John Baez and James Dolan, 2023-08-10

A vector bundle of "subdimension n" is one where the fibers are vector spaces of dimension at most n. Should we think of these as generically having the largest possible dimension, or the lowest possible dimension?
We can treat commutative separable algebras over a commutative algebra A over Q as 'finite sets over Spec(A)'. How can we understand finite sets of cardinality N over Spec(A)? Writing N! for the symmetric group S_N, we can treat them as 2-rig maps from the category of N!-representations over Q to the category of A-modules. We can call these 'N!-torsors over Spec(A)'. (Recall that a '2-rig' over a field k is a symmetric monoidal cocomplete k-linear category, and a map of 2-rigs is a symmetric monoidal cocontinuous k-linear functor.)
We can try to tackle this using props. The prop for special commutative Frobenius monoids is the symmetric monoidal category of cospans of finite sets. This is Prop. 6.1 of
Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, arxiv.org/abs/...
For more on this whole series of conversations, go here:
math.ucr.edu/h...