Erland Arctaedius: Grothendieck's Homotopy Hypothesis and The Homotopy Theory Of Homotopy Theories

Abstract:
We will present two possible models for “infinity-categories”: simplicial set with a horn-filling condition and Kan-complex enriched categories. We present Grothendieck's homotopy hypothesis as a “litmus test” for infinity-categories, and then develop the necessary machinery for explaining the phrase “homotopy theory of homotopy theories”. We also define the maximal Kan-complex contained in a quasi-category, a generalization of the maximal groupoid contained in a category, and prove that it is an adjoint - we believe that this has not been done explicitly (in print) before.