# Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?

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!

• I think of sheafification and categorification as different things, but you seem to be conflating them. Is that so, or are you just referring to the fact that the cited work seems to do both? Jun 7, 2020 at 21:18
• @LSpice I've heard that a topos is a perculiar kind of category, and topos theory is just a a sytnhesis of the different kinds of sheaves (P. Cartier). So, I wouldn't consider my wording (i.e referring to construction of sheaves as "categorification") too conflating. No ? Jun 8, 2020 at 7:16
• @dohmatob: both “sheafification” and “categorification” have acquired rather specific meanings that are a bit different from just “applying the methods of sheaves/categories”. Calling this “sheafification” would sound entirely wrong to me (that means the construction turning a presheaf into a sheaf). Calling this “categorification” sounds much less wrong, but still a bit misleading (that can sometimes mean “reorganising with categorical methods”, but more usually means “replacing set-based structures with analogous category-based structures”). Jun 8, 2020 at 10:30
• @PeterLeFanuLumsdaine OK, thanks for the explanation. Indeed, "categorification" sounds "less wrong" here. Jun 8, 2020 at 10:36
• Probably related? It is a question of mine: mathoverflow.net/questions/357184/… Jun 8, 2020 at 15:25

I'm not an expert on the sheaf-theoretic approach to probability theory, but a quick look at the paper you're asking about shows that it's a 6 page conference proceeding from 2017 that defines a new notion and reads like a set of lecture notes. I think it's a bit early to be asking for big applications to probability theory. That said, if you use Google Scholar to search for who cites Alex Simpson's paper, you find two. The first is about constructive measure theory, so should be of interest to logicians and others wanting a firmer foundation for probability theory. The second is a PhD thesis that goes much more in depth than Simpson's paper about the properties of "probability sheaves" and investigates connections to topos theory.

From the abstract: "In this dissertation, we emphasize how sheaves and monads are important tools for thinking about modern statistical computing." The abstract goes on to advertise applications to hypothesis testing and the analysis of data sets with missing data, pretty important topics. Chapter 3 contains lots of history of previous attempts to bring probability theory under the umbrella of category theory, and also includes applications to probabilistic programming (whatever that is). If you're interested in this field, I think you will want to read these references (plus the blog post by Tao that Simpson cites), and you may need to give it time before super compelling applications arrive.

• Thanks for the detailed input and refs. Jun 8, 2020 at 13:29
• That dissertation looks interesting. A potential application might be that a more categorical approach to probability theory could lead to approaches to statistical computing which are more natural for functional programming, especially with languages like Haskell that are designed with category theory in mind. Jun 8, 2020 at 16:19

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.

• In the sprit of being "constructively curmudgeonly": does the talk explain whether the hope/claim is to refute the quote from Williams's book, or merely to say that Williams's admittedly sweeping claim should not be taken as gospel? Jul 31, 2020 at 3:42
• FWIW I don't think that a synthetic approach to RVs is impossible, but I would need more convincing that the subtleties of stochastic processes can also be handled this way. My inexpert intuition is that continuity/measurability issues for stochastic processes are akin to size issues with constructions in category theory, as discussed at mathoverflow.net/questions/365947/… -- the technical issues can be dealt with, but they are real, and work has to be done to handle them, just as the traditional approaches involve work Jul 31, 2020 at 4:01