vary K in Theorem 6.3.7, we see that the isomorphism classes of stem
extensions of A correspond to regular epimorphisms H2A
 ,2K for any K, which in turn
correspond to subobjects U of H2A by taking the kernels of these regular epimorphisms.
6.3.12 Lemma: Every stem extension of a perfect object A is the regular image of a stem
cover of A.
Proof. As A is perfect, it has a universal central extension H2A
 ,2 ,2C
g  ,2A which is a
stem cover. As g is universal, for every stem extension f there is a map
H2A
 ,2 ,2
δf
_
C
g  ,2

A
K
 ,2 ,2 V
f  ,2 A
where it is easily seen using the Stallings-Stammbach sequence that the induced map
H2A −→ K is indeed (isomorphic to) δf . By the Short Five Lemma, which also holds for
regular epimorphisms (see [Bou2001]), the map C −→ V in the middle is also a regular
epimorphism.
121
Chapter 6. Stem Extensions in the Context of Abelianisation
6.3.13 Remark: The results in Corollaries 6.3.8 and 6.3.11 and in Lemma 6.3.12 are ex-
tensions to general semi-abelian categories of the corresponding results on crossed modules
in [VC2002]. If C is a semi-abelian category, the category A = Grpd(C) of groupoids in
C is also semi-abelian, and its subcategory of abelian objects AbA is RG(AbC), reflective
graphs in AbC. A special case of this is the category of crossed modules when C = Gp the
category of groups. This gives back the results in [VC2002].
122
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126
Index
 ,2 ,2 normal mono, 8
 ,2 normal epi, 8
0 zero-object, zero map, 7
(A ↓ F ) slice category, 85
AbA subcat. of abelian objects, 15
Ab cat. of abelian groups, 17
ab abelianisation functor, 15
ArrkA higher-dimensional arrows, 62
!A map A −→ 0, 71
Centr(A,K) central ext. of A by K, 116
CExtBA cat. of central extensions, 68
cod codomain functor, 73
Q[f ] cokernel object, 8
Coker f cokernel map, 8
A
B cokernel, 8
[G,G] commutator subgroup, 16
G = (G, , δ) comonad, 37
φf,g cooperator, 24
δf connecting homomorphism, 77
DlG lth term of derived series, 18
∆ diagonal, 15
dom domain functor, 78
Endp cat. of endos of p, 101
Êndp enlarged cat. of endos of p, 102
E class of extensions, 62
obE domains and codomains in E, 62
AE full subcat. determined by obE, 62
En class of higher extensions, 63
ExtA category of extensions, 66
ExtnA category of higher extensions, 66
g Lie or Leibniz algebra, 18
gAnn ideal gen. by all [x, x], 19
Γ Galois structure, 20
γf homom. in Everaert sequence, 77
Gp category of groups, 8
HnC homology of a chain complex, 16
HnA — of a simplicial object, 32
Hn(A, I)G comonadic —, 37
Hn(A, I)E — via Hopf formulae, 74
Hn(A, I) — via satellites/limits, 91
I reflector, 17
I1 centralisation functor, 69
I1[f ] domain of I1f , 69
I[f ] image, 9
Im f image inclusion, 9
ιnPn cube (Pn, 0, . . . , 0), 102
J functor giving kernel of unit, 67
J1f kernel of unit for centralisation, 68
J1[f ] domain of J1f , 68
K[f ] kernel object, 7
Ker f kernel inclusion, 7
ker kernel functor, 62
LCmG term in lower central series, 18
LeibK cat. of Leibniz algebras, 19
LieK cat. of Lie algebras, 19
[n] = {0 . . . , n}, 31
Nilm cat. of nilpotent groups, 18
nilm reflector to Nilm, 18
p projective presentation, 73
Pn initial object of n-cube p, 73
rnf map In[f ] −→ Tn[f ], 73
R-Mod cat. of R-modules, 37
R[f ] kernel pair, 21
