Reconsidering MacLane: Coherence for associativity in the infinitary and untyped settings.

This talk is about MacLane's coherence theorem for associativity, and how, in certain extremal settings, the formal and informal statements of MacLane's theorem do not coincide. We describe why this is the case, and give a construction that produces an equivalent "well-behaved" version of the categories in question. Based on this, we give a monic-epic decomposition of MacLane's associativity substitution functor that has a close connection with the categorical theory of self-similarity (the identity S ≡ S ⊗ S). This in turn demonstrates strong unexpected interactions between coherence for associativity, coherence for self-similarity, and untyped monoidal structures such as C-monoids. As well as a coherence theorem we also give a strictification procedure for self-similarity, and describe its rather unusual interaction with the strictification procedure for associativity.

(No in-depth knowledge of category theory is assumed, beyond the very basic definitions. However, the talk does assume an interest in these topics!)