Theory and models of (∞, ω)-categories

Overview & Synopsis

This thesis is divided into two parts. In the first part, we study models of (∞, ω)categories. The main result is to establish a Quillen equivalence between Rezk's complete Segal Θ-spaces and Verity's complicial sets. In the second part, we study the (∞, 1)-category corresponding to these two model structures, denoted by (∞, ω)-cat. Its connection with Rezk's complete Segal Θ-spaces allows us to use the globular language, while its connection with complicial sets gives us access to a fundamental operation, the Gray tensor product. The objective will be to implement standard categorical constructions in the context of (∞, ω)-categories. A special emphasis will be placed on the Grothendieck construction.