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
- 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!