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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