Post #2516089

2026-04-17 16:08 UTC

We put a bunch of examples in the last section of our paper, starting with some of those figure morphisms and gradually building up to more complex examples. Here, I'll just give the final one, because it illustrates a diagram that *doesn't* (generally) commute, but looks at first like it ought to. To start, suppose a is an invertible object in a symmetric monoidal category A, with inverse a'. Then there is a conjugation functor Gₐ: A → A given by z ↦ zᵃ = a' + z + a You can show (using our coherence stuff) that this is a symmetric monoidal functor. Furthermore, you can show (again using coherence) that Gₐ is isomorphic to the identity on A. So, this is a categorification of the fact that conjugation in an abelian group is the identity homomorphism. Of course, conjugation by a' is also a symmetric monoidal functor, and also isomorphic to the identity. The example gets going when you realize that there is a natural isomorphism between these two, with components given by an isomorphism a' + z + a ≅ a + z + a' permuting the summands by a (1 3) permutation. So, is this a *monoidal* natural isomorphism? How could it not be??! (9/11; there are two bonus posts!)

Replies (1)

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

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)

Open ##2516090