(R, pi1, pi2) equivalence relation, 26
dR subdiagonal of equiv. rel., 26
Ran right Kan extension, 82
SESeqA cat. of short exact sequences, 83
A simplicial object, 31
∂i face operator, 31
σi degeneracy operator, 31
NA Moore complex, 32
ZnA object of n-cycles, 32
S simplicial set, 34
AI cocylinder, 56
0(A) cocylinder map, 56
1(A) cocylinder map, 56
s(A) cocylinder map, 56
Solm cat. of solvable groups, 18
solm reflector to Solm, 18
Tn trivialisation functor, 72
TExtnA cat. of trivial extensions, 72
η unit of reflection, 19
ZG,Zb centre of a group/Lie algebra, 22
ZLie(b) Lie centre of Leibniz algebra, 22
ZNilm(B) nil centre, 22
ZSolm(B) solvable centre, 23
127
INDEX
abelian
internal — group, 15
abelian category, 1, 8, 15, 37, 47, 54, 60, 83
abelian group, 97, 116, 117
category of —s, see groups, abelian
abelian object, 3, 14, 26, 32, 67, 79, 99, 111,
116, 117
abelianisation, 15, 16, 17, 20, 28, 95, 96,
111, 114
addition, see abelian object
additive category, 37, 47, 54, 60
additive functor, 83
adjunction, 54, 60, 88, 96
algebras, 54
arrow
double —, 62
higher dimensional —, 62
associated normal subobject, 27
augmented simplicial object, 33, 37, 39, 50
Baer invariant, 102, 104, 105
Barr, 8
Barr-Beck, 1, 31, 37, 47
Barr-Beck homology, 37
Beck, see Barr-Beck
Birkhoff subcategory, 3, 17, 20, 66, 99, 111
strongly E—, 66, 69, 74–80, 85–105
Bourn, 8, 24
bracket, see Leibniz/Lie algebra
category
abelian —, 1, 8, 15, 37, 47, 54, 60, 83
additive —, 37, 47, 54, 60
exact —, 8, 61
— with finite limits, 50
homological —, 10
Mal’tsev —, 10, 21, 34
monadic —, 37
pointed —, 7, 61
protomodular —, 8, 61
regular —, 8, 34
semi-abelian —, 1, 8
sub— of abelian objects, 15, 111
unital —, 10, 24
category of elements, 85
category of endomorphisms, 101, 102
category of extensions of A, 93
central equivalence relation, 28
central extension
finite —, 97
Galois, 21, 29, 67–69, 81, 94, 95, 98,
111–122
higher —, 71
Huq, 25, 28, 29, 111–122
isomorphism class of —s, 116
split —, 24, 76, 78, 117
universal —, 95, 96, 114, 121
weakly universal —, 95
central extension of A by K, 116
sum, 117
central morphism
Huq, 25, 26, 28, 79, 99
central series, 18
centralisation, 68, 69, 70, 73, 75, 85, 88, 95,
96
centralising double relation, 28
centralising equivalence relations, 27
centre
— of a group/Lie algebra, 70
— of group/Lie algebra, 22, 25
Lie —, 22
nil —, 22, 70
solvable —, 23
chain complex, 16, 99
Moore complex, 32, 34
normalised —, see Moore complex
proper —, 16, 32
unnormalised —, 32
Chevalley-Eilenberg, 19, 74
class of extensions, 62, 68
E1, 65
closed under limits, 67
closed under regular quotients, 17
closed under subobjects, 17, 66
cocone, 90
cocylinder, 56
codomain, 73, 86, 89, 91, 101
coefficients, 37, 38, 85, 91
coequaliser, 40
coequaliser iff cokernel, 33, 40
cohomology, 81
comonadic, 37
second — group, 121
cokernel, 8, 9, 40, 58, 61, 102, 117
cokernel iff coequaliser, 33, 40
colimit, 90
comma category, see slice category
128
INDEX
commutative object, 15
commutator, 16, 74, 111
commuting morphisms, 24
comonad, 37, 38, 47, 53, 55
forgetful/free —, 37, 49, 54
projective class of —, 49
relative —, 38, 54, 60
comonadic cohomology, 37
comonadic homology, 2, 19, 37, 38–47, 58–
60
G-acyclicity, 38, 39
G-connectedness, 39
second variable, 38–46, 59
Comparison Theorem, 52, 59
composition of extensions, 62
