I’m working on using Markov categories with conditionals to represent nondeterministic behaviours. The following question popped up: given a Markov category with conditionals C, and a small category I, is it true that the diagram category Fun(I, C), with pointwise operations, has conditionals ?
I suspect it doesn’t hold in general.
The more specific example I am interested in is that of Fun_{cart}(Arity, BorelStoch), the category of strict cartesian functors from the category of arities (dual to that of finite sets/free cartesian in one generator) to the Markov category of standard Borel spaces and measurable Markov kernels.
It would be nice if it had conditionals, however I strongly suspect it does not…