## Polynomials and models of type theory

##### Authors

von Glehn, Tamara

##### Advisors

Johnstone, Peter

##### Date

2015-06-30##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

Magdalene College

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

von Glehn, T. (2015). Polynomials and models of type theory (Doctoral thesis). https://doi.org/10.17863/CAM.16245

##### Abstract

This thesis studies the structure of categories of polynomials, the diagrams that represent polynomial functors. Specifically, we construct new models of intensional dependent type theory based on these categories.
Firstly, we formalize the conceptual viewpoint that polynomials are built out of sums and products.
Polynomial functors make sense in a category when there exist pseudomonads freely adding indexed sums and products to fibrations over the category, and a category of polynomials is obtained by adding sums to the opposite of the codomain fibration.
A fibration with sums and products is essentially the structure defining a categorical model of dependent type theory. For such a model the base category of the fibration should also be identified with the fibre over the terminal object. Since adding sums
does not preserve this property, we are led to consider a general method for building new models of type theory from old ones, by first performing a fibrewise construction
and then extending the base.
Applying this method to the polynomial construction, we show that given a fibration with sufficient structure modelling type theory,
there is a new model in a category of polynomials.
The key result is establishing that although the base category is not locally cartesian closed, this model has dependent product types.
Finally, we investigate the properties of identity types in this model, and consider the
link with functional interpretations in logic.