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In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory formulation of this correspondance. The morphisms associated to von Neumann algebras are simply $*$-homomorphisms. What are the morphisms associated with (localized) measured spaces?

I guess it is some set of equivalence classes of measurable functions, but I can't find any rigorous formulation for this.

Thanks for your help!

[EDIT:] I don't think this question is a duplicate since it is focused on the morphisms associated to measured spaces (those provided with measurable sets and sets of measure 0), not hyperstonian morphisms. I found an explicit answer here (Is there an introduction to probability theory from a structuralist/categorical perspective?): morphisms correspond to maps $(X,M,N)\rightarrow (Y,P,Q)$ such that the preimage of every element of $P$ is a union of an element of $M$ and a subset of an element of $N$ and the preimage of every element of $Q$ is a subset of an element of $N$

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  • $\begingroup$ If I understand the question correctly, then the paper "Two closed categories of filters" by Andreas Blass matwbn.icm.edu.pl/ksiazki/fm/fm94/fm94115.pdf discusses the category of filters which is a special case of the category which you want. $\endgroup$
    – Joseph Van Name
    Commented Sep 6, 2014 at 19:37
  • $\begingroup$ Also, in my new paper "Representations of algebras in varieties generated by infinite primal algebras", I consider a generalization of the category of filters mentioned in Andreas Blass' paper. This category can be used to characterize up-to-equivalence the pro-completion of the category of sets and more general categories. $\endgroup$
    – Joseph Van Name
    Commented Sep 6, 2014 at 19:43
  • $\begingroup$ Of course, I do not believe that we obtain an equivalence of categories between the categories mentioned above unless one restricts the category of measured spaces to an appropriate full subcategory. $\endgroup$
    – Joseph Van Name
    Commented Sep 6, 2014 at 19:49
  • $\begingroup$ Thanks for the comments. In my understanding, there is an equivalence with localized measured spaces. However, I only need to make a rigorous formulation of such a correspondance in a math section of a PhD in physics. Therefore a full understanding of the papers you pointed out goes way beyond my needs. Do you have an idea about what the explicit morphisms associated to localized measured spaces are? $\endgroup$
    – Issam Ibnouhsein
    Commented Sep 6, 2014 at 19:52
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    $\begingroup$ This question was asked before by me, you can find a statement there: mathoverflow.net/questions/23408/…. See also the link there, which explains morphisms of measurable spaces in more detail. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 7, 2014 at 11:35

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