July 28, 2021

Topos Theory and Measurability

Posted by David Corfield

There was an interesting talk that took place at the Topos Institute recently – Topos theory and measurability – by Asgar Jamneshan, bringing category theory to bear on measure theory.

Jamneshan has been working with Terry Tao on this:

• Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration (arXiv:2010.00681)

The topos aspect is not emphasized in this paper, but it seems to have grown out of a post by Tao – Real analysis relative to a finite measure space – which did.

Jamneshan explains in the talk that he is using an internal language for some Boolean toposes, but can rewrite this via a “conditional set theory” (Slide 12). In another of his papers,

• Asgar Jamneshan, An uncountable Furstenberg-Zimmer structure theory (arXiv:2103.17167),

he writes:

we apply tools from topos theory as suggested … by Tao. We will not use topos theory directly nor define a sheaf anywhere in this paper. Instead, we use the closely related conditional analysis… A main advantage of conditional analysis is that it can be setup without much costs and understood with basic knowledge in measure theory and functional analysis…

For the readers familiar with topos theory and interested in the connection between ergodic theory and topos theory, we include several extensive remarks relating the conditional analysis of this paper to the internal discourse in Boolean Grothendieck sheaf topoi.

In the talk itself, after answering some questions, Jamneshan poses a question of his own for topos theorists (at 1:02:20). Perhaps a Café-goer has something to say.

Some other work on Boolean toposes and measure theory is mentioned on the nLab here.

Posted at July 28, 2021 11:44 AM UTC

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Re: Topos Theory and Measurability

Of course, there was a considerable discussion of category-theoretic treatments of probability following Mark Meckes’ post – A Categorical Look at Random Variables. There should surely be some common ground with the work here.

Posted by: David Corfield on July 29, 2021 10:34 AM | Permalink | Reply to this

Re: Topos Theory and Measurability

What’s “uncountable measure theory”? A version of measure theory where you strengthen the usual countable additivity condition?

Posted by: John Baez on August 3, 2021 3:56 AM | Permalink | Reply to this

Re: Topos Theory and Measurability

Perhaps best explained by the opening shot of

• Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration (arXiv:2010.00681)

In this paper we establish various foundational results about the measure theory (and also point set topology and functional analysis) of “uncountable” spaces: topological spaces that are not required to be separable or Polish, measurable spaces that are not required to be standard Borel, measure spaces that are not required to be standard Lebesgue, and $C^{\ast}$-algebras that are not required to be separable. In other work by us and other authors [16, 33, 32] we will use these results to establish various results in “uncountable” ergodic theory (in which the group $\Gamma$ acting on the system is not required to be countable), which in turn can be applied to various “uncountable” systems constructed using ultraproducts and similar devices to obtain combinatorial consequences.

This really could be a bridging two cultures moment, topos theory ends up useful to “problem-solving” Gowers-Tao-Green kind of combinatorics.

From

• Asgar Jamneshan, An uncountable Furstenberg-Zimmer structure theory (arXiv:2103.17167)

Jointly with Tao, we aim at establishing a Host-Kra-Ziegler structure theory for arbitrary (not necessarily countable) abelian group actions on arbitrary (not necessarily separable) probability algebras such as the action of ultraproducts of finite abelian groups on the probability algebra of Loeb probability spaces. The hope is that such a theory (i) may help to unify the Host-Kra-Ziegler structure theory with higher-order Fourier analysis, (ii) lead to new insight in the inverse theorem for the Gowers uniformity norms, and (iii) may establish analogous inverse theorems for a larger class of abelian groups. By proving an uncountable Moore-Schmidt theorem [27] and an uncountable Mackey-Zimmer theorem [28], first steps towards an uncountable Host-Kra-Ziegler structure theorem are accomplished by Tao and the author. The present paper aims to establish another part of this program by developing an uncountable Furstenberg-Zimmer structure theory.

Category theorists should be reaching out to help here.

Posted by: David Corfield on August 3, 2021 9:21 AM | Permalink | Reply to this

Re: Topos Theory and Measurability

I see Neel Krishnaswami had spotted the internal logic aspect a while ago here:

see Terry Tao’s notes on probability theory, which read to me as if he really just wants to work in the internal logic of the presheaf category on the category of probability spaces and extensions.

Posted by: David Corfield on August 5, 2021 10:09 AM | Permalink | Reply to this

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