Coherence and strictification for self-similarity

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.

Significant revisions from previous version: proofs simplified and based on Saavedra units & idempotent splitting, monoidal equivalences made explicit, expository sections significantly revised and shortened, notation and terminology revised and clarified, a clearer criterion for coherence given

References in corpus (3)

• A Characterization of Entropy in Terms of Information Loss
• The homology of Richard Thompson's group F
• Coherence of Associativity in Categories with Multiplication

Cited by in corpus (1)

• Identities in modular arithmetic from reversible coherence operations