A Note on "A Note on the (∞, n)−Category of Bordisms"

Violeta Borges Marques

Key information

Authors:

Violeta Borges Marques (Violeta Borges Marques)

Supervisors:

Gustavo Rui Gonçalves Fernandes de Oliveira Granja (Gustavo Rui Gonçalves Fernandes de Oliveira Granja)

Published in

June 4, 2021

Abstract

The $(\infty,n)-$category of cobordisms $\text{Bord}_n$ is one of the central objects of the Cobordism Hypothesis. In [Lur09b] a sketch of the definition of $\text{Bord}_n$ is given and in [CS19] an attempt is made at giving a rigorous and complete definition. The goal of this dissertation is to understand part of the latter work and propose some corrections. We begin by an overview of the machinery from Abstract Homotopy Theory needed to understand [CS19]. This includes basic properties of model categories, (simplicial) localizations and simplicially enriched categories. We then study a model for $(\infty,1)-$categories, Rezk spaces, and derive some results about these, present in [Hor15], by more explicit methods. We also study $n-$Rezk spaces, that are models for $(\infty,n)-$categories, and how to endow them with a symmetric monoidal structure. The last chapter is devoted to the study of $\text{Bord}_n$. We provide an alternative definition to that in [CS19] and prove that it is indeed an $n-$Rezk space. In particular, we prove that the simplicial sets that compose it are Kan complexes.