Homotopical realizations of infinity Groupoids

Abstract

[en] Grothendieck’s homotopy hypothesis asserts that the study of homotopy types of topological spaces is equivalent to the study of $\infty$-groupoids, illustrating how important ideas in higher category theory stem from basic homotopical concepts. In practice there are distinct models for $\infty$-groupoids, and providing a proof of the homotopy hypothesis is a test for the suitability of any such model. In this thesis, we give a proof of the homotopy hypothesis using topological categories (i.e., categories enriched over topological spaces) as models for $\infty$-groupoids. In the same context, we propose a manageable model for the fundamental $\infty$-groupoids of a topological space.