Morita Cohomology
Julian V. S. Holstein
St John’s College
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Declaration
This dissertation is the result of my own work and includes nothing
which is the outcome of work done in collaboration except where specif-
ically indicated in the text. No part of this dissertation has been submit-
ted for any other qualification.
iii
Summary
This work constructs and compares different kinds of categorified coho-
mology of a locally contractible topological space X. Fix a commutative
ring k of characteristic 0 and also denote by k the differential graded cat-
egory with a single object and endomorphisms k. In the Morita model
structure k is weakly equivalent to the category of perfect chain com-
plexes over k.
We define and compute derived global sections of the constant presheaf
k considered as a presheaf of dg-categories with the Morita model struc-
ture. If k is a field this is done by showing there exists a suitable local
model structure on presheaves of dg-categories and explicitly sheafify-
ing constant presheaves. We call this categorified Cˇech cohomology
Morita cohomology and show that it can be computed as a homotopy
limit over a good (hyper)cover of the space X.
We then prove a strictification result for dg-categories and deduce that
under mild assumptions on X Morita cohomology is equivalent to the
category of homotopy locally constant sheaves of k-complexes on X.
We also show categorified Cˇech cohomology is equivalent to a category
of ∞-local systems, which can be interpreted as categorified singular
cohomology. We define this category in terms of the cotensor action of
simplicial sets on the category of dg-categories. We then show ∞-local
systems are equivalent to the category of dg-representations of chains
on the loop space of X and find an explicit method of computation if X
is a CW complex. We conclude with a number of examples.
iv
Contents
1. Introduction 1
2. Morita cohomology 8
2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . 8
2.2. Further properties of dgCat . . . . . . . . . . . . . . . 22
2.3. Cohomology of presheaves of model categories . . . . 27
2.4. Sheafification of constant presheaves . . . . . . . . . . 38
2.5. Morita cohomology over general rings . . . . . . . . . 45
3. Homotopy locally constant sheaves 46
3.1. Strictification and computation of homotopy limits . . 46
3.2. Strictification for dg-categories . . . . . . . . . . . . . 53
3.3. Restriction to perfect complexes . . . . . . . . . . . . 58
3.4. Homotopy locally constant sheaves . . . . . . . . . . . 59
4. Infinity-local systems 65
4.1. Infinity-local systems . . . . . . . . . . . . . . . . . . 65
4.2. Loop space representations . . . . . . . . . . . . . . . 68
4.3. Cellular computations . . . . . . . . . . . . . . . . . . 72
4.4. Finiteness and Hochschild homology . . . . . . . . . . 79
v
5. Computation and Examples 84
5.1. Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2. Other topological spaces . . . . . . . . . . . . . . . . 87
A. Some technical results on dg-categories 91
A.1. A combinatorial model for dg-categories . . . . . . . . 91
A.2. Simplicial resolutions of dg-categories . . . . . . . . . 93
vi
1. Introduction
The aim of this thesis is to construct and compare categorifications of
the cohomology of topological spaces by considering coefficients in the
category of differential graded categories.
We begin with the calculation of RΓMorita(X, k), for k a field, the coho-
mology of a locally contractible topological space X with coefficients in
a constant sheaf. We categorify by considering the constant sheaf k not
a as a sheaf of rings, but as a presheaf of dg-categories with one object,
where we equip dg-categories with the Morita model structure. In this
model structure k ' Chpe, which is fundamental to our construction.
Hence we call this categorified Cˇech cohomology Morita cohomology.
We writeH M(X) for RΓMorita(X, k).
Let X be a locally contractible topological space. The following
characterization follows once we establish a local model structure on
presheaves of dg-categories.
Theorem 2.21. Given a good hypercover {Ui}i∈I of X one can compute
H M(X) ' holimi∈Iop Chpe.
Remark 1.1. To prove existence of the local model structure we show
that dgCatk is left proper if k is a field. This may be of independent
interest.
1
Under some further assumptions on X we also identify Morita
cohomology with a more intuitive category associated to X.
Theorem 3.15. Let X have a bounded locally finite good hypercover.
Then the dg-categoryH M(X) is quasi-equivalent to the dg-category of
homotopy locally constant sheaves of perfect complexes.
Remark 1.2. The homotopy category of the category of homotopy
locally constant sheaves can be considered as the correct derived
category of local systems on X in the sense that it contains the abelian
category of local systems but its Ext-groups are given by cohomology
of X with locally constant coefficients rather than group cohomology of
the fundamental group.
The proof uses strictification to write the homotopy limit as a
category of homotopy cartesian sections of a constant Quillen presheaf.
Homotopy cartesian sections are then identified with homotopy locally
constant sheaves.
Next we define a category of ∞-local systems on a simplicial set K
by the action of simplicial sets on dg-categories, K 7→ ChdgK . This
construction is well-known to give a Quillen adjunction from sSet to
dgCat. For a topological space X one considers Y (X) B ChdgSing* X,
which can be considered as a categorification of singular cohomology.
We prove the following comparison theorem:
Theorem 4.4. The category H M(X) is equivalent to the category of
∞-local systems (Chpe)Sing*(X).
2
Homotopy invariance and a Mayer–Vietoris theorem are easy to
establish for∞-local systems and hence for Morita cohomology.
The category of ∞-local systems is closely related to the loop space of
X, as is shown by the next result:
Theorem 4.8. If X is a pointed and connected topological space the
category H M(X) is furthermore equivalent to the category of fibrant
cofibrant representations in perfect complexes of chains on the based
loop space ΩX.
We then establish a method of computing ChpeC∗(ΩX) for a CW complex
that allows us to calculate Morita cohomology for a number of
examples.
This characterization allows us to compute Hochschild homology of
Morita cohomology in some cases. It follows for example from some
results available in the literature that for a simply connected space
HH∗(H M(X)) ' H∗(L X), where the right hand side is cohomology
of the free loop space.
The results of Chapters 3 to 5 still hold if k is an arbitrary commutative
ring of characteristic 0. We indicate in Section 2.5 how to relate this to
Chapter 2.
3
Relation to other work
Here we collect some references to ideas and results in the literature
which are related to our constructions. This is not meant to be an
exhaustive list.
The analogous statement to Theorem 3.15 for perfect complexes of
coherent sheaves appears for example as Theorem 2.8 in [41] referring
back to [24] and as an assertion in [53].
Carlos Simpson discusses non-abelian cohomology with coefficients
in a stack as an internal hom-space in geometric stacks, for example
in [40]. This construction also appears in work by Pantev, Toën, Vaquié,
Vezzosi [35] who construct interesting additional structures on these
mapping stacks. In particular in the topological situation they mention
the derived stack Map(M,RPerf), where M is a manifold considered as
a constant stack and RPerf is the moduli stack of perfect complexes.
(One can of course use more general topological spaces, the manifold
condition provides extra structure.) One can consider the construction
of ∞-local systems in Chapter 4 as a non-geometric version of this,
which is already somewhat interesting and more tractable then the
mapping stack.
The comparison of homotopy locally constant sheaves and ∞-local
systems is a linear and stable version of results in [47] or [39], where
the corresponding result for presheaves of simplicial sets is proved by
going via the category of fibrations.
4
An A∞-category of ∞-local systems on a simplicial set is constructed
in [7], where the authors go on to prove a Riemann-Hilbert theorem.
Their explicit formulae can be seen to be equivalent to our construction,
see Proposition A.3.
Outline
After briefly recalling some technical results and definitions in 2.1 we
proceed in 2.2 to show that the two model structures on dg-categories
are cellular and left proper. This allows us to define a local model
structure on presheaves of dg-categories and define their cohomology
in 2.3. We then explicitly sheafify the constant presheaf in 2.4 and use
this to define Morita cohomology H M(X) and write down a formula
as a homotopy limit of a constant diagram with fiber Chpe indexed
by the distinct open sets of a hypercover. Chapter 2 closes with some
comments on the situation if k is not a field in 2.5
We explain how strictification allows to compute a homotopy limit as a
category of homotopy cartesian sections in 3.1 and prove a strictification
result for dg-categories in 3.2. We restrict this correspondence to
sections with perfect fibers to obtain a formula for Morita cohomology
in 3.3. We then use this formula in 3.3 to identify H M(X) with the
category of homotopy locally constant sheaves on X.
Chapter 4 takes a different approach to Morita cohomology. We define
a category of ∞-local systems from the cotensor action of simplicial
sets on dg-categories, and show it es equivalent to H M(X) in 4.1. We
5
use this to identify H M(X) with representations of the loop space,
which is another natural generalization of local systems, in 4.2. As a
by-product we recover the well-known equivalence REndC∗(ΩX)(k, k) '
C∗(X, k). Section 4.3 is then concerned with providing an explicit
method for computing the category of ∞-local systems. In 4.4 we
collect some results about finiteness of H M(X) and show how to
compute Hochschild (co)homology in some cases.
Chapter 5 consists of some example computations. Finally, the
appendix provides a combinatorial model category of dg-categories and
a construction of explicit simplicial resolutions in dgCat from which
we deduce an explicit formula for C K .
Acknowledgements
Firstly, I would like to thank my advisor Ian Grojnowski for many
insightful comments and stimulating discussions as well as never-
ending encouragement. Many thanks to Jon Pridham for some very
helpful conversations.
I am grateful to my colleagues, past and present, in the DPMMS who
make it a wonderful place to do mathematics, and to St John’s College
and Christ’s College which have both provided excellent working
environments as well as great places to live.
During this work I have been supported by a PhD studentship from the
Engineering and Physical Sciences Research Council and by a Blyth
fellowship at Christ’s College.
6
For invaluable non-mathematical support over the the course of this
project I am greatly indebted to my family and friends, sine quibus non.
I want to particularly thank my parents as well as Pit, Sarah and Shaul.
7
2. Morita cohomology
2.1. Preliminaries
In this section we fix our notation and conventions and recall some
definitions and preliminary results. Note that Lemma 2.2 does not seem
to be explicitly available in the literature.
2.1.1. Notation and conventions
We assume the reader is familiar with the theory of model categories,
but will try to recall all the less well-known facts about them that we
use.
In any model category we write Q for functorial cofibrant replacement
and R for functorial fibrant replacement. We write mapping spaces
(with values in sSet) in a model categoryM as MapM (X,Y). All other
enriched hom-spaces in a category D will be denoted as HomD (X,Y).
In particular we use this notation for differential graded hom-spaces,
internal hom-spaces and hom-spaces of diagrams enriched over the
target category. It will always be clear from context which category
we enrich in.
8
We will work over the (underived) commutative ground ring k. We
assume characteristic 0 in order to freely use differential graded
constructions.
Remark 2.1. If k is not a field, for example if we we work over k = Z,
then the category of dg-categories is not automatically k-flat, i.e. hom-
spaces might not be cofibrant as chain complexes (see below). This
means some technical results in Chapter 2 are unavailable, see Section
2.5.
Ch = Chk will denote the model category of chain complexes over
the ring k equipped with the projective model structure where fibrations
are the surjections and weak equivalences are the quasi-isomorphisms.
Note that we are using homological grading convention, i.e. the
differential decreases degree. The degree is indicated by a subscript
or the inverse of a superscript, i.e. Ci = C−i.
We write Chpe for the subcategory of perfect complexes in Ch.
Chdg denotes the dg-category whose object are fibrant and cofibrant
objects of Ch. Note that there is a natural identification of the
subcategory of compact objects in Chdg with Chpe. This follows
since an object X in Chdg is compact if HomHo(Chdg)(A,−) commutes
with arbitrary coproducts. Then compact objects are precisely perfect
complexes, i.e. bounded complexes which are level-wise projective. But
perfect complexes are automatically cofibrant in the projective model
structure.
There is a natural smart truncation functor τ≥0 from Ch to Ch≥0, the
category of non-negatively graded chain complexes, which naturally has
9
the projective model structure. The functor τ≥0 is right Quillen with left
adjoint the inclusion functor.
2.1.2. Differential graded categories
Basic references for dg-categories are [29] and [49]. Many technical
details are proven in [44].
Let dgCat denote the category of categories enriched in Ch. Given
D ∈ dgCat we define the homotopy category H0(D) as the category
with the same objects as D and HomH0(D)(A, B) = H0HomD (A, B). If
D is a model category enriched in Ch we define LD as its subcategory
of fibrant cofibrant objects. We say D is a dg-model category if the
two structures are compatible, that is if they satisfy the pushout-product
axiom, see for example the definitions in Section 3.1 of [49]. Then
Ho(D) ' H0(LD), where we take the homotopy category in the sense
of model categories on the left and in the sense of dg-categories on the
right. We mostly consider dg-categories with unbounded hom-spaces,
but there is a natural truncation functor τ≥0 : dgCat → dgCat≥0 that is
just truncation on hom-spaces.
Recall that there are two model structures on dgCat. Firstly there is
the Dwyer–Kan model structure, denoted dgCatDK . Weak equivalences
are quasi-equivalences, i.e. dg-functors that induce weak equivalences
on hom-spaces and are essentially surjective on the homotopy category.
Fibrations are those dg-functors F that are surjective on hom-spaces and
have the property that every homotopy equivalence F(a) → b′ in the
10
codomain of F lifts to a homotopy equivalence a → b with F(b) = b′.
A set of generating cofibrations is given by the following.
• ∅ → k
• S (n)→ D(n) for all n ∈ Z.
Here S (n) is the linearization of the category a
g→ b where g has
degree n and is the only non-identity morphism. D(n) equals S (n)
with additional morphisms k. f in degree n + 1 such that d f = g.
