Formalization of dependent type theory: The example of CaTT

by   Thibaut Benjamin, et al.

We present the type theory CaTT, originally introduced by Finster and Mimram to describe globular weak ω-categories, and we formalise this theory in the language of homotopy type theory. Most of the studies about this type theory assume that it is well-formed and satisfy the usual syntactic properties that dependent type theories enjoy, without being completely clear and thorough about what these properties are exactly. We use the formalisation that we provide to list and formally prove all of these meta-properties, thus filling a gap in the foundational aspect. We discuss the key aspects of the formalisation inherent to the theory CaTT, in particular that the absence of definitional equality greatly simplify the study, but also that specific side conditions are challenging to properly model. We present the formalisation in a way that not only handles the type theory CaTT but also all the related type theories that share the same structure, and in particular we show that this formalisation provides a proper ground to the study of the theory MCaTT which describes the globular, monoidal weak ω-categories. The article is accompanied by a development in the proof assistant Agda to actually check the formalisation that we present.


page 1

page 2

page 3

page 4


Globular weak ω-categories as models of a type theory

We study the dependent type theory CaTT, introduced by Finster and Mimra...

Graded Modal Dependent Type Theory

Graded type theories are an emerging paradigm for augmenting the reasoni...

External univalence for second-order generalized algebraic theories

Voevodsky's univalence axiom is often motivated as a realization of the ...

Monoidal weak omega-categories as models of a type theory

Weak ω-categories are notoriously difficult to define because of the ver...

Hybrid dynamical type theories for navigation

We present a hybrid dynamical type theory equipped with useful primitive...

A Type Theory for Strictly Associative Infinity Categories

Many definitions of weak and strict ∞-categories have been proposed. In ...

Staged Compilation with Two-Level Type Theory

The aim of staged compilation is to enable metaprogramming in a way such...

Please sign up or login with your details

Forgot password? Click here to reset