Globular weak ω-categories as models of a type theory

We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak ω-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 ω-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak ω-groupoids: those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an ω-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.