Coherence and strictification for self-similarity

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 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, and give a simple intuitive interpretation of this characterisation.