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Stream: learning: questions

Topic: Comodules in the category of convex sets and its relatives

fosco (May 04 2024 at 08:06):

Consider the following three monads:

• the finite distribution monad $D$, whose algebras are convex sets
• the finite subdistribution monad $D_\le$, where probabilities don't have to sum to 1, obtained from $D$ and from the exception monad via a distributive law. Algebras: "subconvex" sets.
• the unnormalized distribution monad, where "probability distributions" can sum to any nonneggative real number. Algebras: semimodules over the semiring $[0,\infty)$.

Each of these categories has a monoidal product, because the associated monad is suitably lax monoidal. The first is the tensor product of convex sets. The last one is the "tensor product" of semimodules over $[0,\infty)$, that behaves in many (but not all) respects as the tensor product of modules over rings.

I am now interested in three things:

1. Check whether the monoidal product of convex sets and subconvex sets preserves equalizers: this means, each $A\otimes-$ should preserve equalizers. Context is not very relevant, this is just a technical requirement to apply a theorem. I think I proved it for semivector spaces, but I lack intuition about the tensor of (sub)convex sets and I can't reproduce the argument.
2. Understand what is a comonoid in $\bf Cvx$ and $\bf subCvx$ with respect to their tensor products, and (for example) understand whether the free convex set on a set is a comonoid. I think the answer is yes to all three, for the same reason that $D$ and $D_\le$ being colax monoidal send comonoids (in Set) to comonoids in their EM-category. Am I correct or did I mix up some assumptions/direction of arrows?
3. Describe the monoidal product in the category of comodules for a fixed comonoid $K$ in $\bf Cvx$ and $\bf subCvx$.

Is there any resource where such a problem is already investigated?

fosco (May 04 2024 at 08:43):

The monoidal structure on comodules is dual to the one on modules for monoids: instead of coequalizing two actions, one equalizes two coactions: say $X\to X\otimes C$ and $Y\to C\otimes Y$ are comodules over the comonoid $C$, then their "tensor coproduct" is the equalizer of

$X\otimes Y \underset{X\otimes y}{\overset{x\otimes Y}\rightrightarrows} X\otimes C\otimes Y$