I recently discovered an nlab articleon categorified probability theory. It features my old advisor as the central reference. I have been trying to put a finger on a concept I call “category update”. This “category update” is meant to capture the notion of Bayesian update when we are using a category as a model rather than a probability space. A typical update would be to add a morphism to a category, or to conclude that two morphism can be composed. Thus a category update is a functor . At this point, I don’t know what properties I want the functors to have as I only have an intuition of what “update” means. Regardless, I am going to assume that this “update” is a categorification of Bayesian inference. This assumption allows me to start by looking at a category of probability spaces as a place to begin categorifying Baye’s rule. Clearly, in a category of probability spaces, Bayes rule is a morphism and the question is whether or not it is the most general morphism. If it were, this would mean that the category of probability spaces is a category with probability spaces as objects and Bayesian inference as morphisms. Now looking at the nlab article, we see that Panangaden chooses conditional probability densities as morphisms which looks something like Bayes rule. Regardless, what i am interested in is n-ifying this by exactly one layer in that I want to exchange the probability spaces for categories. A question might be, once we have done this and we have some collection of categories with updates between them (functors of special nature) what can we say about this collection? For instance, it must be a category, but what rules will the morphisms obey? What, if any are the axioms of the category?
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