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Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on $W^\ast$-algebras in the topos language really nicely…).
I have to rush to the dinner now. But to remind myself, I have added cross-links between Boolean topos and measurable space and for the moment pointed to
for more. Simon Henry’s thesis will be out soon.
Have to rush now…
Simon Henry’s thesis is out now:
(it’s actually written in English, for the most part).
There seems to be an interesting line of development going on with Asgar Jamneshan teaming up with Terry Tao to formulate a category-theoretic pointless measure theory.
The topos aspect is not emphasized here, but it seems to have grown out of a post by Tao – Real analysis relative to a finite measure space.
In a talk at the Topos Institute – Topos theory and measurability – Jamneshan explains that he is using an internal language for some Boolean toposes, but can hide this via a “conditional set theory” (Slide 12).
I’ll add this paper to his page:
In it 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.
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