Recall the functor D 7→ D-Mod sending a dg-category to its model
category of modules, i.e. D-Mod is the category of functors D → Ch
and strict natural transformations. This is naturally a cofibrantly
generated model category enriched in Ch whose fibrations and weak
equivalences are given levelwise. We usually consider its subcategory
of fibrant and cofibrant objects, L(D-Mod).
Remark 2.2. The construction of the model category D-Mod follows
Chapter 11 of [23], but there are some changes since we are considering
enriched diagrams. Let I and J denote the generating cofibrations
and generating trivial cofibrations of Ch. The generating (trivial)
cofibrations of D-Mod are then of the form hX ⊗ A → hX ⊗ B for
A → B ∈ I (resp. J), where hX denotes the contravariant Yoneda
embedding. As in Theorem 11.6.1 of [23] we transfer the model
structure from Chdiscrete(D). This works since hX is compact in D-Mod
and so are its tensor products with the domains of I, ensuring condition
(1) of Theorem 11.3.2 holds. For the second condition we have to check
that relative J ⊗ hX-cell complexes are weak equivalences. Pushouts are
constructed levelwise. The generating trivial cofibrations of Ch are of
11
the form 0 → D(n). Since the pushout U ← 0 → D(n) is weakly
equivalent to U we are done.
Note that cofibrations in this model category need not be levelwise
cofibrations, unless hom-spaces in D are cofibrant, in which case the
map Hom(α, β)⊗A→ Hom(α, β)⊗B is a cofibration and the proof goes
through just like in [23]. In fact, if hom-spaces are cofibrant this works
for categories of functors enriched in any monoidal model category V .
Remark 2.3. In order to satisfy the smallness assumption we will always
assume that all our dg-categories are small relative to some larger
universe.
The homotopy category of the model category LDop-Mod is called the
derived category of D and denoted D(D).
Definition. We denote by dgCatMor the category of dg-categories with
the Morita model structure, i.e. the Bousfield localization of dgCatDK
along functors that induce equivalences of the derived categories, see
Chapter 2 of [44].
Fibrant objects in dgCatMor are dg-categories A such that the
homotopy category of A is equivalent (via Yoneda) to the subcategory
of compact objects of D(A ) [29]. We can phrase this as: every
compact object is quasi-representable. An object X ∈ D(A ) is called
compact if HomD(A )(X,−) commutes with arbitrary coproducts. We
denote by ()pe the subcategory of compact objects. Morita fibrant dg-
categories are also called triangulated since their homotopy category is
an (idempotent complete) triangulated category.
12
With these definitions D 7→ L(Dop-Mod)pe, often denoted the
triangulated hull, is a fibrant replacement, for example k 7→ Chpe.
The category dgCat is symmetric monoidal with tensor product
D ⊗ E given as follows. The objects are ObD × ObE and
HomD⊗E ((D, E), (D
′, E′)) B HomD (D,D
′) ⊗ HomE (E, E′). The unit is
the one object category k, which is cofibrant in either model structure.
While dgCat is not a monoidal model category there is a derived
internal Hom space and the mapping spaces in dgCatMor can be
computed as follows [48]: Let RHom(C ,D) be the dg-category of right-
quasirepresentable C ⊗ Dop-modules, i.e. functors F : C ⊗Dop → Ch
such that for any c ∈ C we have that F(c,−) is isomorphic in
the homotopy category to a representable object in Dop-Mod and
moreover cofibrant. Then RHom is right adjoint to the derived tensor
product ⊗L. Moreover Map(C ,D) is weakly equivalent to the nerve of
the subcategory of quasi-equivalences in RHom(C ,D). We will quote
further properties of this construction as needed.
We will need the following lemma relating the two model structures.
The definition of homotopy limits will be recalled in Section 2.1.4
Lemma 2.1. Fibrant replacement as a functor from dgCatMor to
dgCatDK preserves homotopy limits.
Proof. We know that dgCatMor is a left Bousfield localization of
dgCatDK , hence the identity is a right Quillen adjunction and its derived
functor, given by fibrant replacement, preserves homotopy limits.
13
This means we can compute homotopy limits in dgCatMor by com-
puting the homotopy limit of a levelwise Morita-fibrant replacement in
dgCatDK .
We will abuse notation and write R for the dg-algebra by the same name
as well as for the 1-object dg-category with endomorphism space R
concentrated in degree 0.
Recall that there is a model structure on differential graded algebras
over k with unbounded underlying chain complexes, which can be
considered as the subcategory of one-object-categories in dgCatDK .
Note that all objects of dgCatDK are fibrant and hence dgCatDK is right
proper (i.e. pullbacks along fibrations preserve weak equivalences),
while dgCatMor as a left Bousfield localization need not be and in fact
is not, cf. Example 4.10 of [46]. We will consider the question of left
properness (i.e. whether the pushout along a cofibration preserves weak
equivalences) in 2.5.
Recall also the category sModCatk of categories enriched over
simplicial k-modules and the natural Dold–Kan or Dold–Puppe functor
DK : dgCat≥0 → sModCat that is defined hom-wise. DK and its
left adjoint, normalization, form a Quillen equivalence between non-
negatively graded dg-categories and sModCat. For details see section
2.2 of [41] or [45].
Remark 2.4. While we are working with differential graded categories
we are facing some technical difficulties for lack of good internal hom-
spaces. It would be interesting to know if another model of stable linear
(∞, 1)-categories could simplify our treatment.
14
2.1.3. Model V -categories and simplicial resolutions
Model categories are naturally models for∞-categories and in fact have
a notion of mapping spaces. Even if a model category is not enriched
in sSet one can define mapping spaces in Ho(sSet). One way to do this
is by defining simplicial resolutions, which we will make extended use
of.
Let ∆ be the simplex category and consider the constant diagram functor
c : M → M ∆op . Then a simplicial resolution M• for M ∈ M is a
fibrant replacement for cM in the Reedy model structure onM ∆
op
. (For
a definition of the Reedy model structure see for example Chapter 15
of [23].)
For example, this construction allows us to compute mapping spaces: If
cB→ B˜ is a simplicial framing inM ∆op and QA a cofibrant replacement
in M then Map(A, B) ' Hom•(QA, B˜) ' R(Hom•(−, c−)), where the
right-hand side uses the bifunctor Hom• : M op ×M ∆op → Set∆op that is
defined levelwise. The dual notion is a cosimplicial resolution.
Recall V is a symmetric monoidal model category if it is both
symmetric monoidal and a model category and the structures are
compatible, to be precise they satisfy the pushout-product axiom, see
Definition 4.2.1 in [25]. This means in particular that tensor and internal
Hom give rise to Quillen functors. We then call the adjunction of two
variables satisfying the pushout-product axiom a Quillen adjunction of
two variables.
15
Similarly a model V -category M is a model category M that is
tensored, cotensored and enriched over V such that the pushout product
axiom holds. We call a model Ch-category a dg-model category.
For example a model sSet-category, better known as a simplicial
model category, M consists of the data (M ,Map,⊗,map) where the
enrichment Map : M op ×M → sSet, the cotensor (or power) map :
sSetop ×M → M and the tensor ⊗ : sSet ×M → M satisfy the
obvious adjointness properties (in other words, they form an adjunction
of two variables). The pushout-product axiom says that the natural map
fg : A ⊗ L qA⊗K B ⊗ K → B ⊗ L is a cofibration if f and g are and is
acyclic if f or g is moreover acyclic.
While not every model category is simplicial, every homotopy category
of a model category is enriched, tensored and cotensored in Ho(sSet).
In fact, M can be turned into a simplicial category in the sense that
there is an enrichment Map and there are a tensor and cotensor which
can be constructed from the simplicial and cosimplicial resolutions. Let
a cosimplicial resolution A• ∈ M ∆ and a simplicial set K be given.
Consider ∆K, the category of simplices of K, with the natural map
u : ∆K → ∆ sending ∆[n] 7→ K to [n]. We define A• ⊗ K = colim∆K An
to be the image of A• under colim ◦ u∗ : C ∆ → C ∆K → C . Similarly
there is AK which is the image of the simplicial resolution A• ∈ M ∆op
under lim ◦ v∗, where v : ∆Kop → ∆op. This can also be written as
AK = limn(
∏
Kn An).
IfM is a simplicial category one can use (RA)∆
n
for An and (QA) ⊗ ∆n
for An.
16
Remark 2.5. Note that AK can also be written as a homotopy limit,
holim∆Kop An. (The definition of a homotopy limit is recalled below.)
This follows for example from Theorem 19.9.1 of [23], the conditions
are satisfied by Propositions 15.10.4 and 16.3.12.
The functor (A,K) 7→ AK is adjoint to the mapping space construction
A, B 7→ Hom(B, A•) ∈ sSet. Similarly (B,K) 7→ B ⊗ K is adjoint
to the mapping space construction A, B 7→ Hom(B•, A) ∈ sSet, see
Theorem 16.4.2 in [23]. Hence on the level of homotopy categories
the two bifunctors together with Map give rise to an adjunction of two
variables. This is of course not a Quillen adjunction, but it is sensitive
enough to the model structure to allow for certain derived functors. We
will quote further results about this construction as needed.
2.1.4. A very short introduction to homotopy limits
Ordinary limits in a model category are not very well behaved, in
particular they are not invariant under weak equivalence. A much better
notion is provided by homotopy limits.
Let I be a small category, M a model category and C : I → M
a diagram. On the category of diagrams M I we can often define
the injective model structure with levelwise weak equivalences and
cofibrations. Limits are right adjoints of the constant diagram functor
and with the injective model category structure on diagrams they
become Quillen adjoints. Then homotopy limits are just their right
derived functors.
17
in general, the injective model structure only exists if M is combina-
torial (or if the index category is direct) and the dual projective model
structure still needs M to be cofibrantly generated (or the index cate-
gory to be inverse). But even if we only have M I as a category with
weak equivalences, i.e. as a homotopical category, we can still define
derived functors, see for example [15]. Simply put, the homotopy limit
is the right adjoint to the constant functor Ho(M ) → Ho(M I) on the
level of homotopy categories.
Note that the derived functors of right Quillen functors preserve
homotopy limits, and so do all Quillen equivalences. (The reason for the
latter is that Quillen equivalences induce equivalences of the homotopy
categories of diagram categories.)
To compute homotopy limits explicitly there is a number of formulae
available. Let us assume Ci is levelwise fibrant, and replace fibrantly
if that is not the case. (Sometimes taking the homotopy limit of a
levelwise fibrant replacement is called the corrected homotopy limit.)
Then we will use the following, cf. e.g. Definition 19.1.4 in [23].
holim
i
Ci = eq
∏
i∈I
CN(I↓i)i ⇒
∏
h→ j
CN(I↓h)j
Here we use a simplicial cotensor defined using a simplicial frame (Ci)•
for Ci. Since Ci is assumed fibrant this is just a simplicial resolution
such that Ci → (Ci)0 is an isomorphism, and since the construction is
invariant under weak equivalences between fibrant objects we can take
any simplicial resolution.
18
By contrast, for a much more computable example, let us consider a
homotopy pullback. It is provided by replacing the target fibrantly and
both maps by fibrations before taking the limit, i.e.
holim(A→ B← C) ' lim(A˜→ B˜← C˜)
where A˜ → B˜ and C˜ → B˜ are fibrations and A˜ ' A etc. If the model
category is right proper, i.e. pushout along fibrations preserves weak
equivalences, it suffices to replace one map by a fibration.
Similarly we construct homotopy ends of bifunctors. Recall that an end
is a particular kind of limit. Let α(I) denote the twisted arrow category
of I: Objects are arrows, f : i → j, and morphisms are opposites of
factorizations, i.e. ( f : i → j) ⇒ (g : i′ → j′) consists of maps i′ → i
and j → j′ such that their obvious composition with f equals g. Then
there are natural maps s and t (for source and target) from α(I) to Iop
and I respectively. For a bifunctor F : Iop × I → C one defines the end∫
i
F(i, i) to be limα(I)(s × t)∗F. Then the homotopy end is:∫ h
i
F(i, i) B holim
α(I)
(s × t)∗F
Details on this view on homotopy ends can be found (dually) in [27].
The canonical example for an end is that natural transformations from
F to G can be computed as
∫
A
Hom(FA,GA). A similar example of
the use of homotopy ends is provided by the computation of mapping
spaces in the diagram category of a model category Map(A•, B•) '∫ h
i
Map(Ai, Bi). The case of simplicial sets is dealt with in [17].
19
In general, we have the following lemma. Assume M I exists with the
injective model structure and let Q and R denote cofibrant and fibrant
replacement in this model category.
Lemma 2.2. Consider a right Quillen functor H : M op ×M → V .
Then there is a natural Quillen functor (F,G) → ∫
i
H(Fi,Gi) from
(M I)op ×M I to V whose derived functor is
(F,G) 7→
∫
i
H(QFi,RGi)
which is weakly equivalent to
(F,G) 7→
∫ h
i
RH(Fi,Gi)
Proof. The V -structure exists by standard results in [30]. We have a
model V -category by Lemma 2.3 below. Hence the derived functor is
(F,G) 7→ ∫
i
H(QFi,RGi).
On the other hand
∫
i
H(Fi,Gi) is the composition of levelwise hom-
spaces with the limit,
lim ◦ Hα(I) ◦ (s × t)∗ : (M I)op ×M I → (M op ×M )α(I) → V α(I) → V
But then the derived functor is the composition of derived functors,∫ h
i
RH(Fi,Gi).
