Post #2516082

2026-04-17 15:58 UTC

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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@nilesjohnson@mathstodon.xyz 2026-04-17 15:59

One version of our main theorem can be explained in Pₓ, the free symmetric monoidal category on an invertible object x. It says that morphisms in Pₓ are characterized by the *parity* of how many instances of 8ₓ they have. In particular, the composite or sum of two figure eights is the identity! Our fantastic choice of notation expresses this fact as follows: 8ₓ∘8ₓ = 8ₓ+8ₓ = 1₀. So, this fact implies that any composite or sum of figure eights can be reduced to just *odd* or *even*. The main theorem says, moreover, that *every* morphism in Pₓ boils down to some composite or sum of figure eights. A little later in this thread I'll give some more precise (more comprehensible) versions of the same result. [Aside: I want to pause and note that this might sound familiar to some readers, because these facts have been known in some form or other for a *long time*. They are very well studied! Our paper has a "Relation to Literature" subsection that addresses some of this, and I'll make some further comments below, but this thread is mostly for people who haven't seen it before, or have seen it but would like to see an alternative explanation because it's neat.] (3/9)

Open ##2516083