Post #2516090

2026-04-17 16:10 UTC

Continuing the previous post, here is the monoidal naturality diagram for two objects z and w: Checking the a-parity, one composite is even but the other is odd. So, the two composites around the diagram are not generally equal. In particular, they are not equal when z and w are the unit object, 0, and a is a free invertible generator. The paragraph after the diagram gives this explanation: conjugation by a and a' are both symmetric monoidal functors, and are both monoidal naturally isomorphic to the identity. So, they are monoidal naturally isomorphic to each other, but the (1 3) permutation above is *not* that isomorphism. Instead, that isomorphism factors through the identity functor, so it involves just de/cancellation morphisms with no permutations of the object a past its inverse a'. I think that makes sense in retrospect, but also could be a source of confusion. (It definitely was for me!! One day while we were working on this I sent Nick a sequence of increasingly frantic/confused emails, followed the next morning by a long explanation of how useful it is to get a good night sleep.) (10/11)

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@nilesjohnson@mathstodon.xyz 2026-04-17 16:10

Finally, I want to conclude with a mention of the related literature. Monoidal categories where every morphism and every object is assumed to be invertible are sometimes called *2-groups*. When the monoidal structure is symmetric, they're called *symmetric 2-groups* or *Picard categories*. These have been studied *a lot*, for a long time. There are various coherence theorems in the literature, for both the non-symmetric and symmetric cases. Highlights include work of Laplaza [1], Baez-Lauda [2], Kelly-Laplaza[3], and Dugger [4]. (More detail in our "Relation to literature" subsection.) So, why do we need another version some decade(s) later?? Well, one honest reason is that we had a hard time understanding the older versions. Even the more recent ones depend crucially on the early Laplaza and Kelly-Laplaza work. We tried to explain them in a way we could understand, and wound up with the independent (2-monadic) approach I mentioned above. So here we are. Yes, serious people have known all about the essential computational facts for decades, but our version adds a nontrivial and (we think) useful perspective! [1]: Laplaza, Coherence for categories with group structure: An alternative approach (1983) https://dx.doi.org/10.1016/0021-8693(83)90081-9 [2]: Baez-Lauda, HDA V: 2-groups (2004) http://tac.mta.ca/tac/volumes/12/14/12-14abs.html [3]: Kelly-Laplaza, Coherence for compact closed categories (1980) https://dx.doi.org/10.1016/0022-4049(80)90101-2 [4]: Dugger, Coherence for invertible objects and multigraded homotopy rings (2014) https://dx.doi.org/10.2140/agt.2014.14.1055 (11/11)

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