This is a little subtle, since we do not want to fibrantly replace at the
level of diagram categories. However, levelwise RH from (M op)I ×M I
to V α(I) is a derived functor. This is the case since levelwise fibrant
replacement gives a right deformation retract in the sense of 40.1 in [15]
20
since (s× t)∗ preserves all weak equivalences and levelwise H preserves
weak equivalences between levelwise fibrant objects.
Remark 2.6. A slight modification of the lemma implies the formula for
mapping spaces. We just have to replace H by Hom• : M op ×M ∆op →
sSet and adjust the proof accordingly.
Lemma 2.3. M I is a model V -category ifM is.
Proof. Given cofibrations f : V → W in V and g : A → B in M I we
have to show that whenever f and g are cofibrations then so is
fg : (V ⊗ B) qV⊗A (W ⊗ A)→ W ⊗ B
and fg is an acyclic cofibration if f or g is.
Cofibrations and weak equivalences inM I are defined levelwise so it is
enough to check ( fg)i. Colimits in diagram categories are also defined
levelwise, so it is enough to check f(gi). But by assumption M is a
model V -category.
These results generalise verbatim to categories of presections which will
be defined in Section 3.1.
Homotopy colimits etc. can be defined entirely dually.
21
2.2. Further properties of dgCat
In this section we will show that dgCatMor is cellular and, if k is a field,
left proper. This will be used to localize diagrams of dg-categories. It
may also be of independent interest.
Proposition 2.4. The categories dgCatDK and dgCatMor and their
small diagram categories are cellular.
Proof. Recall that a model category is cellular if it is cofibrantly gener-
ated with generating cofibrations I and generating trivial cofibrations J
such that the domains and codomains of the elements of I are compact,
the domains of the elements of J are small relative to I and the cofibra-
tions are effective monomorphisms. See Chapter 10 of [23] for precise
definitions.
Left Bousfield localization preserves being cellular, and so does taking
the category of diagrams indexed by a small category I with the
projective model structure, see Theorem 4.1.1 and Proposition 12.1.5
of [23]. So it is enough to show dgCatDK is cellular.
The domains and codomains of elements of I are categories with at
most two objects and perfect hom-spaces, so maps from these objects
to relative I-complexes factor through small subcomplexes. So domains
and codomains of I are compact.
Similarly the domains of the elements of J have two objects and perfect
hom-spaces. Hence taking maps from a domain of J commutes with
filtered colimits. So domains of J are small relative to I.
22
We are left to check that relative I-cell complexes, i.e. transfinite
compositions of pushouts of generating cofibrations, are effective
monomorphisms, i.e. any relative I-cell complex f : X → Y is the
equalizer of Y ⇒ Y qX Y . Note that we form the pushout along a
generating cofibration by attaching maps freely. If we form C ′ and C ′′
from C by attaching maps freely then the equalizer will have the same
objects and the hom-spaces are given by considering morphisms of the
pushout that are in the image of both C ′ and C ′′. But these are precisely
the hom-spaces of C .
Proposition 2.5. If k is a field the categories dgCatDK and dgCatMor
and their small diagram categories are left proper.1
Proof. Left Bousfield localization preserves left properness, see Propo-
sition 3.4.4 of [23], and so does taking the category of diagrams indexed
by a small category I (with the injective or projective model structure)
since pushout and pullback are constructed levelwise. So it is enough
to show dgCatDK is left proper.
The main work is showing that pushout along the generating cofibra-
tions preserves quasi-equivalences.
To see this suffices note firstly that transfinite compositions are just
filtered colimits, and filtered colimits preserve quasi-equivalences
as follows: A filtered colimit of categories can be computed set-
theoretically on objects and morphisms. Now filtered colimits preserve
weak equivalences of simplicial sets (since S n is compact) and hence
1Thanks to Jon Pridham for helpful discussions about this result.
23
of mapping spaces. They also preserve the homotopy category since
a filtered colimit of equivalences of categories is an equivalence of
categories and taking the homotopy category commutes with filtered
colimits.
Secondly, if pushout along some map preserves weak equivalences then
so does pushout along a retract by functoriality of colimits. Since
all cofibrations are retracts of transfinite compositions of generating
cofibrations, it does indeed suffice to check generating cofibrations.
It is clear that pushout along ∅ → k preserves quasi-equivalences.
So consider the generating cofibration S (n) → D(n) with a map
j : S (n) → C and a quasi-equivalence F : C → E . Let the objects of
S (n) be denoted a, b and the non-identity morphism g. Then in forming
the pushforward we adjoin a new map f with d f = j(g). We call the
resulting category C ′. Then let E ′ be the pushout of S (n) → D(n)
along F ◦ j.
The pushout along j has the same objects as C . The morphism space
is obtained by collecting maps from C to D, graded by how often
they factor through j(a) → j(b). Write C (A, B) etc. for the enriched
hom-spaces HomC (A, B) etc. Then the hom-spaces in C
′ are given as
follows:
C ′(C,D) = Tot⊕
(
C (C,D)⊕ (C ( j(b),D)⊗ k. f ⊗T ⊗C (C, j(a)))) (2.1)
Here T =
∑
n(C ( j(b), j(a)) ⊗ k. f )⊗n and we introduce a horizontal
degree n with C (C,D) in degree −1. The right hand side has a vertical
differential given by the internal differential and a horizontal differential
24
given by f 7→ j(g) ∈ Hom( j(b), j(a)) composed with the necessary
compositions.
If the functor F is not the identity on objects from C to E we factor
F = Q ◦ H : C → D → E where D has as objects the objects of C but
HomD (A, B) = HomE (FA, FB). Then H is identity on objects and Q is
an isomorphism on hom-spaces. We form the pushforward and obtain
the factorization F′ = Q′ ◦ H′ through D ′.
So it suffices to prove the following two lemmas.
Lemma 2.6. Q′ defined as above is a quasi-isomorphism if Q is.
Proof. Q′ is quasi-essentially surjective if Q is since both D → D ′ and
E → E ′ are essentially surjective as pushout along j does not change
the set of objects.
We filter the formula 2.1 for hom-spaces in E ′ by columns, i.e. by
n. This filtration is bounded below and exhaustive for the direct sum
total complex and hence the spectral sequence converges. Now we
use the fact that Q induces isomorphisms on hom-spaces to obtain
an isomorphism of spectral sequences, which in turn induces an
isomorphism D ′(C,D) E ′(QC,QD), so Q′ gives weak equivalences
on mapping spaces.
Note that since S (n) maps to E via D all the hom-spaces involved in
computing E ′(QC,QD) are indeed images of hom-spaces in D .
Lemma 2.7. H′ defined as above is a quasi-isomorphism if H is.
25
Proof. H′ is quasi-essentially surjective if H is for the same reason that
Q′ is.
To consider the effect of H′ on mapping spaces proceed as in the
previous lemma.
Now H only induces weak equivalences on hom-spaces, but we know
all hom-spaces are flat over k as k is a field. Hence the tensor product
in 2.1 preserves weak equivalences. So the same spectral sequence
argument applies.
Remark 2.7. Note that Dwyer and Kan prove left properness for
simplicial categories on a fixed set of objects in [16].
Remark 2.8. If we have some non-flat hom-spaces then things go wrong
even in the subcategory of dg-algebras. Consider Example 2.11 in [37].
In that paper the existence of a proper model for simplicial k-algebras is
proven. A similar result for dg-categories may or may not be true, but
is certainly beyond the scope of this work
Hence the DK-model category of dg-categories is only left proper if all
dg-categories are k-flat, i.e. if and only if k has flat dimension 0, which
is the case if and only if k has no nilpotents and Krull dimension 0. In
practice we may as well assume k is a field. See Section 2.5 for how to
interpret some of the results in the remainder of this chapter for more
general k.
26
2.3. Cohomology of presheaves of model categories
Now we will define what we mean by cohomology of a sheaf with
coefficients in a model category.
Let us assumeM is a cellular and left proper model category. The case
we are interested in isM = dgCatMor.
We will consider the category M J of presheaves on a category Jop
with values in a model category M . We will denote by M Jpro j the
projective model structure on M J with levelwise weak equivalences
and fibrations, and whose cofibrations are defined by the lifting
property. IfM is cofibrantly generated this is well-known to be a model
structure, which is cellular and left proper ifM J is.
We are interested in enriching the model category M Jpro j. Let us start
by recalling the case where the construction is straightforward. Let
V be a monoidal model category and assume that it has a cofibrant
unit. V = sSet,Ch are examples. Then if M is a model V -category,
then so is M J. In particular if M is monoidal then M J is a model
M -category. We can write Hom for the enriched hom-spaces, and
the functor Hom : (M J)op × M J → M is right Quillen and there
is a pleasant derived functor RHom obtained by fibrant and cofibrant
replacement, see Lemma 2.2
If M is monoidal and a model category, but not a monoidal model
category, then we can still construct an M -enrichment of M Jpro j
as a plain category, which will of course not be a model category
enrichment. We define HomMJ (A, B) =
∫
j
Hom(A( j), B( j)), see [30].
27
Note that this enrichment is not in general derivable, i.e. weak
equivalences between cofibrant and fibrant pairs of objects do not
necessarily go to weak equivalences. So defining a suitable substitute
for RHom takes some care, see the proof of Lemma 2.9.
We have to consider this case since our example of interest is M =
dgCat, which a symmetric monoidal category and a model category,
see [49], but not a symmetric monoidal model category. (The tensor
product of two cofibrant objects need not be cofibrant.)
Now fix a locally contractible topological space X, for example a
CW complex, and consider presheaves on Op(X). We consider the
Grothendieck topology induced by the usual topology on X and write
the site as (SetOp(X)
op
, τ). In other words τ is just the collection of
maps represented by open covers. (We will not use any more general
Grothendieck topologies or sites.) We let J = Op(X)op. Our aim is to
localize presheaves on Op(X) with respect to covers in τ.
Recall that a left Bousfield localization of a model category N at
a set of maps S is a left Quillen functor N → NS that is initial
among left Quillen functors sending the elements of S to isomorphisms
in the homotopy category. We need to know that left Bousfield
localizations of M I exist. There are two general existence results: If
M is combinatorial and left proper or if M is cellular and left proper.
We have shown that diagrams in dgCat satisfy the latter condition in
Section 2.2.
Remark 2.9. In the appendix we prove that there is an equivalent
subcategory of dgCatDK that is combinatorial, see Section A.1. Then
28
the existence of Bousfield localizations follows from Jeff Smith’s
theorem, which is for example proven as Theorem 2.11 in [1].
Lemma 2.8. Assume N is a cellular and left proper model category
and let S be a set of maps. Then NS exists. The cofibrations are equal
to projective cofibrations, weak equivalences between are S -local weak
equivalences and fibrant objects are S -local objects.
Proof. This is Theorem 4.1.1 of [23].
Recall for future reference that an object P is S -local if it is fibrant inN
and every f : A → B ∈ S induces a weak equivalence MapN (B, P) '
MapN (A, P). A map g : C → D is an S -local weak equivalence if
it induces a weak equivalence MapN (D, P) ' MapN (C, P) for every
S -local P.
Given a set N we write N ·M B qN M ∈M for the tensor over Set and
extend this notation to presheaves.
Definition. LetM Jτ B (M Jpro j)Hτ denote the left Bousfield localization
ofM Jpro j with respect to
Hτ = {S · 1M → hW · 1M | S → hW ∈ τ}
Here h− denotes the covariant Yoneda embedding X 7→ Hom(−, X).
We have assumed M and hence M J is cellular and left proper. Since
Hτ is a set the localizationM Jτ exists.
29
We have now localized with respect to Cˇech covers. We are interested
in the local model structure which is obtained by localizing at all
hypercovers.
Remark 2.10. By way of motivation see [12] for the reasons that
localizing at hypercovers gives the local model structure on simplicial
presheaves, i.e. weak equivalences are precisely stalk-wise weak
equivalences.
Definition. A hypercover of an open set W ⊂ X is a simplicial presheaf
U∗ on the topological space W such that:
1. For all n ≥ 0 the sheaf Un is isomorphic to a disjoint union of a
small family of presheaves representable by open subsets of W.
We can write Un = qi∈InhU(i)n for a set In where the U
(i)
n ⊂ W are
open.
2. The map U0 → ∗ lives in τ, i.e. the U (i)0 form an open cover of W.
3. For every n ≥ 0 the map Un+1 → (cosknU∗)n+1 lives in τ. Here
(cosknU)n+1 = MWn U is the n-th matching object computed in
simplicial presheaves over W.
Intuitively, the spaces occurring in U1 form a cover for the intersections
of the U (i)0 , the spaces in U2 form a cover for the triple intersections of
the U (i)1 etc. To every Cˇech cover one naturally associates a hypercover
in which all Un+1 → (cosknU∗)n+1 are isomorphisms.
Note that despite the notation Un is not an open set but a presheaf on
open sets that is a coproduct of representables.
30
We denote by I = ∪In the category indexing the representables making
up the hypercover.
Associated to any hypercover of a topological space is the simplicial
space n 7→ qi∈InU in which is also sometimes called a hypercover.
Hypercovers are naturally simplicial presheaves. We work with
presheaves with values in a more general model category. The obvious
way to associate to a simplicial object in a model category a plain object
is to take the homotopy colimit.
Definition. Let the set of hypercovers inM J be defined as
Hˇτ = {hocolim
I
(U∗ · 1M )→ hW · 1M | U∗ → hW a hypercover}
where we take the levelwise tensor and the homotopy colimit in M J
with the projective model structure. Since disjoint union commutes
with cofibrant replacement we could equivalently take the limit of Un
over ∆op, the opposite of the simplex category.
