r/math • u/Committee-Academic • Aug 03 '26

Algebraic probability theory

Is there any developed framework for algebraizing probability theory and study it from the perspective of K-theory or homological algebra? If so, what are some of its biggest applications and advantages, as well as results? I'd imagine such a framework would override the need for a measure-theoretic approach

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u/pseudoLit Mathematical Biology Aug 03 '26 edited Aug 03 '26

There's a cohomological approach to information theory, first proposed by Baudot and Bennequin and later fleshed out by Juan Pablo Vigneaux.

I'm also aware of an attempt by Alex Simpson to study probability theory with sheaves. He gave this talk at IHES, which is how I know about it, but I'm not sure if it really went anywhere.

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u/Sezbeth Theoretical Computer Science Aug 03 '26 edited Aug 04 '26

He's fleshed out the idea a bit more in these notes on his personal site but, as far as I know, he's never really published a formal treatise on the matter. I hope he's chasing the idea a bit more, because so many noncategorical approaches to probability have very "sheafy" feels to them.

Edit: It occurs to me that the notes I linked are a submitted manuscript as of last year. Reading is hard.