#### Title

Local Higher Category Theory

#### Degree

Doctor of Philosophy

Mathematics

Jardine, J.F.

#### Abstract

The purpose of this thesis is to give presheaf-theoretic versions of three of the main extant models of higher category theory: the Joyal, Rezk and Bergner model structures. The construction of these model structures takes up Chapters 2, 3 and 4 of the thesis, respectively. In each of the model structures, the weak equivalences are local or ‘stalkwise’ weak equivalences. In addition, it is shown that certain Quillen equivalences between the aforementioned models of higher category theory extend to Quillen equivalences between the various models of local higher category theory.

Throughout, a number of features of local higher category theory are explored. For instance, at the end of Chapter 3, descent for the local Joyal model structure is given a simple characterization in terms of descent for the Jardine model structure. Interestingly enough, this characterization is a consequence of the Quillen equivalence between the local complete Segal model structure and the local Joyal model structure.

The right properness of the local Bergner model structures means that one has an attractive theory of cocycles and torsors. This theory can be interpreted as defining non-abelian $H^{1}$ with coefficients in an $\infty$-groupoid, generalizing the results found in \cite[Chapter 9]{local}. This is explored in the last section of Chapter 4.

Throughout, a number of features of local higher category are explored. For instance, at the end of the Chapter 3, descent for the local Joyal model structure is given a simple characterization in terms of descent for the Jardine model structure. Interestingly enough, this characterization is a consequence of the Quillen equivalence between the local complete Segal model structure and the local Joyal model structure.

The right properness of the local Bergner model structures means that one has an attractive theory of cocycles and torsors. This theory can be interpreted as defining non-abelian $H^{1}$ with coefficients in a $\infty$-groupoid, generalizing the results found in Jardine's `Local Homotopy Theory.' This is explored in the last section of Chapter 4.

COinS