$1/Kl(D)$ is the comma category of the one element set in the Kleisli category of the distribution monad. There is mention of it here. The objects are probability distributions called states and the maps are just the Kleisli maps.
The thing I want to focus on is the following:
- $1/Kl(D)$ is a Copy-Delete category
- It also has a dagger
- The CD properties then have a set of axioms that are co-axioms, like product and unit
The CD objects with their co-axioms might form a bialgebra. I believe this is not Frobenius, but there may be an interaction. Does anyone know what form of interaction is satisfied by the CD-coCD objects? Is this a known bialgebra type? Does this known bialgebra type appear in any applied math context? For instance, what logic is supported by this known bialgebra?
Edit: I might have come up with the product definition here