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Stream: theory: categorical probability

Topic: Gromov's functorial probability theory

Fawzi Hreiki (Nov 17 2020 at 17:49):

In 2014 Gromov gave a very interesting series of lectures on probability theory (https://www.youtube.com/watch?v=aJAQVletzdY). The slides/notes are here Six-Lectures-on-Probabiliy-Symmetry-Linearity.pdf.

One of his main ideas is that probability theory needs a Grothendieck-style refoundation.
He compares Kolmogorov's approach to probability to Weil's approach to algebraic geometry in two ways:

• Both are "manifestly set theoretic".
• Both rely on an arbitrary absolute domain over which everything takes place (a given probability space in the case of Kolmogorov vs. a given field in the case of Weil).

Gromov states that probability spaces should instead be certain functors from the category $P$ = (finite probability measure spaces and reductions) to $\text{Set}$ akin to how schemes are just certain functors from commutative rings to sets.

There's a lot more there and it's worth watching the lectures (and reading through the notes). My knowledge of probability theory is practically nonexistent so I was curious if anyone knows whether this approach works and whether it's been pursued at all.

Reid Barton (Nov 17 2020 at 18:00):

This looks related to https://drops.dagstuhl.de/opus/volltexte/2017/8051/pdf/LIPIcs-CALCO-2017-1.pdf except I'm not sure what to make of the fact that (based on a quick skim of the notes) Gromov seems to consider covariant functors rather than contravariant ones.

Fawzi Hreiki (Nov 17 2020 at 18:05):

Well a scheme is a certain functor $\text{Aff}^\text{op} \rightarrow \text{Set}$, but $\text{Aff}$ is defined to be $\text{CRing}^\text{op}$ so Gromov's category $P$ plays the 'algebraic' role akin to the role of commutative rings in AG.

Reid Barton (Nov 17 2020 at 18:10):

But compared to Alex Simpson's notes, the category $P$ is the full subcategory of $\Omega$ on the finite probability spaces, right?

Reid Barton (Nov 17 2020 at 18:11):

I'm not sure how to see $P$ as on the algebra side rather than the geometry/spatial side. I guess I will need to take a closer look at what Gromov does.

Fawzi Hreiki (Nov 17 2020 at 18:18):

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Fawzi Hreiki (Nov 17 2020 at 18:23):

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Fawzi Hreiki (Nov 17 2020 at 18:27):

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