Remark 2.11. Note that the homotopy colimit does not change if instead
we use the localised model structureM Jτ . Left Bousfield localization is
left Quillen and hence preserves homotopy colimits.
Definition. Let the left Bousfield localization ofM Jτ at the hypercovers
Hˇτ be denoted byM Jτˇ and call it the local model structure.
The localization exists just as before. The fibrant objects are the Hˇτ-
local objects ofM J.
Note that Hom(hW ,F ) ' F (W) if the model structure on M J is
enriched over M . So we sometimes write hypercovers as if they
31
are open sets. For example given a hypercover U∗ and a presheaf
F ∈M J we writeF (Un) for Hom(Un,F ) etc. In particularF (Un) =
F (qiU (i)n ) B ∏iF (U (i)n ).
Definition. We call a presheaf F a sheaf (sometimes called hyper-
sheaf ) if it satisfies
F (W) ' holim
Iop
F (U∗) for every hypercover U∗ of every open W ⊂ X
(2.2)
The limit is over Iop = ∪In; we could write it holimn holimi∈In F (U (i)n )
which can be considered as holimn∈∆F (Un) using the convention
above. This condition is also called descent with respect to hypercovers.
Remark 2.12. If we give, say, the category of abelian groups the trivial
model structure (only identities are weak equivalences and all maps are
fibrations and cofibrations) we recover the usual notion of sheaf.
Note that in general being a hypersheaf is a stronger condition than
being a sheaf of underived objects.
For the next Lemma we needM to have a certain homotopy enrichment
over itself. For simplicity we specialise toM = dgCatMor.
Lemma 2.9. Levelwise fibrant sheaves are fibrant in the above model
structure.
Proof. We need to show that for a levelwise fibrant presheaf F the
sheaf condition on F implies that F is Hˇτ-local, i.e. that whenever
: hocolim(U∗ ·1)→ hW ·1 is in Hˇτ we have Map(hocolim(U∗ ·1),F ) '
32
Map(hW · 1,F ). We will show that both sides are weakly equivalent to
MapdgCatMor (1,F (W)).
We need a suitable derived hom-space between sheaves of dg-
categories with values in dg-categories. We define RHom′(A•, B•) B∫ h
V
RHom(AV , BV), where RHom is Toën’s internal derived Hom of dg-
categories.
First note that
RHom′(hW · 1,F ) '
∫ h
V⊂W
RHom(1,F (V)) ' holim
V⊂W
F (V)
The first weak equivalence holds since hW(V) is just the indicator
function for V ⊂ W and the second since the homotopy end over a
bifunctor that is constant in the first variable degenerates to a homotopy
limit, by comparing the diagrams. Then we observe holimV⊂W F (V) '
F (W) ifF satisfies the sheaf condition.
We claim that this implies Map(hW · 1,F ) ' Map(1,F (W)). Note that
in dgCat we have Map(A, B) ' Map(1,RHom(A, B)), see Corollary 6.4
of [48]. Moreover the mapping space in diagram categories is given by
a homotopy end, see Lemma 2.2.
Putting these together we see Map(A•, B•) =
∫ h
V
Map(1,RHom(AV , BV)).
Then the claim follows since Map(1,−) commutes with homotopy lim-
its and hence homotopy ends.
Similarly we have
Map(hocolim
i
(Ui · 1),F ) ' holim
i
Map(1,RHom′(Ui · 1,F ))
' Map(1, holim
i
holim
V⊂Ui
F (V))
33
which is Map(1,F (W)) again by applying the sheaf condition twice.
Remark 2.13. The theory of enriched Bousfield localizations from [1]
says that in the right setting M Jτˇ is an enriched model category and
fibrant objects are precisely levelwise fibrant sheaves. However, this
theory requires that we work with a category M that is tractable,
left proper and a symmetric monoidal model category with cofibrant
unit. The characterization of fibrant objects in particular depends on
the enriched hom-space being a Quillen bifunctor. While dgCatMor is
left proper by Proposition 2.5 and equivalent to a combinatorial and
tractable subcategory by Section A.1, it is well-known dgCatMor is not
symmetric monoidal. Tabuada’s equivalent category Lp of localizing
pairs has a derivable internal Hom object, but is not a monoidal model
category either. In fact, tensor product with a cofibrant object is not left
Quillen. Recall the dg-categoryS (0) that is the linearization of a→ b.
The exampleS (0) ⊗S (0) in dgCat gives rise to
(∅ ⊂ S (0)) ⊗ (∅ ⊂ S (0)) ' (∅ ⊂ S (0) ⊗S (0))
which is again a tensor product of cofibrant objects that is not cofibrant.
Then of course Hom(S (0),−) cannot be Quillen either.
Lemma 2.10. Let M = dgCat. Assume that for two presheaves F
andF ′ there is a hypercover V∗ on whichF andF ′ agree and which
restricts to a hypercover of W for every open W. Then F and F ′ are
weakly equivalent inM Jτˇ .
34
Proof. We need to show that there is a Hˇτ-local equivalence between
F andF ′, i.e. MapM J (F ,G ) ' MapM J (F ′,G ) for any fibrant G.
Specifically, we consider sets V in the hypercover of agreement con-
tained in W. Then we know Map(F (V),G (V)) ' Map(F ′(V),G (V)).
To compute Map(F ,G ) =
∫ h
W
Map(F (W),G (W)) note that the homo-
topy end can be computed as follows:∫ h
W
Map(F (W),G (W)) '
∫
Hom((Q•F )(W),RG (W))
Here we use fibrant replacement and a cosimplicial frame in M J.
But now holimV G (V) = limV RG (V) by fibrancy of the diagram
RG . So it suffices to consider
∫
W
Hom(Q•F(W),RG (W)) where
RG (W) = limV⊂W RG (V)). But an end is just given by the collection
of all compatible maps, and every map from QiF (W) to RG (W) is
determined by the maps from QiF (W) to RG (V), which factor through
QiF (V). So the end over the V is the same as the end over all W and
Map(F ,G ) '
∫
V
Hom(Q•F (V),RG (V))
'
∫
V
Hom(Q•F ′(V),RG (V)) ' Map(F ′,G )
This completes the proof.
Remark 2.14. If M is a symmetric monoidal model category then by
Remark 2.13 fibrant objects are precisely levelwise fibrant sheaves and
are again determined on a hypercover and Lemma 2.10 holds again.
To compute cohomology we need to compute the derived functor of
global sections. First we need to know that pushforward is right Quillen.
35
Lemma 2.11. Consider a map r : C → D of diagrams and a model
category M. Then there is a Quillen adjunction r! : MCpro j M
D
pro j : r
∗.
Proof. We define r∗ by precomposition. Then r! exists as a Kan
extension. Clearly r∗ preserves levelwise weak equivalences and
fibrations.
Lemma 2.12. Given any map r : (C, τ)→ (D, σ) that preserves covers
and hypercovers we get a Quillen adjunction r! : MCτˇ M
D
σˇ : r
∗. The
same adjunction works if we only localize with respect to Cˇech covers.
Proof. To prove the result for the localization with respect to covers we
use the universal property of localization applied to the map MC →
MD → MDσ which is left Quillen and sends hypercovers to weak
equivalences and hence must factor through MC → MCτ in the category
of left Quillen functors, giving rise to r! ` r∗.
To prove the result for the localization at hypercovers we repeat the
same argument for MCτ → MCτˇ etc.
The arguments in the proofs of these two lemmas are Propositions 1.22
and 3.37 in [1].
Consider now locally contractible topological spaces X and Y with sites
of open sets (Op(X), τ) and (Op(Y), σ). Given a map f : X → Y
consider f −1 : (Op(X), τ) → (Op(Y), σ). Then f∗ B ( f −1)∗ and by
the above it is a right Quillen functor. (This is the usual definition of
pushforward: f∗F (U) B F ( f −1u).)
As usual we write Γ or Γ(X,−) for (piX)∗ where piX : X → ∗.
36
Definition. Let C be a presheaf with values in a model categoryM and
let C # be a fibrant replacement for C in the local model category M Jτˇ
defined above. Then we define global sections as
RΓ(X,C ) = C #(X)
In Section 2.4 we will compute C # if C is constant.
Since a sheaf satisfies F (X) = holimiF (Ui) for some cover {Ui} we
can also think of global section as a suitable homotopy limit. A concise
formulation of this is Theorem 2.21.
Definition. Consider the presheaf k that is constant with value k ∈
dgCat and let k# be a fibrant replacement for k. Then we define Morita
cohomology as
RΓMorita(X, k) B RΓ(X,Chpe) = k#(X)
in Ho(dgCatMor). We writeH M(X) B RΓMorita(X, k).
We will also consider the version with unbounded fibers, RΓMorita(X,Ch).
Remark 2.15. As usual RΓ(∅, k) ' 0, the terminal object of dgCat.
Remark 2.16. The term cohomology is slightly misleading as our
construction corresponds to the underlying complex and not the
cohomology groups. The closest analogue to taking cohomology is
probably semi-orthogonal decomposition, see for example [8].
37
2.4. Sheafification of constant presheaves
Our aim now is to compute a sheafification of the constant presheaf with
values in a model category.
We assume X is a locally contractible topological space. Fix a model
category M that is cellular and left proper and that is moreover
homotopy enriched over itself and has a cofibrant unit. We will also
need that the derived internal hom-space commutes with homotopy
colimits.
The example we care about is M = dgCatMor. The fact that
holim RHom(Ai, B) ' RHom(hocolim Ai, B) in dgCat follows from
Corollary 6.5 of [48]. The one object dg-category k is a cofibrant unit.
We write P for the constant presheaf with fiber P ∈M .
First we will need two lemmas about comparing homotopy limits.
Given a functor ι : I → J, recall the natural map e j : ( j ↓ ι) → J from
the undercategory, sending (i, j→ ι(i)) to ι(i).
Lemma 2.13. Let ι : I → J be functor between small categories
such that for every j ∈ J the overcategory (ι ↓ j) is nonempty with
a contractible nerve and let X : J → M be a diagram. Then the map
holimJ X → holimI ι∗X is a weak equivalence.
Lemma 2.14. Let ι : I → J be a functor between small categories and
let X : J → M a diagram with values in a model category. Suppose
that the composition
X j → lim
( j↓ι)
e∗j(X)→ holim( j↓ι) e
∗
j(X)
38
is a weak equivalence for every j. Then the natural map holimJ X →
holimI ι∗X is a weak equivalence.
Proofs . For topological spaces these are Theorems 6.12 and 6.14 of
[11] and the proofs (in section 9.6) do not depend on the choice of model
category.
We will also rely on the following results of [13]. The first statement
is Theorem 1.3 and the second is a corollary of Proposition 4.6 as any
basis is a complete open cover.
Proposition 2.15. Consider a hypercover U∗ → X of a topological
space as a simplicial space. Then the maps hocolim U∗ → |U∗| → X
are weak equivalences in Top.
The colimit here is over the category ∆op, but recall that hocolim∆op Un '
hocolimI U in.
Proposition 2.16. Consider a basis U of a topological space X as
a simplicial space. Then the map hocolimU∈U U → X is a weak
equivalence in Top.
Let X be locally contractible. Then we can define the (nonempty) set
{Us}s∈S of all bases of contractible sets for X.
Definition. Fix a basis of contractible sets Us for X. Let P be a constant
presheaf with fiber P ∈M and define a presheafL sP by
L sP (U) = holimV⊂U,V∈Us
RP(V)
39
where P → RP is a fibrant replacement inM . Denote the natural map
by λ : P → L sP . The restriction maps are induced by inclusion of
diagrams.
This construction proceeds via constructing rather large limits, so even
the value ofL s on a contractible set is hard to make explicit.
We will mainly be interested inL sk ' L sChpe .
The following lemma is the first step towards showing that our
construction does indeed give a sheaf.
Lemma 2.17. Consider a constant presheaf P with fibrant fiber P ∈M
on Op(X). Then on any contractible set U ⊂ Op(X) we have L sP (U) '
P.
Proof. Consider U as a category. We need to show that holimUop P ' P.
The crucial input is that the weak equivalences V → ∗ give rise to
U ' hocolimV⊂U V ' hocolimU ∗ via Proposition 2.16.
Now consider any N ∈ M and a cosimplicial resolution N•. Then
we have the functor K 7→ N ⊗ K defined in the introduction which is
left Quillen, as is shown in Corollary 5.4.4 of [25]. Hence it preserves
homotopy colimits and we have:
N = N ⊗ hocolim
U
∗ ' hocolim
U
(N ⊗ ∗) = hocolim
U
N
40
Finally, we use the fact that M has internal hom-spaces. Replace N
above be the cofibrant unit. Then we conclude:
holim
U∈Uop
P(U) ' holim
Uop
RHom(1, P(U))
' holim
Uop
RHom(1(U), P) ' RHom(hocolim
U
1, P)
' RHom(1, P) ' P
In the second line we use the fact that RHom(−, P) sends homotopy
colimits to homotopy limits.
Proposition 2.18. For two choices Ut and Us there is a chain of quasi-
isomorphisms between L tP and L
s
P . Hence there is a presheaf LP well
defined in the homotopy category.
Proof. By considering the union of of Us and Ut it suffices to show the
result if Ut is a subcover of Us. By Lemma 2.14 it then suffices to fix
Ui ∈ Ut and check that holimi/ι P ' P where ι is the natural inclusion
map. But the arrow category stands for the opposite of the category of
all the elements of Us contained in Ui. These form a basis and hence the
homotopy limit is given be Lemma 2.17.
Proposition 2.19. For any choice of Us the presheaf L sP is fibrant, i.e.
it is Hˇ-local.
