The Homotopy Hypothesis Via Kan Complexes
A final project I wrote in my junior year as an undergrad for my homotopy theory class (Math 215B at UC Berkeley). The paper begins with a brief introduction to the basics of model categories and Quillen equivalences, followed by a review of the foundations of simplicial sets and Kan complexes. Finally, I use the Kan-Quillen model structure on simplicial sets to prove the homotopy hypothesis – that is, that the homotopy category of topological spaces is equivalent to the homotopy category of simplicial sets, and hence the infinity groupoid of a space captures all of its topological structure.