Coherence for Skew-Monoidal Categories

TL;DR

This paper introduces a new coherence theorem for skew-monoidal categories, a variation of monoidal categories where certain laws are not isomorphisms, and formalizes the proof in Agda.

Contribution

It presents the first coherence theorem for skew-monoidal categories and formalizes the proof in a dependently typed language.

Findings

Coherence is characterized by uniqueness of normalizing rewrites.

Formal proof of the coherence theorem in Agda.

Extension of monoidal category theory to skew-monoidal categories.

Abstract

I motivate a variation (due to K. Szlach\'{a}nyi) of monoidal categories called skew-monoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the well-known monoidal case. In my (to my knowledge new) version, it becomes a statement of uniqueness of normalizing rewrites. I present a proof of this coherence theorem and also formalize it fully in the dependently typed programming language Agda.