Ivan Kobe: (ω,1)-categories vs. topological categories

Abstract: We are going to specialize the definition of Grothendieck-Maltsiniotis $(\omega,n)$-categories to the case $n = 1$ and take a closer look at the structure of $(\omega,1)$-categories. In particular, we will derive a composition operation which is unital and associative up to homotopy, as well as homotopical inverses of cells of dimension $> 1$.

Next, we will describe the close relationship between $(\omega,1)$-categories and categories enriched in topological spaces, which takes the form of a realization $\dashv$ nerve construction. The homotopically relevant data of a topological category (i.e. its homotopy category and the homotopy gorups of its hom-spaces) can be reconstructed from its nerve, showing that the adjunction offers a good comparison between the two models of $\infty$-categories.

Joint work with Bastiaan Cnossen.