cone, 85, 86, 92, 95, 97–98, 104, 112, 114
connected equivalence relations, 28
connecting homomorphism, 77, 79, 86, 99,
112, 114, 118
contractibility
relative —, 51, 52
contractible simplicial object, 33, 39
homology, 33
contractible simplicial set, 50
contraction, 33
cooperating morphisms, 24, 26, 79
cooperator, 24
coproducts, 61
crossed modules, 8, 17, 21, 54, 122
cycles, see object of n-cycles
degeneracy operator, 31, 37
derived functor, 82
derived series, 18
diagonal, 15, 23, 117
diagram (with limit), 85, 86, 90, 92–95, 97–
98, 102, 104, 105, 112, 114
diagram lemmas
Five Lemma, 11, 43, 45
Short —, 8, 116, 122
3× 3 Lemma, 11, 71, 103, 115
Noether’s Iso. Theorem, 12, 75, 80, 104
Snake Lemma, 12, 32
direct image, 13, 16, 79, 99
domain, 78, 90, 103
double arrow, 62
double extension, 63, 64, 66, 80
kernel is extension, 64, 68
symmetry, 64, 65
Eckmann-Hilton, 38
Eilenberg, see Chevalley-Eilenberg
endomorphism of p over A, 104–108
enlargement of domain, 34, 57
enough projectives, 49, 49, 74–81, 85–91,
101–109, 114
epimorphism
normal —, 8
regular —, 8–11, 13, 60, 63, 64, 67, 122
pullback-stable —s, 8
split —, 62, 65
equaliser, 34
equivalence relation, 26
centralising —s, 27
connected —s, 28
normal to an —, 27
Everaert, 5, 31, 36, 61, 74
Everaert sequence, 2, 77, 81, 85, 86, 90, 99,
112
exact category, 8, 61
exact sequence, 9, 99
long —, see homology, —
short —, see short exact sequence
examples
abelianisation functor, 16
Birkhoff subcategories, 17
central extensions
Galois, 21
Huq, 25
centralisation functor, 70
class of extensions, 63
comonads, 37
cooperating morphisms, 25
Galois structures, 20
homology
finite cyclic groups, 105
finite groups, 96
generators and relations, 106
Heyting semi-lattices, 100
— of projective objects, 95
reflection is identity, 92
— of zero, 95
zero reflection, 98
Hopf formula, 74
Kan projective class, 54
regular projectives, 49
129
INDEX
relative Kan property, 50
satellite, abelian, 83
semi-abelian categories, 8
strongly E-Birkhoff subcategories, 67
two comonads on R-Mod, 60
Ext groups, 38, 60
extension
central —
finite —, 97
Galois, 21, 29, 67–69, 81, 94, 95, 98,
111–122
higher —, 71
Huq, 25, 28, 29, 111–122
isomorphism class of —s, 116
split —, 24, 76, 78, 117
universal —, 95, 96, 114, 121
weakly universal, 95
central — of A by K, 116
sum, 117
class of —s, 62, 68
E1, 65
composition of —s, 62
double —, 63, 64, 66, 80
symmetry, 64, 65
double — has kernel an —, 64, 68
Galois, 20, 68
higher —, 63, 66, 67, 94
higher central —, 71
higher trivial —, 72
kernel of —, 63
la´ma´ra —, 113, 114, 120
n-fold —, see higher —
normal —, 21
pullback-stable —s, 62, 65, 94
regular epimorphism, 9
split —, 95, 97
stem —, 96, 113, 114, 120
stem cover, 113, 114, 120, 121
trivial —, 21, 24, 78, 98, 112, 114, 117,
120
higher —, 72
extension preserving functor
I, 19
J , 67
extensions of A, 93, 105
higher —, 94
face operator, 31, 37
filler of a horn, see horn filler
finite groups, 96
Five Lemma, 11, 43, 45
Short —, 8, 116, 122
fixed points, 81, 105–108
forgetful/free comonad, 37, 49, 54
G-acyclicity, 38, 39
G-connectedness, 39
G-exact sequence, 39, 40–45
G-left derived functors, 39, 42
G-projective object, 49
Galois structure, 20, 28, 68, 69, 72
Gran, 5, 61
groups, 8, 15–17, 19–21, 25, 37, 50, 54, 61,
67, 70, 74, 96, 98, 105, 106
