Topological proofs of categorical coherence

TL;DR

This paper presents topological proofs of categorical coherence theorems, using CW complexes and combinatorial homotopy, offering new geometric insights into Mac Lane's coherence results.

Contribution

It introduces novel topological methods, including CW complex analysis and Morse theory, to prove coherence theorems in category theory, extending to higher categories.

Findings

Topological proof of coherence for categorified non-symmetric operads

A new geometric proof of Mac Lane's coherence theorem for symmetric monoidal categories

Morse theory shows the second proof is less general than the first

Abstract

We give a short topological proof of coherence for categorified non-symmetric operads by using the fact that the diagrams involved form the 1-skeleton of simply connected CW complexes. We also obtain a "one-step" topological proof of Mac Lane's coherence theorem for symmetric monoidal categories, as suggested by Kapranov in 1993. Our analysis is based on a notion of combinatorial homotopy, which we further study in the special case of polyhedral complexes, leading to a second geometrical proof of coherence which is very close to Mac Lane's original argument. We use Morse theory to show that this second method is (strictly) less general than the first. We provide a detailed analysis of how both methods allow us to deduce these two categorical coherence results and discuss possible generalizations to higher categories.