Proof. By Lemma 2.9, it is enough to show L sP is levelwise fibrant
(immediate from definition) and satisfies the sheaf condition.
Given a hypercover {Wi}i∈I of U we may assume that any element of
Us is a subset of one of the Wi. Then we consider for every i the basis
41
of contractibles Us(i) for Wi of elements of Us that are contained in Wi.
Then we obtain the following:
holim
Iop
L sP (Wi) ' holimi∈Iop holimU∈Us(i)op P(U)← holimU∈Usop P(U)
And our aim is to show the arrow on the right is a weak equivalence.
By considering RHom(1⊗hocolim ∗, P) as in the proof of Lemma 2.17 it
suffices to show hocolimi∈I hocolimV∈Us(i) V → hocolimU∈Us U is a weak
equivalence. But if we apply Proposition 2.16 this is weakly equivalent
to hocolimi∈I Wi → X, which is a weak equivalence by Proposition
2.15.
If X is locally contractible then it has a basis of contractible open sets.
Moreover one can associate a hypercover to any basis. For details on
the construction see Section 4 of [13] and note that a basis is a complete
cover.
Proposition 2.20. If P is constant then the natural map P → LP is
a weak equivalence of presheaves.
Proof. To show that L resolves P it is enough to observe that
LP(U) 'P for contractible U by Lemma 2.17. Now the contractible
opens give rise to a hypercover on which P and LP agree and that
restricts to a hypercover on every open set. By Lemma 2.10 that suffices
to prove the proposition.
Remark 2.17. An objectF inM Jτˇ is called a homotopy locally constant
presheaf if there is a hypercover U∗ such that all restrictions F |U(i)n are
weakly equivalent to constant presheaves.
42
IfP is only homotopy locally constant we still have the construction of
LP and Proposition 2.20 holds as well as Lemma 2.17 for small enough
contractible open sets. We expect that P → LP will still be a fibrant
replacement. However, the proof of Proposition 2.19 relies explicitly
on the fact that we are considering constant presheaves, so it does not
readily adapt to the more general case.
With Proposition 2.20 we can compute RΓ(X, P) as LP(X). Note that
since we have not used functorial factorization this is not a functor
on the level of model categories but only on the level of homotopy
categories.
Definition. We will call a cover in (SetOp(X)
op
, τ) a good cover if
all its elements and all their finite intersections are contractible.
Correspondingly a good hypercover is a hypercover such that all its
open sets U (i)n are contractible.
We will now consider a good hypercover {Ui}i∈I . For computations
it is easier not to consider the full simplicial presheaf given by open
sets in the cover but only the semi-simplicial diagram of nondegenerate
open sets, i.e. leaving out identity inclusions. This becomes particularly
relevant when we consider locally finite covers in the Chapter 3.
Theorem 2.21. Let U∗ → hX be a good hypercover of a topological
space X. Let P be a constant presheaf on X. Then RΓ(X, P) = holimIop0 P
where I0 indexes the distinct contractible sets of U∗.
43
Proof. We consider a fibrant replacement LP as in Definition 2.4. Let
I index the connected open sets of U∗. Then we have:
RΓ(X, P) ' LP(X) ' holimLP(U∗)
' holim
Iop
LP(U (i)n ) ' holimIop P
Here we use Lemma 2.17 to identify LP(U
(i)
n ) and P. Now consider
ι : Iop0 ⊂ Iop and note that all the overcategories ι ↓ i are trivial (any i ∈ I
is isomorphic to some j ∈ I0) so by Lemma 2.13 we have
RΓ(X,U) ' holim
Iop0
P
Remark 2.18. Note that we can of course take the hypercover associated
to a Cˇech cover in this theorem. In fact, since we are concerned with
locally constant presheaves it makes very little difference whether we
use the Cˇech or local model structure for computations. Considering
hypercovers simplifies the theory and Cˇech covers make for simpler
examples.
We conclude this section with some results on functoriality.
Lemma 2.22. Let f : X → Y be a continuous map and let PX or Y
denote the constant presheaf with fiber P on X or Y. Then RΓ(X, P) '
RΓ(Y,R f∗(P)).
Proof. The fact that RΓ◦R f∗ ' RΓ follows immediately from piY,∗ ◦ f∗ =
piX,∗ and the fact that all these maps preserve fibrations.
Lemma 2.23. In the setting of the previous lemma there is a functor
RΓ(Y, PY)→ RΓ(X, PX).
44
Proof. Γ is a covariant functor. From Lemma 2.22 we have a natural
weak equivalence RΓ(Y,R f∗(PX))→ RΓ(X, PX).
Let P• →P• be a fibrant replacement. It is then enough to construct a
map f • : PY → R f∗(PX) of sheaves on Y . On any open set U this is
given by PY(U) = P→ f∗PX(U)→ f∗PX(U).
Remark 2.19. With P = Chpe this gives functoriality for Morita
cohomology if we use functorial factorizations.
Note that our computation using good covers is not functorial unless
we pick compatible covers. However, if X and Y have bases of
contractible sets which are suitably compatible we can just write down
the comparison map between homotopy limits.
2.5. Morita cohomology over general rings
Everything we have done since Section 2.2 was built on the assumption
that dgCat is left proper, which is only the case if k is of flat dimension
zero, see Remark 2.8.
Nevertheless, we can consider the question of what Morita cohomology
should be over other ground rings. The obvious way out is to use
Γ(X,LChpe) as our definition of Morita cohomology if k has positive
flat dimension. All pertinent results in the remainder of the chapter then
still apply, in particular Theorem 2.21, and we can prove equivalence
with the category of homotopy locally constant sheaves in Chapter 3
and with∞-local systems in Section 4.1.
45
3. Homotopy locally constant sheaves
3.1. Strictification and computation of homotopy
limits
In this chapter we will show that Morita cohomology of X is equivalent
to the category of homotopy locally constant sheaves of perfect
complexes on X.
We begin in this section with generalities on strictification and the
computation of homotopy limits.
But first we set up the situation with which we will be concerned. Fix
throughout this chapter a topological space X and assume that X has a
good hypercover U = {Ui}i∈I satisfying certain finiteness conditions.
Specifically we assume that U satisfies the following two conditions,
which we sum up by saying U is bounded locally finite.
• U is locally finite. (Every point has a neighbourhood meeting only
finitely many elements of U.)
• There is some positive integer n such that no chain of distinct
open sets in U has length greater than n.
46
Remark 3.1. If X is a finite-dimensional CW complex it has a bounded
locally finite cover. One can show this by induction on the n-skeleta
using collaring, see Lemma 1.1.7 in [20], to extend a bounded locally
finite hypercover on Xn to one on a neighbourhood in Xn in Xn+1. Then
one extends over the n + 1-cells.
Note that by Theorem 2.21 we can compute cohomology as the
homotopy limit of a diagram indexed by I0 ⊂ I, the category of non-
degenerate objects. In the next section we will use strictification to
compute this small homotopy limit explicitly as a category of homotopy
cartesian sections.
Let us consider the fiber Ch instead of Chpe at first, which has the
advantage of being a model category. Model categories are often a
convenient model to do computations with ∞-categories. However,
as the category of model categories is not itself a model category
there exist no homotopy limits of model categories. Instead one can
compute categories of homotopy cartesian sections and strictification
results compare them to homotopy limits of the∞-categories associated
with the model categories in question.
Generally speaking, using strictification to compute a homotopy limit
proceeds as follows. As ingredients we need some localization
functor L : MC → ∞Cat from model categories to some model of
(∞, 1)-categories and a method, call it hsect, of computing homotopy
cartesian sections of a Quillen presheaf, i.e. of a suitable diagram
of model categories. Then given a diagram (Mi) of model categories
indexed by I one proves holimi∈Iop LMi ' L hsect(I,Mi).
47
We will proceed by adapting the strictification result for inverse
diagrams of simplicial categories from Spitzweck [42] to dg-categories.
J is an inverse category if one can associate to every element a non-
negative integer, called the degree, and every non-identity morphism
lowers degree. This is certainly the case for I0 if U is bounded locally
finite.
We then have to restrict to compact objects in the fibers to compute
RΓ(X,Chpe) rather than RΓ(X,Ch).
Remark 3.2. There is a wide range of strictification results in the
literature: For simplicial sets [18, 52], simplicial categories [42], Segal
categories (Theorem 18.6 of [24]) and complete Segal spaces [4, 5].
Most of the above results make fewer assumptions on the index cate-
gory, for example Theorem 18.6 of [24] proves strictification of Segal
categories with general Reedy index categories, and a generalization
to arbitrary small simplicial index categories is mentioned in Theorem
4.2.1 of [51]. But since it is unclear how to adapt this proof to the
dg-setting and since the existence of a bounded locally finite good hy-
percover for X does not seem a very restrictive assumption we stay with
it.
We will deal with model categories that are already enriched in some
symmetric monoidal model category V and our ∞-categories will be
V -categories. (Think V = sSet or Ch.)
Definition. Denote by L the localization functor L : V MC → V Cat
that sends M to Mc f , the subcategory of fibrant cofibrant objects of M.
48
The fibrant cofibrant replacement is necessary to ensure that the V -hom
spaces are invariant under weak equivalences. In the case V = sSet
compare the homotopy equivalence between LM and the Dwyer–Kan
localization of M.
Let us set up the machinery:
Definition. A left Quillen presheaf on a small category I is a
contravariant functor M• : I → Cat, written as i 7→ Mi such that for
every i ∈ Ob(I) the category Mi is a model category and for every map
f : i→ j in I the map f ∗ : M j → Mi is left Quillen. (One can similarly
define right Quillen presheaves.)
Definition. The constant left Quillen presheaf with fiber M, denoted as
M is the Quillen presheaf with Mi = M for all i and f
∗ = 1M for all f .
Remark 3.3. One can define Quillen presheaves in terms of pseudofunc-
tors instead of functors, see [42]. The complicated definition is in [25].
One then rectifies the pseudofunctor to turn it into a suitable functor, i.e.
into a left Quillen presheaf as defined above.
Definition. Let M• be a left Quillen presheaf of model categories. We
define a left section to be a tuple consisting of (Xi, φ f ) for i ∈ Ob(I)
and f ∈ Mor(I) where Xi ∈ Mi and φ f : f ∗X j → Xi, satisfies
φg ◦ (g∗φ f ) = φ f◦g : ( f ◦ g)∗Xk → Xi for composable pairs g : Xi → X j,
f : X j → Xk.
A morphism of sections consists of mi : Xi → Yi in Mi making the
obvious diagrams commute. We write the category of sections of
49
M• as psect(I,M•). The levelwise weak equivalences make it into a
homotopical category.
Definition. A homotopy cartesian section is a section for which all the
comparison maps φ f : R f ∗X j → Xi are isomorphisms in Ho(Mi). We
write the category of homotopy cartesian sections of M as hsect(I,M•).
If I is an inverse category or M is combinatorial then the category of left
sections psect(I,M) has an injective model structure, just like a diagram
category, in which the weak equivalences and cofibrations are defined
levelwise, cf. Theorem 1.32 of [1].
We write L hsect(I,M•) for the subcategory of homotopy coherent
sections whose objects are moreover fibrant and cofibrant.
Note that hsect(I,M•) is not itself a model category since it is not in
general closed under limits.
Remark 3.4. One would like homotopy cartesian sections to be the
fibrant cofibrant objects in a suitable model structure. If we are
working with the projective model structure of right sections then
(under reasonable conditions) there exists a Bousfield localization, the
so-called homotopy limit structure (cf. Theorem 2.44 of [1]). The
objects of L hsectR(I,M) (which are projective fibrant) are precisely the
fibrant cofibrant objects of (psectR)holim(I,M).
The homotopy limit structure on left sections is subtler. It is the subject
matter of [3]. Assuming the category of left sections is a right proper
model category Bergner constructs a right Bousfield localization where
the cofibrant objects are the homotopy cartesian ones in Theorem 3.2
50
of [5]. Without the hard properness assumption the right Bousfield
localization only exists as a right semimodel category, cf. [2].
Note that we will still use model category theory, all we are losing
is a conceptually elegant characterization of the subcategory we are
interested in.
In the remainder of this section we recall the construction of
enrichments of presections and presheaves that will be used in the proof
of strictification.
Assume that V is a symmetric monoidal model category and that the
we are given a left Quillen presheaf such that all the Mi are model V -
categories. Note that V will be the category Ch in our application.
Lemma 3.1. If M• is as above and the comparison functors are V -
functors then psect(I,M•) is a model V -category.
Proof. Tensor and cotensor can be defined levelwise.
We define Hompsect(X•,Y•) as the end
∫
i
Hom(Xi,Yi). The same
reasoning as in diagram categories applies, see the discussion before
Lemma 2.2.
Since cofibrations and weak equivalences in psect(I,M•) are defined
levelwise the pushout product axiom holds, cf. Lemma 2.3, and we
have a model V -structure.
It follows that the derived internal hom-spaces can be computed by cofi-
brantly and fibrantly replacing source and target, RHompsect(X•,Y•) =∫
i
Hom((QX)i, (RY)i), cf. Lemma 2.2.
51
In particular if all Mi are dg-model categories then psect(I,M) is a dg-
model category.
Definition. If M is enriched in V let V Psh(M) be the category of V -
functors from M to V , i.e. functors such that the induced map on hom-
spaces is a morphism in V .
V Psh(M) is a model category if V = Ch or if V has cofibrant hom-
spaces, see Remark 2.2.
Lemma 3.2. V Psh(M) is enriched, tensored and cotensored over V .
