### 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.

## 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.