abelian —, 16, 17, 21, 37, 67, 70
integral homology of —, 1, 19, 37, 74,
78, 96, 105–108
nilpotent —, 18, 21, 22, 70
perfect —, 96
solvable —, 18, 21, 22, 70
Heyting semi-lattices, 8, 100
higher central extension, 71
higher dimensional arrow, 62
higher extension, 63, 66, 67, 94
higher presentation, 73, 75
higher trivial extension, 72
Hochschild, 38, 60
Hom functor, 38, 60
homological category, 10
homology
abelian object, 32, 79, 99
Barr-Beck —, 37
— of chain complexes, 16
comonadic —, 2, 19, 37, 38–47, 58–60
G-acyclicity, 38, 39
G-connectedness, 39
second variable, 38–46, 59
contractible simpl. obj., 33
Everaert sequence, 2, 77, 81, 85, 86, 90,
99, 112
fixed points, 81, 105–108
— via Hopf formulae, 2, 19, 74, 86, 111–
122
integral group —, 1, 19, 37, 74, 78, 96,
105–108
— via Kan extensions, 4, 19, 91
130
INDEX
— via limits, 4, 19, 91, 111
central extensions, 95, 112
equivalent diagrams, 92
with projectives, 104
long exact sequence
abelian, 83
Everaert sequence, 2, 77, 81, 85, 86,
90, 99, 112
second variable, 39
simplicial objects, 33, 39
object of B, 98
— of a projective object, 95
— with projectives, 101–109
— without projectives, 91–101
— via satellites, 4, 19, 91
semi-abelian —, 17
— of a simplicial object, 32, 56
— is zero, 92, 95, 98, 100
— of zero, 95
homotopic maps, 52, 58
— have same homology, 58
homotopy, 53, 55
homotopy equivalent, 55
Hopf formula, 2, 61, 73, 74, 80, 96, 101,
102, 105, 111
alternative —, 75–77
Hopf homology, see homology
horn, 34, 52, 53, 58
horn filler, 34, 52, 53, 58
Huq central, 25, 26, 28, 79, 99
image, 9, 41, 60, 99
direct —, 13, 16, 79, 99
image factorisation, 9, 13, 41
initial functor, 93, 94
initial object, 95
initial object of n-cube, 72, 73, 75, 102
initial subcategory, 93, 104
integral group homology, 1, 19, 37, 74, 78,
96, 105–108
internal abelian group, 15
internal Kan property, 34, 50
isomorphism class of central extensions, 116
isomorphism class of stem covers, 120
Jacobi identity, 19
Janelidze, 20, 81, 82
jointly epic pair, 10, 15, 24
Kan extension, 81–83, 86, 91, 99
Kan projective class, 53, 55, 58
Kan property, 34, 52, 56
internal —, 34, 50
relative —, 49, 53
Kan simplicial set, 34, 50
kernel, 7, 25, 27–29, 41, 75, 95, 96, 102
kernel functor, 62, 88
kernel of double extension, 64, 68
kernel of extension, 63
kernel pair, 23, 26, 51
la´ma´ra extension, 113, 114, 120
Leibniz algebras, 8, 18, 20, 22, 25, 70
Leibniz identity, 18
Lie algebras, 8, 16, 18–20, 22, 25, 70, 74
Lie-centre, 22, 71
limit, 85, 86, 90, 92–95, 97–98, 102, 104, 105,
112, 114
closed under —s, 67
long exact homology sequence
abelian, 83
Everaert sequence, 2, 77, 81, 85, 86, 90,
99, 112
second variable, 39
simplicial objects, 33, 39
Mal’tsev category, 10, 21, 34
modules, 37, 54, 60
monadic category, 37
monomorphism
normal —, 8, 9, 13, 27
normal to an equiv. rel., 27
Moore complex, 32, 34
object of n-cycles, 32, 57
multiplication, see abelian object
n-fold extension, see higher extension
nil-centre, 22, 70
nilpotent groups, 18, 21, 22, 70
3× 3 Lemma, 11, 71, 103, 115
Noether’s Iso. Theorem, 12, 75, 80, 104
normal epimorphism, 8
normal extension, 21
normal monomorphism, 8, 9, 13, 27
to an equivalence relation, 27
normal subobject, 13, 15, 26, 117
associated —, 27
normalisation (of an equiv. rel.), 27
131
INDEX
normalisation functor, 32
