DeepAI AI Chat
Log In Sign Up

A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

by   Eric Finster, et al.

Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets of variables. They can be organized into a cartesian bicategory, which unfortunately fails to be closed for essentially two reasons, which we address here by suitably modifying the model. Firstly, a naive closure is too large to be well-defined, which can be overcome by restricting to polynomials which are finitary. Secondly, the resulting putative closure fails to properly take the 2-categorical structure in account. We advocate here that this can be addressed by considering polynomials in groupoids, instead of sets. For those, the constructions involved into composition have to be performed up to homotopy, which is conveniently handled in the setting of homotopy type theory: we use it here to formally perform the constructions required to build our cartesian bicategory, in Agda. Notably, this requires us introducing an axiomatization in a small universe of the type of finite types, as an appropriate higher inductive type of natural numbers and bijections.


page 1

page 2

page 3

page 4


Cubical informal type theory: the higher groupoid structure

Following a project of developing conventions and notations for informal...

Dialectica models of type theory

We present two Dialectica-like constructions for models of intensional M...

On Higher Inductive Types in Cubical Type Theory

Cubical type theory provides a constructive justification to certain asp...

The Integers as a Higher Inductive Type

We consider the problem of defining the integers in Homotopy Type Theory...

Jacobi polynomials and design theory I

In this paper, we introduce the notion of Jacobi polynomials with multip...

Clubs and their applications

Clubs of rank k are well-celebrated objects in finite geometries introdu...

Quotients of Bounded Natural Functors

The functorial structure of type constructors is the foundation for many...