Proof. We can tensor and cotensor levelwise. For the enrichment we
have to define an object in V of V -natural transformations between
two V -functors F,G : A → B. Recall that a V -natural transformation
is, for every object A ∈ A a morphism 1V → B(FA,GA). And
morphism spaces in V live themselves in V so Nat(F,G) is a limit
(to be specific, an end) in V . This is of course entirely standard, see
Chapter 1 of [30].
Lemma 3.3. There is an enriched Yoneda embedding M → V Psh(M).
If V has a cofibrant unit and fibrant hom-spaces then the Yoneda
embedding factors through the subcategory of fibrant cofibrant objects.
Proof. For the existence of the embedding see 2.35 in [30]. It is clear
from the projective model structure that the objects in the image are
fibrant. To see the image consists of cofibrations, we just note that the
maps 0→ hX ⊗ 1 are generating cofibrations.
The conditions of the lemma are satisfied in Ch.
52
3.2. Strictification for dg-categories
Our goal now is to prove the following theorem:
Theorem 3.4. Let I be a direct category. Let Mi be a presheaf of model
categories enriched in Ch. Then L hsect(I,M•) holimi∈Iop LMi in
Ho(dgCatDK).
This theorem allows us to characterize Morita cohomology of X. From
Theorem 2.21 we immediately obtain the following:
Corollary 3.5. Let {Vi}i∈I be a locally finite good hypercover of X. Then
RΓ(X,Chdg) ' L hsect(I0,Ch).
We are mainly interested in restricting attention to Chpe. In this case we
have to be careful about defining the right hand side. We will consider
this situation in Section 3.3.
To show Theorem 3.4 we closely follow the method of proof in
[42], replacing enrichments in simplicial sets by enrichment in chain
complexes wherever appropriate. For easier reference we write in terms
of V -categories, where V = Ch for our purposes and V = sSet in [42].
One simplification is that we are assuming the model categories we start
with are already enriched in Ch, so that we can use restriction to fibrant
cofibrant objects instead of Dwyer–Kan localization as the localization
functor.
There are two times two steps to the proof: First we define homotopy
embeddings ρ1 and ρ2 of the two sides into L psect(I,V PS h(RLM•)).
53
We then show that their images are given by homotopy cartesian section
whose objects are in the image of Mi. The first pair of steps are
quite formal. The second pair is given by explicit constructions using
induction along the degree of the index category.
The proof of the strictification result depends on setting up a comparison
between the limit construction and presections. Since the fibrant
replacement of LM• is not a Quillen presheaf we have to embed
everything into a presheaf of enriched model categories. This is
achieved by using the Yoneda embedding.
We write RLM• for i 7→ (RLM)i, where R stands for fibrant replacement
in the injective model structure on diagrams of V -categories and L is
taking fibrant cofibrant objects of every Mi.
Lemma 3.6. Let Di be an Iop-diagram of V -categories. We have a
canonical full V -embedding:
ρ2 : holim D• = lim RD• ↪→ L psect(I,V PS h(RD•))
Proof. The map to psect(I,V Psh(D f•)) is obtained by composing the
Yoneda embedding with the map of V -categories limi Ci → psect(I,C•)
that sends a to {pii(a)} if pii : lim j C j → Ci are the universal maps. (C•
is not a model category, but we can still take psect with the obvious
meaning, the comparison maps are identities by definition.)
Recall that Ob(lim Ci) consists of collections {ci ∈ ObCi} such that
C( f )(c j) = ci for f : i → j and Mor(lim Ci) consist of collections
{gi ∈ Mor(Ci)} such that C( f ) ◦ gi = g j ◦ C( f ), i.e. the hom-space
54
between {ci} and {di} is given by
∫
i
Hom(ci, di). If the Ci are enriched
then Homlim Ci(a, b) =
∫
Hom(pii(a), pii(b)) in V .
The universal property is clear. Hence the objects are a subset of the
objects of presections, namely the ones with identities for comparison
maps.
By definition of the morphisms in psect(I,D•) we have the following
isomorphisms of hom-spaces:
Hompsect(c•, d•)
∫
i
HomV Psh(RD•)i(ci, di)
∫
i
HomRDi(ci, di)
Homlim RD•({ci}), {di})
Here we use that the enriched Yoneda embedding is indeed an enriched
embedding. We abusively write ci for the image of ci in V Psh(RDi).
This proves there is an embedding of the homotopy limit into psect(I, •).
To show this embedding factors through fibrant cofibrant objects note
first that cofibrations are defined levelwise. For fibrations one uses the
fibrancy of RLM•, this is Lemma 6.3 of [42].
It follows from this embedding that homotopy equivalences in the
homotopy limit are determined levelwise since in L psect homotopy
equivalences are weak equivalences and weak equivalences are defined
levelwise. This is Corollary 6.5 in [42].
From now on we will write ρ2 for the case Di = LMi.
We also have the following:
55
Lemma 3.7. There is a natural homotopy V -embedding
ρ1 : L hsect M• ↪→ L psect(I,V PS h(RLM•))
Proof. We have an embedding hsect ↪→ psect and homotopy
embeddings Mi ↪→ V Psh(RLMi) which give a homotopy embedding
when we apply L psect(I,−) since the hom-spaces of presections
between fibrant cofibrant objects are given by homotopy ends, which
are invariant under levelwise weak equivalence.
Remark 3.5. Note that the situation is a little more intricate in
[42] where simplicial localization and restriction to fibrant cofibrant
objects are a priori distinct and need to be compared through another
embedding.
Next we have to identify the images of ρ1 and ρ2. The explicit
computation is done in Lemma 6.6 of [42]. The only use of special
properties of the category sCat made in this lemma (and the results
needed for it) is the characterization of fibrations in terms of lifting
homotopy equivalences. But this characterization is also valid for
fibrations in dgCatDK .
We provide the argument here for convenience and future reference.
Lemma 3.8. The image of ρ1 consists of homotopy cartesian sections
X• ∈ L psect(I,V Psh(RLM•)) such that all Xi are in the image of Mi.
Proof. We proceed by induction on the degree of the indexing category.
Let X• be given as in the statement and assume by induction we have a
56
levelwise equivalence Y 1. In other words
S (n − 1) provides a model for singular chains on ΩS n equipped with
the Pontryagin product. This is of course well-known: H∗(ΩX) is a
polynomial algebra on a generator in degree n − 1. A computation via
the James construction is in 3.C and 4.J of [22]. So there is a natural
map S (n− 1) H∗(ΩS n)→ NΩS n which is a quasi-isomorphism. (We
will see in Example 5.2 how to show directly that S (n − 1) ' NΩS n.)
We also need to know that there is a map D(n) → NΩBn compatible
with S (n − 1) → D(n). This follows by the lifting property of
the cofibration S (n − 1) → D(n) with respect to the trivial fibration
NΩBn → ∗.
These are the building blocks needed to associate to any connected
CW-complex X without 1-cells a dg-algebraB(X) quasi-isomorphic to
C∗(ΩX) that approximates the way X is glued from cells.
Lemma 4.10. The inclusion of dg-algebras as dg-categories with one
object preserves pushouts.
Proof. Constructing the pushout of dg-categories with one object we
obtain a dg-category with a single object. Furthermore maps from a
one-object dg-category to other dg-categories are just maps from the
dg-algebra of endomorphism to the dg-algebra of endomorphisms of
the image.
74
In the proof of the next theorem we need to compute some homotopy
pushouts. We assume k is a field so that the model category dgAlgk is
proper and we can compute homotopy pushouts as pushouts whenever
one of the constituent maps is a cofibration.
Theorem 4.11. Assume k is a field. Associated to every connected CW
complex X with cells in dimension ≥ 2 there is a cofibrant dg-algebra
B(X) with one generator in degree n − 1 for every n-cell, that is quasi-
equivalent to N(ΩX). In particular Y (X) ' ChB(X).
Proof. We proceed by induction on the cells, let α be the index. We
begin with the 2-skeleton of X which by assumption is a wedge of s
spheres. Since the derived versions of N, k⊗−, G and Sing* all preserve
homotopy colimits, so does NΩ on cofibrant spaces in CGHauss. This
gives NΩ(
∨
s S 2) ' ⊗sk[x1], i.e. a free dg-algebra on s generators in
degree 1. We defineB(X2) B ⊗sk[x1].
From here on we can compute N(ΩX) inductively by forming, for every
pushout X≤α = colim(en+1α ← S n → X<α), the diagram of dg-categories
N(ΩBn+1) ← N(ΩS n) → N(ΩX<α). Our aim is to show the pushout of
the second diagram is weakly equivalent to N(ΩX≤α). Since CGHauss
and dgCatDK are proper the pushouts of both diagrams are homotopy
pushouts and hence matched up by NΩ.
Hence N(ΩX≤α) is determined by the map N(ΩS n) → N(ΩX<α). Since
the left-hand side is weakly equivalent to a free dg-algebra on a single
generator in degree n − 1 it suffices to specify its image, which is a
homology class on the right-hand side, i.e. an element of Hn−1(ΩX<α).
75
Inductively assume there is a weak equivalence B(X<α) → N(ΩX<α).
In particular there is an isomorphism on Hn−1 and hence there is a map
S (n − 1) → B(X<α) corresponding to the attachment map S n → X<α.
This gives a weak equivalence between pushout diagrams. We define
B(X≤α) to be colim(D(n) ← S (n − 1) → B(X<α)). By the previous
lemma the pushout computed in dgAlg is the same as the one in
dgCat. Since the pushouts are homotopy pushouts of diagrams which
are levelwise weakly equivalent they agree up to homotopy and we have
B(X≤α) ' N(ΩX≤α).
To extend to infinite CW-complexes we have to check the same
argument goes through for filtered colimits. Since the maps X<α → X≤α
are cofibrations the filtered colimit is a homotopy colimit and commutes
with NΩ. So NΩX≤λ ' hocolimα<λ NΩXα and we can defineB(X≤λ) as
colimα<λB(X≤α).
Proposition 4.12. Let k be a commutative ring of characteristic 0.
Assume the CW complex X as above is such that all attachment maps
are cofibrations. ThenB(X) constructed as above is weakly equivalent
to NΩ(X).
Proof. The only place we used that k is a field was in asserting that
pushouts are homotopy pushouts if one of the maps is a cofibration.
With our new assumption both N(ΩBn) → N(ΩS n) and N(ΩS n) →
N(ΩXα) are cofibrations, so the pushout is a homotopy pushout without
assuming properness.
76
Remark 4.6. To use this computation in practice we need to identify the
degree n−1 element y ofB(X<α) that corresponds to the image of S n−1.
Then we adjoin a new generator x with dx = y. This can of course be
quite non-trivial. There are some examples in the next section.
Next we deal with the case of 1-cells.
Remark 4.7. The main difficulty in considering ChS
1
arises as follows.
It is clear that C∗(ΩS 1) ' k[Z]. However, k[Z] is not a cofibrant dg-
algebra, so cannot be used for computing homotopy pushouts.
Theorem 4.13. Associated to every connected CW complex X there is
a dg-algebra B(X) with one generator in degree n − 1 for every n-cell
with n ≥ 2, and with two inverse generators in degree 0 for every 1-cell,
such that Y (X) ' ChB(X).
Proof. Let us define S ∗(0) = k[a, a−1] and D∗(1) = k[a, a−1, b 7→ a − 1]
and consider the cofibration S ∗(0) ↪→ D∗(0). Of course D∗(0) ' k.
Then we have compatible quasi-isomorphisms NΩS 1 → S ∗(0) and
NΩB2 → D∗(1). The first is induced by projection to connected
components G Sing* S
1 → Z, the second map exists since D∗(1)→ 0 is
a trivial fibration and NΩS 1 → NΩB2 is a cofibration.
Let X1 be the 1-skeleton of X and defineB(X1) = B(
∨
s S 1) := ⊗sS ∗(0)
which is weakly equivalent to C∗(Ω(
∨
s S 1)) There is an obvious map
from S ∗(0) to B(X1) for any attachment map S 1 → X1. Assume first
that X is obtained from X1 by attaching a 2-cell. Then we define
B(X) = colim (D∗(1)← S ∗(0)→ B(X1))
77
Now Y (X) is the homotopy pullback of Y (B2) ← Y (S 1) → Y (X1).
But this diagram is weakly equivalent to ChNΩB
2 → ChNΩS 1 ← ChNΩX1
which is in turn weakly equivalent to ChD
∗(1) → ChS ∗(0) ← ChB(X1).
These are all pullback diagrams of fibrant objects with one map
a fibration, hence they are homotopy pullbacks as dgCatDK is
right proper. Since the diagrams are levelwise quasi-equivalent
their pullbacks are quasi-equivalent, and thus also isomorphic in
Ho(dgCatMor). But since D 7→ ChD sends colimits to limits it also
follows that
Y (X) ' holim
(
Y (B2)→ Y (S 1)← Y (X1)
)
' holim
(
ChD
∗(1) → ChS ∗(0) ← ChB(X1)
)
' lim
(
ChD
∗(1) → ChS ∗(0) ← ChB(X1)
)
' Chcolim(D∗(1)←S ∗(0)→B(X1))
The colimit in the exponent is how we have definedB(X).
Now consider the general case. First to obtain B(X2) note that any
attachment map from S 1 factors through X1, so we can repeat the
previous step as often as required. Attachment of higher-dimensional
cells works in exactly the same manner, we just have to replace S ∗(0)
by S (n − 1) and D∗(1) by D(n).
The argument extends to filtered colimits just like in the proof of
Theorem 4.11.
