Categorical approach to probability?

[icantadd]

I am interested in, and try to understand category theory. At the same time, I am taking my first (real) probability class. I am wondering if there is a way to understand probability theory through categories, and more importantly, if so, would it be interesting? It seems that there would only be very trivial limits, i.e. only categories with monadic diagrams would have limits.

 

[Hurkyl]

You can do the usual translations, but I haven't seen anyone present probability theory in a purely category theoretic manner.

One little point is, in my opinion, I think there's merit in syntactically treating random variables as generalized elements.

 

[icantadd]

One little point is, in my opinion, I think there's merit in syntactically treating random variables as generalized elements.

What do you mean, syntactically treating random variables as generalized elements. Are you talking about discrete random variables, continuous random variables, or all being generalized in some way? If the latter, I would love to see this expounded upon, maybe if you could, some examples.