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

TL;DR

This thesis establishes a deep connection between different models of $( abla, ext{omega})$-categories, linking Rezk's Segal spaces and Verity's complicial sets, and explores their categorical structures and operations.

Contribution

It proves a Quillen equivalence between two models of $( abla, ext{omega})$-categories and investigates their associated $( abla, ext{omega})$-category, enabling categorical constructions like the Grothendieck construction.

Findings

Quillen equivalence between Rezk's Segal $ heta$-spaces and Verity's complicial sets

Development of the $( abla, ext{omega})$-category framework

Implementation of categorical constructions such as the Grothendieck construction

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.