Strictification and non-strictification of monoidal categories

TL;DR

This paper surveys various methods for strictifying and non-strictifying monoidal categories, detailing their models, relationships, and connections to higher categorical structures and coherence theorems.

Contribution

It provides explicit models for (non-)strictification, shows their role as free constructions, and connects them to Power's coherence theorem and bicategorical Yoneda embedding.

Findings

Explicit models for strictification and non-strictification

These models form free-forgetful 2-adjunctions

Connection to Power's coherence theorem and bicategorical Yoneda embedding

Abstract

In this survey paper we give account of several approaches to the strictification and non-strictification of monoidal categories, which are constructions that turn a monoidal category into a (non-)strict one monoidally equivalent to the original category, and how they are related to analogous notions in higher categorical structures. We first provide explicit, elementary models for the (non-)strictification and show that these two constructions give the free (non-)strict monoidal category generated by a monoidal category. Moreover, we prove in detail that these two constructions are part of a pair of free-forgetful 2-adjunctions. We later show that these constructions can be recovered from Power's general coherence theorem for 2-monads. Lastly we describe another model for the strictification based on right-module endofunctors and provide a detailed, self-contained proof that this is a particular instance of strictification of bicategories via the bicategorical analogue of the Yoneda embedding.