Title: Discrete homotopy groups of cubical sets
Abstract:
Cubical sets are cubical analogs of simplicial sets and carry a nice homotopy theory. They also play important roles in higher category theory and discrete homotopy theory of (un)directed graphs. However, the homotopy groups are only defined for a Kan complex rather than a general cubical set. In this talk, we introduce the discrete homotopy groups of a general cubical set in a purely combinatorial manner, which extends the discrete homotopy groups of an undirected graph.
We will show that the discrete homotopy groups of a cubical set provide a pure combinatorial description of the homotopy groups of its geometric realization. If the cubical set is closed under permutations of cubical coordinates, which is called quasisymmetric (and is not necessarily a Kan complex), we prove the Hurewicz Theorem for the discrete homotopy groups. As an application, we will prove the Hurewicz Theorem for directed graphs, generalizing the Hurewicz Theorem for undirected graphs as recently proved by Carranza and Kapulkin.
This is a joint work with Daisuke Kishimoto (Kyushu University, Kyushu, Japan).