— is exact, 32, 36
normalised chain complex, see Moore com-
plex
object of n-cycles, 32, 57
Ω-groups, 8
P-exact simplicial object, 51
P-resolution, 50, 52, 59
perfect group, 96
perfect object, 95, 96, 114, 114–122
pointed category, 7, 61
pointwise satellite, 85, 85–91
presentation, see projective presentation
presentation of A, 73, 81, 83, 86, 102, 104,
105, 111
prism, 45
projective class, 49
Kan —, 53, 55, 58
projective class of a comonad, 49
projective presentation, 61, 73, 75, 78, 81,
83, 86, 105, 111
projectives, 49, 73, 81, 95
enough —, 49, 49, 74–81, 85–91, 101–
109, 114
regular —, 49, 54
proper chain complex, 16, 32
proper morphism, 9
protomodular category, 8, 61
pullback, 71, 72, 93, 98
— cancellation, 10, 13
— reflects monos, 10
isomorphic kernels, 12, 64, 76
mono between cokernels, 12, 64, 69, 75–
77, 103, 117
pullback-stable
— extensions, 62, 65, 94
— regular epimorphisms, 8
pushout, 57
generalised regular —, 14, 57
isomorphic cokernels, 13, 57
regular —, 13, 19, 64
quotient, see regular quotient
R-modules, 37, 54, 60
reflection, see reflector
reflective subcategory, 17, 66, 67, 72
reflector, 17, 20, 68, 72, 85, 91, 94, 95, 105
identity, 92
zero, 98
regular category, 8, 34
regular epimorphism, 8–11, 13, 60, 63, 64,
67, 122
pullback-stable —s, 8
regular image, see regular quotient
regular projectives, 49, 54
regular pushout, 13, 19, 64
generalised —, 14, 57
regular quotient, 8, 9, 115, 121
closed under —s, 17
relative comonad, 38, 54, 60
relative contractibility, 51, 52
relative Kan property, 49, 53
rings, 8, 54
satellite, 81, 82
composing —s, 88
homology, abelian, 83
homology, semi-abelian, 86
composed, 89
pointwise —, 85, 85–91
Schur multiplier, 96
semi-abelian category, 1, 8
examples, 8
short exact sequence, 63, 77, 83, 103
— of functors, 67–69, 111
Short Five Lemma, 8, 116, 122
simplicial group, 54
simplicial identities, 31
simplicial kernel, 51, 52
simplicial object, 31, 55
augmented —, 33, 37, 39, 50
contractible —, 33, 39
homology, 33
degeneracy operator, 31, 37
face operator, 31, 37
homology, 32
homotopic maps, 52, 58
— have same homology, 58
homotopy equivalent —s, 55
simplicial resolution, 37, 50, 52, 59
simplicial set, 34
contractible —, 50
Kan —, 34, 50
simplicially homotopic, 52, 58
132
INDEX
slice category, 85, 93, 95, 104
Smith central, 27
Snake Lemma, 12, 32
solvable centre, 23
solvable groups, 18, 21, 22, 70
split central extension, 24, 76, 78, 117
split epimorphism, 62, 65
split extension, 95, 97
Stallings-Stammbach sequence, 3, 61, 112,
115, 118
stem cover, 113, 114, 120, 121
stem extension, 96, 113, 114, 120
strongly E-Birkhoff, 66, 69, 74–80, 85–105
subdiagonal, 26, 29
subobject
closed under —s, 17, 66
sum of central extensions, 117
surjection, 60, 96, 105
symmetry of double extension, 64, 65
tensor, 38, 60
terminal object, 90
terminal object of n-cube, 94, 102
theory of G-left derived functors, 39, 42
Tierney-Vogel, 50, 59
Tor groups, 38, 60
trivial extension, 21, 24, 78, 98, 112, 114,
117, 120
higher —, 72
trivialisation, 72, 75
unital category, 10, 24
universal central extension, 95, 96, 114, 121
unnormalised chain complex, 32
Van der Linden, 5, 31, 36, 48, 61, 82
variety of algebras, 37, 55
Vogel, see Tierney-Vogel
weakly universal central extension, 95
zero map, 7
zero-object, 7
zigzag, 56
133