Coherence and strictification for self-similarity

TL;DR

This paper explores the limitations and possibilities of strictification in self-similar structures within semi-monoidal categories, providing coherence results and a method to relate strict and non-strict forms.

Contribution

It introduces a coherence theorem for self-similarity and presents a strictification procedure linking strict and non-strict self-similar categories.

Findings

Strict self-similarity cannot coexist with strict associativity.

A coherence result for arrows exhibiting self-similarity is established.

A strictification procedure relates strict and non-strict self-similar categories.

Abstract

This paper studies questions of coherence and strictification related to self-similarity - the identity $S\cong S\otimes S$ in a (semi-)monoidal category. Based on Saavedra's theory of units, we first demonstrate that strict self-similarity cannot simultaneously occur with strict associativity -- i.e. no monoid may have a strictly associative (semi-)monoidal tensor, although many monoids have a semi-monoidal tensor associative up to isomorphism. We then give a simple coherence result for the arrows exhibiting self-similarity and use this to describe a `strictification procedure' that gives a semi-monoidal equivalence of categories relating strict and non-strict self-similarity, and hence monoid analogues of many categorical properties. Using this, we characterise a large class of diagrams (built from the canonical isomorphisms for the relevant tensors, together with the isomorphisms exhibiting the self-similarity) that are guaranteed to commute.