Post #2516085

2026-04-17 16:00 UTC

As often happens, our main theorem is a little easier to state with some additional background. I'll add that now. In Pₓ, the free symmetric monoidal category on the invertible object x, the morphisms are generated as sums and composites of six basic morphisms, with the following parities: There are two de/cancel morphisms, and they have even parity: ηₓ: 0 → x'+x εₓ: x+x' → 0 Then, there are four basic braiding morphisms βₚ,ₛ: p+s → s+p where s and p are each either x or x'; each of these has odd parity. These parities follow from the parities of the "figure morphisms", 8, C, and H above. Then, parity for any other morphisms in Pₓ are computed from these: parity is additive on sums or composites of morphisms. Our main theorem says that any parallel morphisms with the same parity are equal. (5/9)

Replies (1)

@nilesjohnson@mathstodon.xyz 2026-04-17 16:00

With even more background, I can give an even easier statement of our main theorem, and finally an explanation of how it's proved. The required background involves a cute little category that we call the *Super Integers*. This is a symmetric monoidal category, Z, whose objects are the integers, and where each object has two automorphisms called "odd" and "even" or denoted ±1; that's the "super" part. There are no morphisms between non-equal objects. [Aside: yes, this name is too hip, but I've come to terms with it.] You can think of the Super Integers like the integers with "virtual permutations": it's symmetric monoidal, so you can make sums and permute summands, but each permutation is characterized only by its *sign*. (These generating objects could also be denoted ±1, but then I get confused by having the same notation for objects and morphisms, so I'll avoid that here!!) (6/9)

Open ##2516086