TITLE: Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
QUESTION [14 upvotes]: Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with.
So, my understanding is that category theory and related fields of higher mathematics are meant to (A) better organise knowledge within a fields of mathematics (B) build powerful bridges between otherwise dislocated branches of mathematics (C) elevate concepts to a higher level of generality / abstraction so as to expose them to a more unified treatment (which can give one a better understanding of problems / questions in these fields, etc.).
I found Simpson - Probability sheaves and the Giry monad, in which a sheaf for probability theory was constructed. In my layman's quick reading, the manuscript doesn't seem to (1) tell anything new / non-trivial about probability theory (2) help in anyway organize concepts already present within probability theory. So my question is
Question

How does the construction of a sheaf for probability theory (see paper cited above) help probability theory? 
Has any such attempts (to "categorify" probability theory) been made before?

Thanks in advance for any enlightenment!

REPLY [4 votes]: Coincidentally, a day after you asked this question, Alex Simpson gave a nice talk (video, slides) where he gave a synthetic formulation of probability theory. In this formulation, random variables are a primitive notion, not maps from a sample space to a measurable space. Hence there's no need to keep track (or even mention) sample spaces at all. That's basically how probability theory was done informally, long before it was encoded in set theory. Several prominent mathematicians (Rota, Tao, Mumford) had suggested that such a reformulation of probability theory would be desirable.
I'd consider this an application of "categorical" probability theory, since I suspect that Simpson arrived at this axiomatisation via the categorical sheaf model he had constructed earlier.