symmetric monoidal (∞,1)-category of spectra
A von Neumann algebra that is commutative.
The category of commutative von Neumann algebras is a full subcategory of that of von Neumann algebras and has many special properties.
The following five categories are equivalent:
This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.
The opposite category of commutative von Neumann algebras is not a topos because categorical products with a fixed object do not always preserve small colimits. See Theorem 6.4 in Andre Kornell’s Quantum Collections.
The opposite category of commutative von Neumann algebras admits a non-cartesian closed monoidal structure, where the monoidal product corresponds to the spatial product of measurable spaces.
Sur certains espaces considérés par M. H. Stone. Summa Brasiliensis Mathematicae 2 (1951), 151–182. PDF.
Equivalences of measure spaces. American Journal of Mathematics 73:2 (1951), 275–313. doi:10.2307/2372178.
Gelfand-type duality for commutative von Neumann algebras. 2005.05284
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