Measure theory over boolean toposes

TL;DR

This paper develops a measure theory framework over boolean toposes, paralleling noncommutative measure theory, and establishes a topos-theoretic analogue of the modular time evolution of von Neumann algebras.

Contribution

It introduces a topos-theoretic version of modular time evolution, linking topos theory with noncommutative geometry and von Neumann algebra concepts.

Findings

Established a topos-theoretic modular time evolution

Constructed a canonical R+*-principal bundle over boolean toposes

Bridged topos theory with noncommutative measure theory

Abstract

In this paper we develop a notion of measure theory over boolean toposes which is analogous to noncommutative measure theory, i.e. to the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebra which take the form of a canonical R+*-principal bundle over any integrable locally separated boolean topos.