Theory and models of $(\infty,\omega)$-categories

Abstract

This thesis is divided into two parts. In the first part, we study models of $(\infty,\omega)$-categories. The main result is to establish a Quillen equivalence between Rezk's complete Segal $\Theta$-spaces and Verity's complicial sets. In the second part, we study the $(\infty,1)$-category corresponding to these two model structures, denoted by $(\infty,\omega)$-cat. Its connection with Rezk's complete Segal $\Theta$-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 $(\infty,\omega)$-categories. A special emphasis will be placed on the Grothendieck construction.