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By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict to separable predual as I think it would prevent the existence of finite products).

Of course, I don't really believe it is a topos. I would honestly be very surprised if it were but so far I can't find a clear-cut argument that would show that it is not a topos. Part of the problem is that products in this category are a bit tricky to understand.

On the other hand, it is not so far-fetched to suggest this, since this category has some topos-like properties: it's definitely a "category of spaces", it has a subobject classifier given by $\mathbb{C}^2$, subobjects of each object form a complete boolean algebra, it is extensive, and it might very well be a regular category (not so clear).

I don't think it is exact or cartesian closed (if it were any of these, that would make it a topos), but that's not something completely inconceivable either.

Does anyone have a clean argument to show that this category is not a topos ?

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    $\begingroup$ Counter-question (based on my own ignorance): Every abelian von Neumman algebra on a separable Hilbert space is isomorphic to $L^\infty(X,\mu)$ for some $\sigma$-finite measure $\mu$. Measurable spaces are not Cartesian closed (by a result of Aumann). Have I missed some subtlety? $\endgroup$
    – user1504
    Commented Feb 18, 2021 at 21:20
  • $\begingroup$ @user1504 : I'm not familiar with the result you mention, it highly depends on what category it applies too exactly. Do you have a reference ? The restriction to separable Hilbert space might also be a big problem, as far as I understand, the coproduct of abelian Von Neuman algebra already don't preserve this conditions. $\endgroup$
    – Simon Henry
    Commented Feb 18, 2021 at 21:24
  • $\begingroup$ This one: projecteuclid.org/journals/illinois-journal-of-mathematics/… I learned of it in this paper: arxiv.org/abs/1701.02547 $\endgroup$
    – user1504
    Commented Feb 18, 2021 at 21:26
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    $\begingroup$ @TimCampion: Monics and epics in commutative von Neumann algebras are precisely injective and surjective homomorphisms, which answers your question in the affirmative. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 18, 2021 at 22:04
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    $\begingroup$ @user1504: Properties like cartesian closedness only transfer along equivalences of categories. The correct formulation of this equivalence has only been written down recently in arxiv.org/abs/2005.05284, see also mathoverflow.net/questions/20740/…. In particular, the fact that the category is not cartesian closed does not follow from Aumann's result. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 18, 2021 at 22:27

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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.

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