Globular weak $\omega$-categories as models of a type theory

Abstract

We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak $\omega$-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories. Our main contribution is a formal proof that the models of this type theory correspond precisely to weak $\omega$-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak $\omega$-groupoids: Those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an $\omega$-category. This comparison is established by proving the initiality conjecture for the type theory CaTT, in a way which suggests the possible generalization to a nerve theorem for a certain class of dependent type theories

References in corpus (2)

• Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories
• On the homotopy groups of spheres in homotopy type theory

Cited by in corpus (2)

• Bicategorical type theory: semantics and syntax
• A Type Theory for Strictly Associative Infinity Categories