Remark 4.8. By constructionB(X) is Morita-equivalent to NΩX, but it
does not follow from the construction whether the two dg-algebras are
isomorphic in Ho(dgAlg).
78
4.4. Finiteness and Hochschild homology
In this section we consider conditions for Morita cohomology to satisfy
various finiteness properties, and determine Hochschild homology in
some cases by quoting relevant results from the literature.
Let us first make some definitions. Here R denotes fibrant replacement
in dgCatMor. Specifically, RB = L(Bop-Mod)pe.
We say a dg-category D is locally proper if the hom-space between
any two objects is a perfect complex. D is proper if moreover the
triangulated category H0(RD) has a compact generator, i.e. a compact
object which detects all objects.
Recall an object X in a model category is homotopically finitely
presented if Map(X,−) commutes with filtered colimits. D is smooth
if it is homotopically finitely presented as a Dop ⊗ D-module. D is
saturated if it is smooth, proper and Morita fibrant.
D is of finite type if there is a homotopically finitely presented dg-
algebra B such that RD ' R(Bop).
These definitions are Morita-invariant (except for the condition of being
Morita fibrant). Toën shows in Lemma 2.6 of [50] that a dg-category has
a compact generator if and only if RD ' RBop for some dg-algebra B
and is moreover proper if and only if the underlying complex of B is
perfect. Moreover any dg-category of finite type is smooth (Proposition
2.14 of [50]).
79
Remark 4.9. One reason to be interested in these finiteness conditions is
that if a dg-category is saturated there is a nice moduli stack of objects,
this is the main result of [50].
Proposition 4.14. Y u(X) is triangulated and has a compact generator.
If X is a finite CW-complex without 1-cells then Y u is smooth. If
moreover H∗(ΩX) is of finite type then Y u(X) is saturated.
Proof. Note first that as a homotopy limit Y u(X) is fibrant and the
compact generator is given by C∗(ΩX).
Theorem 4.13 implies that in the absence of 1-cells the dg-algebraB(X)
is homotopically finitely presented. So the category Y u(X) is of finite
type and hence smooth. If H∗(ΩX) is of finite type, then B(X) is a
perfect complex over k, and Y u is moreover proper and we find that
Y u(X) is saturated.
By contrast if X is an infinite CW-complex then B(X) is usually not
homotopically finitely presented. For example consider B(CP∞) '
k[x1]/(x21). Any cofibrant replacement as infinitely many generators and
hence the identity does not factor through any subobject of finite type.
Next we consider properness for Y (X). The categoryH M(X) is locally
proper if all cohomology groups of X with coefficients in local systems
are finite dimensional and concentrated in finitely many degree. This is
for example the case if X has a finite good cover. Then the hom-spaces
are finite limits of perfect chain complexes.
80
This is in contrast to Ext-groups of local systems which can be large
even if X is very well behaved, for example if X is a smooth projective
variety [10].
The example X = S 1 shows that we cannot expect Y (X) to be proper in
general. ChS
1
is the category of complexes of Z-representations, with
infinitely many connected components, see Example 5.1.
Proposition 4.15. If pi1(X) has only finitely many irreducible repre-
sentations which are all finite dimensional then there exists a com-
pact generator A and Y (X) ' L(End(A)op-Mod)pe. Y (X) is proper
if C∗(X,End(A)) is a perfect complex.
Proof. We define A to be the sum of all the irreducibles. Then A maps
to the lowest nontrivial homology group of any object in Y (X) and
hence generates the dg-category since objects with trivial homology are
quasi-isomorphic to 0.
By Lemma 2.6 of [50] L(Y (X)op-Mod) ' L(EndY (X)(A)op-Mod). Since
Y (X) ' L(Y (X)op-Mod)pe we deduce that Y (X) is the subcategory of
compact objects in End(A)-Mod.
The second statement is clear.
The proposition applies for example if the fundamental group is finite.
Then we can take A to be the group ring.
Example 4.2. Let X be simply connected. Then we can take A = k and
find End(A) ' RHom
ΩX(k, k) ' C∗(X, k) by earlier results. In particular
Y (X) ' C∗(X, k) in dgCatMor. Then Y (X) is proper if and only
81
if C∗(X, k) is a perfect complex. If C∗(X, k) is homotopically finitely
presented then Y (X) is moreover smooth and saturated.
If Y (X) has a compact generator it becomes much easier to compute
secondary invariants. In particular we can compute Hochschild
homology and cohomology. For definitions and a summary of results
see [29]. Since Hochschild homology and cohomology are Morita-
invariant we can compute them on a generator of a dg-category if there
is one.
Example 4.2 implies the following proposition. Here HH stands for
either HH∗ or HH∗.
Proposition 4.16. Let X be simply connected then HH(Y (X))
HH(C∗(X)).
So we can compute Hochschild (co)homology of Morita cohomology
from minimal models (in the sense of Sullivan).
Proposition 4.17. HH(Y u(X)) HH(C∗(ΩX)) HH(B(X)).
Proof. The second isomorphism follows since Hochschild (co)homology
is Morita-invariant.
The following applications follows from results readily available in the
literature.
Proposition 4.18. Let X be simply connected then HH∗(Y (X))
H∗(L X). If M is a simply connected closed oriented manifold of
82
dimension d then HH∗(Y (M)) H∗+d(LM) as graded algebras with
the Chas-Sullivan product on the right hand side.
Proof. If X is simply connected it is well known (see [32]) that
HH∗(C∗(X, k)) H∗(L X) whereL X is the free loop space.
The second part follows since the Hochschild cohomology ring of
singular cochains on M (with the cup product) is isomorphic to its loop
homology with the Chas-Sullivan product, cf. [9].
Proposition 4.19. HH∗(Y u(X)) H∗(L X). If X is simply connected
HH∗(Y u(K)) H∗(L X) as graded algebras.
Proof. We find HH∗(Y u(X)) HH∗Ω(X) H∗(L X) from 7.3.14
in [31].
For a the result that HH∗ Sing* ΩX H
∗(L X) (as graded algebras) for
a simply connected CW-complex X, see [34].
Note that we do not expect Hochschild homology of Y (X) to be
particularly tractable if X is not simply connected. For example
Y (S 1) has |k∗| simple objects with no morphisms between them, cf.
Example 5.1. Hence it follows from the explicit definition in [29] that
Hochschild homology consists of |k∗| copies of HH∗(k[y]) where y lives
in degree 1 and has square 0.
83
5. Computation and Examples
In this chapter we compute some examples of Morita cohomology. In
the following whenever an element has a subscript, this will denote its
degree.
For simplicity we will often not mention the restriction to fibrant
cofibrant objects in representations of C∗(ΩX) orB(X).
5.1. Spheres
Example 5.1. We begin with the case X = S 1. Clearly H M(S 1) is
equivalent to the category of representations of Z ' ΩS 1.
We can also view this as the category of bounded chain complexes
of local systems on S 1 since S 1 is K(Z, 1) and the cohomological
dimension of Z is 1, so that all non-trivial extensions in chain complexes
of Z-modules are extensions in the abelian category of Z-modules.
We can also characterizeH M(S 1) as the explicit limit
(Chpe)I ×Chpe×Chpe Chpe
Here ChpeI is the path object in dg-categories, which is (Chpe)1 in
the simplicial resolution constructed in the appendix, see Example A.2.
84
The limit then comes out as the category of pairs (M, φ ∈ Aut(M))
with morphisms ( f , g, h) : (M, φ) → (N, ψ) in Hom(M,N)⊕2 ⊕
Hom(M,N)[−1] with differential
( f , g, h) 7→ (d f , dg, dh − (−1)|g|gφ + ψ f )
In particular Hom∗(k, k) k⊕ k[1], which is exactly cohomology of S 1,
as predicted.
We can also compute H M(S 1) with a Cˇech cover. Using a good cover
by three open sets and their intersections and an explicit model for Ch1,
see above, we find that an object is given by three chain complexes with
weak equivalences between them. We can use two weak equivalences
to identify the complexes and are left with a single homotopy invertible
map.
As we have stated before, the categoryH M(S 1) is highly disconnected,
in fact isomorphism classes of simple objects are naturally in bijection
with k∗. Of course k∗ has a geometric structure, and one way of
interpreting large sets of isomorphism classes of objects is to consider
a moduli stack of objects of H M(X). We will not follow this direction
here.
Example 5.2. If n > 1 then H M(S n) ' ChpeS (n), i.e. the category of
perfect chain complexes with an endomorphism in degree n− 1 that are
fibrant and cofibrant as such modules.
Proof 1. This is immediate from the quasi-isomorphism S (n) →
N(Ω Sing* S
n) which was mentioned in the last section.
85
Proof 2. We can also compute B(S 2) using the method of Theorem
4.13 by gluing two copies of B2 along S 1. The resulting dg-algebra
has one invertible generator with two trivialising homotopies, which is
quasi-isomorphic to k[x1] = S (1).
Once we know the case n = 2 we can inductively compute S n =
Dn qS n−1 Dn and note that S (n) ' D(n) ⊗LS (n) D(n).
Note that we can use this construction ofB(S n) in the proof of Theorem
4.13. There is no circularity as we only need a model for spheres in
smaller dimensions to computeB(S n).
Example 5.3. Next consider some more detail for n = 2. Since k is
a generator and REndC∗(ΩS 2) ' C∗(S 2) ' k[x2, x3
d→ x22] =: A we can
characterize Y (S 2) as compact objects in A-Mod.
An example of an object of RΓMorita(S 2, k) is the chain complex
associated to the Hopf fibration p : S 3 → S 2. As a homotopy
locally constant sheaf we can consider this as Rp∗ Sing*(S
3). As a
representation of ΩS 2 this can be written as k⊕k[−1] with the canonical
map of degree 1.
Since pi1(S 2) is trivial, we can also view RΓMor(S 2, k) as generated by
the trivial local system and the information H∗(S 2,−) provides about
(iterated) extensions. This provides a slightly different viewpoint on
Morita cohomology.
Specifically, consider the forgetful map D → Ch. The objects
in the fibre over M
⊕
Mi[−i] are all the homotopy locally
constant sheaves with homology M. They can be determined
86
iteratively. For example, over M0 ⊕ M1[−1] the fiber is parametrized
by C∗(X,Hom1(M1[−1],M0).
5.2. Other topological spaces
Example 5.4. H M(BG) is just the dg-category of perfect complexes
with an action of G.
Example 5.5. RΓ(RP2,Chpe) is given by representations of B(RP2)
on perfect complexes, and B(RP2) has generators a0, a−10 , b1 such that
db1 = a0 ◦ a0 − 1. This follows from Theorem 4.13. The identification
db1 = a0 ◦ a0 − 1 is induced by the attaching map from the boundary of
the 2-cell to RP1.
If we are working over the field Q Morita cohomology has certain
similarities to rational homotopy theory, cf. the duality between C∗(ΩX)
and C∗(X) in the simply connected case. On the other hand we see
that RP2 has trivial minimal model, but its Morita cohomology is a dg-
category with two simple objects corresponding to the irreducible reps
of Z/2.
We can obtainB(RP3) fromB(RP2) by adding c2 with dc2 = 0.
Example 5.6. Next we compute the map p∗ : H M(S 2) → H M(S 3)
induced by the Hopf fibration. On the level of loop spaces we see that
the map is induced by Ωp∗ : H∗(ΩS 3) → H∗(ΩS 2) which is given by
x2 7→ y21 on the generators.
87
With this in mind we can work outH M(CP2) explicitly by considering
the following diagram:
H M(B4)
i∗−→H M(S 3) p
∗
←−H M(CP1)
On the level of dg-algebras we have
D(3)
i∗←− S (3) p∗−→ B(S 2) S (2)
The attaching map p∗, is induced by the Hopf fibration. As we have just
seen it corresponds to the map H∗(ΩS 3) → H∗(ΩS 2) given by sending
x2 7→ y21. Hence we find:
B(CP2) ' k[α1, α3 | dα3 = α21]
Example 5.7. We can generalise this to CPn, every extension over a
2i-cell corresponding to another map α2i−1 in degree 2i − 1. We find
d : α3 7→ α21;α5 7→ α3α1 + α1α3;α7 7→ α5α1 + α23 + α1α5 etc.
Let us compare this with a computation of H∗(ΩCPn), which is done
e.g. in [38]. The fibration ΩS 2n+1 → ΩCPn → S 1 is a direct product.
Hence H∗(ΩCPn) Λ(y1) ⊗ k[y2n] as a Hopf algebra, in particular
the Pontryagin products agree. C∗(ΩCPn) is moreover formal since
C∗(ΩS 2n+1) and C∗(ΩS 1) are.
To relate this to the above description identify y2n = α2n−1α1 + · · · +
α1α2n−1. The dg-algebra B(X) is larger since it is quasi-free (i.e. the
underlying graded associative algebra is free), while H∗(ΩCPn) is only
quasi-free as a commutative dg-algebra.
88
Example 5.8. Taking the limit we findB(CP∞).
Of course the homology algebra of ΩCP∞ is just that of S 1. Indeed
k[α1, α3, . . . ] is a quasi-free model for k[z1].
We conclude with a few examples that give some insight into what
Morita cohomology (doesn’t) tell us about a space.
From [47] we know that a CW-complex X is determined up to weak
equivalence by the ∞-category of homotopy locally constant sheaves
of spaces with its fiber functor. We can think of this category as
RΓ(X, sSet) and it is natural to compare with RΓ(X,Chpe). The main
differences are linearization, stabilization and restriction to compact
fibers.
