Post #1421413

2026-04-17 15:57 UTC

Nick Gurski and I have a new paper out, about another coherence problem! Invertibility and parity in symmetric monoidal categories https://arxiv.org/abs/2604.15142 For someone who doesn&#39;t enjoy coherence theorems, I seem to spend a lot of time on them. This one is about coherence for *invertible* objects in a symmetric monoidal category, using an invariant that we call *parity*. In the thread below, I&#39;ll explain - what our main results say, in a few different ways, - what technology we use to prove them (spoiler: it&#39;s 2-monads, again), and - a few different examples, including one that is nontrivial. [Note: The attached picture is a crop of the cover art by Pablo Delcan for Jeff VanderMeer&#39;s book Annihilation. The book is sort of related to coherence, in a non-mathematical sense, and anyway I liked it. That boar looks like it knows a thing or two about invertible objects.] (1/9)

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@nilesjohnson@mathstodon.xyz 2026-04-17 15:58

To explain the main ideas in our paper, consider a symmetric monoidal category (A,+,0,β). So, the monoidal sum is denoted +, the monoidal unit is denoted 0, and the braiding (a.k.a. symmetry) is denoted β. We assume that the unit and associativity isomorphisms are identities, so the monoidal structure is strict. An invertible object x in A has a weak inverse, x', with morphisms ε:x+x' ≅ 0 (cancel) and η:0 ≅ x'+x (decancel) satisfying triangle identities that make the functors x+(-) and x'+(-) adjoint inverse equivalences. Using the braiding, β, each invertible x gives us an automorphism of the unit 0 -η-> x'+x -β-> x+x' -ε-> 0 This composite is sometimes called the _trace_ of 1ₓ or the _Euler characteristic_ of x. We call it the _figure eight on x_ and write 8ₓ because the string diagram looks like a figure eight. (2/9)

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