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

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