Modeling the homotopy theory of spaces via posets

Abstract

[en] The aim of this project is to study the basics of Quillen model structures as an essential tool in algebraic topology and abstract homotopy theory.

In the first part, we will focus on the necessary background on category theory and homotopy theory in order to understand the notion of model structure and some fundamental constructions and tools within this framework.

The second part will deal with particular examples of model structures. Namely, we will study Thomason's model structure on the category of small categories and how it relates to Kan-Quillen's model structure on simplicial sets via an equivalence of homotopy categories, providing a model for the homotopy theory of topological spaces.

Finally, we will describe how the category of partially ordered sets inherits this model structure, offering yet another model for the homotopy theory of spaces. Moreover, we will analyze the relation between this structure and $T_0$ Alexandroff spaces.