Homotopy groups of cubical sets

TL;DR

This paper develops and compares multiple definitions of homotopy groups for cubical sets, proving their equivalence and consistency with topological counterparts, and provides combinatorial proofs of classical theorems in this setting.

Contribution

It introduces four equivalent definitions of homotopy groups for cubical sets and proves their alignment with topological homotopy groups, along with combinatorial proofs of key theorems.

Findings

Four equivalent definitions of homotopy groups for cubical sets

Homotopy groups agree with topological homotopy groups via geometric realization

Combinatorial proofs of classical homotopy theorems

Abstract

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead's theorem. This is a companion paper to our "Cubical setting for discrete homotopy theory, revisited" in which we apply these results to study the homotopy theory of simple graphs.