Example 5.9. An example of a space with trivial Morita cohomology
is provided by the classifying space of Higman’s 4-group H. This is
known to be a finite CW complex and H is an acyclic group without
non-trivial finite dimensional representations. For references and other
examples see e.g. [6].
To show that the Morita cohomology of BH is trivial we have to show
it is Morita equivalent to Ch. Now given an object M of H M(BH)
we can take homology. As H has no non-trivial finite-dimensional
representations this is a direct sum of shifted trivial representations. We
can now show by induction that M must be a trivial extension, as there
is no cohomology H>0(H, k). Since quasi-isomorphisms are detected
on the underlying complex an object with trivial homology is quasi-
isomorphic to 0. It follows thatH M(BH) Ch in Ho(dgCatMor).
89
Example 5.10. Whenever a group G and its algebraic completion Galg
have the same cohomology with coefficients in their finite-dimensional
representations then they have the same Morita cohomology. By
definition this is true for algebraically good groups, see [28].
Example 5.11. Next we consider an example of information in the
category of simplicial fibrations with fiber of finite type that is not
detected by Morita cohomology.
Consider the outer automorphism group of the free group on four
generators, Out(F4). This group is know to be finitely presented (since
Aut(F4) is), and nonlinear [19]. It is also isomorphic to pi0Map(F, F)
if F =
∨
4 S 1. Hence there is a homotopy locally constant sheaf of
simplicial sets of finite type (with fiber F) on BOut(F4) that does not
descend to any BG for G a quotient of Out(F4). On the other hand the
associated complex of local systems must descend to the classifying
space of the image of Out(F4) in its linearization.
Remark 5.1. Contrast this with Y u(X) which loses little information
about the weak homotopy type of X. Since ΩX is an H-space it is
nilpotent and there is a Whitehead theorem for integral homology for
nilpotent spaces. So if NΩX and NΩY are quasi-equivalent (rather than
just Morita equivalent) then X ' Y .
90
A. Some technical results on dg-categories
A.1. A combinatorial model for dg-categories
For some technical questions it is convenient to work with combinato-
rial model categories. (For example it is easier to prove the existence
of localizations and injective model structures.) One does not expect
dgCatMor to be combinatorial as it is too large, but we show in this
section that it is equivalent to a subcategory that is combinatorial.
Definition. A model category is combinatorial if the underlying
category is locally presentable.
Being locally presentable is a finiteness condition.
Definition. Let λ be a regular cardinal. An object A in a category D
is λ-presentable if it is small with respect to λ-filtered colimits, i.e.
if for every λ-filtered colimit colim Bi the map colim Hom(A, Bi) →
Hom(A, colim Bi) is an isomorphism. We say A is presentable if it is
λ-presentable for some λ. A cocomplete category is locally presentable
if for some regular cardinal λ it has a set S of λ-presentable objects such
that every object is a λ-directed colimit of objects in S .
91
Proposition A.1. The categories dgCatDK and dgCatMor are Quillen
equivalent to combinatorial subcategories.
Proof. This follows immediately from the proof of the main theorem
of [36]. Let D denote either of the two model structures. Let S be the
collection of objects that are domains or codomains of the generating
cofibrations and generating trivial cofibrations. Clearly S is a set. Let
S denote the full subcategory of D with objects S . Define ηS (X) to
be the colimit of the forgetful diagram (s → A) 7→ s indexed by the
overcategory S ↓ A. Then an object A ∈ D is S -generated if it is
isomorphic to ηS (X).
Now by the proof of Theorem 1.1 in [36] the subcategory of S -
generated objects of D is a model category DS which is Quillen
equivalent to the original one. Moreover, by Proposition 3.1 of [36],DS
is locally presentable if every object in S is presentable. But this is clear
since they have finitely many objects and generating morphisms.
Remark A.1. Note that Vopenka’s principle is not needed here since the
objects of S are presentable.
We get another finiteness condition for free. A model category is
called tractable if the generating cofibrations and generating trivial
cofibrations can be chosen to have cofibrant domains.
Corollary A.2. The categories DS are tractable.
92
Proof. By Corollary 1.12 in [1] it is enough to check all the generating
cofibrations can be chosen to have cofibrant domains. This is immediate
in our example.
A.2. Simplicial resolutions of dg-categories
In this section we will construct explicit simplicial resolutions C 7→ C•
in dgCatDK to improve our understanding of the homotopy theory of
dgCatDK . Such a resolution can be used for explicit, if unwieldy,
computations. The resolutions we construct will be simplicial frames
for a fibrant replacement.
Our construction is directly motivated by Simpson’s construction of
global sections of a presheaf of dg-categories as a dg-category of
Maurer–Cartan elements, cf. section 5.4 of [41].
Remark A.2. In fact, the construction of Cn below corresponds to
considering the constant presheaf of dg-categories on a covering of |∆n|
by n + 1 open sets (corresponding to leaving out one of the faces).
Define Cn as follows. We think of this as C ∆
n
, but recall that the precise
definition of C ∆
n
differs by a a limit over degenerate simplices.
Definition. AssumeC is fibrant, replace fibrantly otherwise. ThenCn is
a dg-category with objects given by pairs (E, η) where E is a collection
E0, . . . , En ∈ ObC and η is a collection of ηI = η(I) ∈ Homk−1(Ei0 , Eik)
for all multi-indices I = (i0, . . . , ik) with 1 ≤ k ≤ n. The case k = 0
is subsumed by the differential on E. (We interpret η(i) = 0 where it
93
comes up in computation.) These pairs must satisfy the Maurer–Cartan
condition: δη + η2 = 0, explained below. We also demand that all
ηi ∈ Hom(Ei, Ei) are weak equivalences in C .
Let us spell out the Maurer–Cartan condition. Intuitively, η provides
all the comparison maps as well as homotopies between the different
compositions. We define the differential
(δη)(i0, . . . , ik) B d(η(i0, . . . , ik)) + (−1)|η|
k−1∑
j=1
(−1) jη(i0, . . . , î j, . . . , ik)
which lives in Homk(Ei0 , Eik). We write δ = d + ∆. Here we define
|η| = 1. The product is:
(φ ◦ η)(i0, . . . , ik) B
k∑
j=0
(−1)|φ| jφ(i j, . . . , ik) ◦ η(i0, . . . , i j)
Both definitions follow section 5.2 of [41], with some corrections to the
signs. We leave out the terms in δη corresponding to leaving out i0 and
ik as they do not live in the correct hom-spaces.
One can now check that ∆d = −d∆ (and hence δ2 = 0) and we have the
following Leibniz rule:
δ(φ ◦ η) = (−1)|η|(δφ) ◦ η + φ ◦ (δη)
The same equation holds for the summands d and ∆. (The unusual sign
appears because of the backward notation for compositions.)
Example A.1. For n = 1 we have (δη+ η2)01 = d(η01) + 0, the expected
cycle condition. For n = 2 we have for example
(δη + η2)012 = d(η012) + η02 − η12 ◦ η01 ∈ Hom1(E0, E2)
94
So an element of D2 is of the form (E, η) where E = (E0, E1, E2) and
η = (η01, η02, η12; η013) satisfies dη+η2 = 0, which comes out to dηi j = 0
and dη012 = −η02 + η12 ◦ η01. This agrees with our intuition that η012 is
a homotopy from η12 ◦ η01 to η02.
Morphisms from (E, η) to (F, φ) are as follows.
Hom−mCn ((E, η), (F, φ)) = {a(i0, . . . , ik)}
where a(i0, . . . , ik) ∈ Homm−k(Ei0 , Fik). (Here C−m = Cm.) We write
m = |a| for the degree of a morphism. We have a differential dη,φ defined
by
(dη,φ(a))(i0, . . . , ik) = δ(a) + φ ◦ a − (−1)|a|a ◦ η
where composition and differential are defined as above. The Maurer–
Cartan condition on η and φ together with the Leibniz rule ensures
(dη,φ)2 = 0.
Example A.2. For example C1 agrees with the path object in dgCat
as constructed in Section 3 of [46]. Indeed, objects are homotopy
invertible morphisms η : A → B and morphisms from η to φ are given
by triples (a0, a1, a01) with differential
δ : (a0, a1, a01) 7→ (da0, da1, da01 + φ ◦ a0 − (−1)|a0 |a0 ◦ η)
Notation. Given an object or morphism α and a positive integer k we
write α[k] for the collection of all αi0...ik .
Before we embark on the somewhat technical proof that C• is a
simplicial resolution, we note the following application. We can extend
95
the definitions of the differentials and composition to functions defined
on general simplices. (That is, we replace “leaving out the i-th term” by
the map induced by ∂i etc.)
Proposition A.3. Given a simplicial set K we can construct C K as the
dg-category with objects (E, η) where E ∈ (ObC )K0 and η assigns to
every k-simplex in K≥1 a map in Homk−1(E(∂
k
0σ), E(∂
k
maxσ)) satisfying
the Maurer–Cartan equations. Hom-spaces are defined similarly to
hom-spaces in C•.
Proof. This follows from the construction of C K = lim∆Kop C•.
Remark A.3. This shows that the construction of ∞-local systems as
C 7→ C K corresponds to the ∞-local systems of [7], or the A∞-functor
of [26] if K is the nerve of a category, or to the Cˇech globalization
in [41] if K is the nerve of an open covering.
Proposition A.4. The inclusion from the constant simplicial dg-
category cC to C• is a levelwise weak equivalence.
Proof. We have to check the inclusion map ι : cC → Cn is a quasi-
equivalence.
Let us first show that ι induces weak equivalences on hom-complexes.
We have to show that HomC ∆n ((E, η), (F, φ)) ' HomC (E, F) when both
η and φ are of the form (1, 0), i.e. the constituent morphisms in degree
0 are the identity and all others are 0.
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Write (H, dH) B Hom(E, F) and note that from the definitions we can
write
Hom((E, 0), (F, 0)) ' (H[1] ⊗
∧
〈e0, . . . , en〉,D)
Here the ei all have degree 1 and we identify H.ei0 ∧ · · · ∧ eik with
the a(i0, . . . , ik). The differential D is dH + ι∑ ei where the second term
denotes contraction. This complex is a resolution of (H, dH).
Next we show ι is quasi-essentially surjective, i.e. show that any object
(E, η) is equivalent to an object (F0, (1, 0)) where F0 is of the form
(F0, . . . , F0).
We can deduce this if we can show that every (E, η) is equivalent to
some (F, φ) such that all compositions which agree up to homotopy by
δφ+φ2 = 0 agree strictly, i.e. φ = (φ[0], 0), and that any such (F, (φ[0], 0))
is equivalent to (F0, (1, 0)). The second part of this is immediate:
We define a map from (F0, (1, 0)) to (F, (φ[0], 0)) by sending F0 to Fi
via φ(0, i) = φ(i − 1, i) · · · φ(0, 1). Since all φ( j, j + 1) are homotopy
invertible there is a homotopy inverse.
We will now show that any (E, η) is equivalent to (F, φ) where φ has no
higher homotopies. Let F = E and let φ(i, j) = η( j − 1, j) · · · η(i, i + 1).
We may assume by induction on n that all η(i0, . . . ik) with ik < n are 0.
We define the homotopy equivalence H : (E, η)→ (E, φ) as follows:
H(i) = 1
H(i0, . . . , ik) = (−1)k−1η(i0, . . . , ik−1, n − 1, n) if ik = n and in−1 , n − 1
H(i0, . . . , ik) = 0 otherwise
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And define H− to be equal to H in degree 0 and −H in degree > 0, i.e.
the sign of H(i0, . . . , ik) is always (−1)k.
Then it is clear that H and H− are inverses. Since H(i0, . . . , in) is zero
unless in = n there are no nontrivial compositions and the composition
1 ◦ H(. . . ) and H−(. . . ) ◦ 1 cancel in degrees greater than 0.
So it remains to show that dH = dH− = 0 to show we have a genuine
homotopy equivalence.
We consider H first. Putting together our definitions we find the
following. Let us first assume ik−1 , n − 1 and ik = n. To obtain
the correct signs recall that |H| = 0 and |η| = |φ| = 1.
(dH)(i0, . . . , ik) = d(H(i0, . . . , ik)) +
k−1∑
j=1
(−1) jH(i0, . . . , iˆ j, . . . , in)
+
k∑
j=0
(−1) jφ(i j . . . in) ◦ H(i0, . . . , i j)
−
k∑
j=0
H(i j, . . . , ik) ◦ η(i0, . . . , i j)
= (−1)k−1dη(i0, . . . , n − 1, n)
+ (−1)k−2
∑
j
(−1) jη(i0, . . . , iˆ j, . . . , n − 1, n)
+ 0 − (−1)k−2η(i1, . . . , n − 1, n) ◦ η(i0, i1) − 1 ◦ η(i0, . . . , ik)
= 0
The last equality holds since the penultimate term is of the form
(−1)k−1(δη + η2)(i0, . . . , ik−1, n − 1, n)
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This becomes clear if we write η(i0, . . . , ik) = η(i0, . . . , n̂ − 1, n) and
observe that all the other terms we expect in δη + η2 are 0.
The other cases are easier. If ik , n all terms in the differential are 0
and if ik−1 = n − 1 and ik = n there are only two nonzero terms, which
cancel.
When we consider dH− the sign of the term η(i0, . . . , ik) changes, as it
now comes from η ◦H and not H ◦ η. This cancels the effect of the sign
of H(i) also changing by a factor of −1. There are no other occurrences
of the sign of H(i) unless k = 1 when all but the last two terms are zero
and the last two terms cancel.
Proposition A.5. C• is Reedy fibrant.
Proof. Write
η