Coherence for bicategories, lax functors, and shadows

TL;DR

This paper revisits Mac Lane's coherence proof for monoidal categories using the Grothendieck construction, extending it to bicategories with shadows and their functors, providing streamlined proofs and new coherence results.

Contribution

It introduces a generalized approach to coherence theorems for bicategories with shadows, building on Mac Lane's original proof and the Grothendieck construction.

Findings

Efficient proofs of standard coherence theorems

New coherence results for bicategories with shadows

Generalization of Mac Lane's proof technique

Abstract

Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.