This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or noncommutative monoids in some (symmetric) monoidal category. Tom Leinster gave a complete answer for the commutative case, which turned out to be quite different from what I expected, because conditions on the underlying symmetric monoidal category were too weak.

I want to revisit this question for the opposite category of the category of smooth manifolds, which is a full subcategory of the category of commutative real algebras. This time I want to consider only additive symmetric monoidal categories.

I have in mind the classical example of the category of affine schemes, whose opposite category is equivalent to the category of commutative monoids of the category of abelian groups equipped with the standard tensor product.

Perhaps the additive structure should be compatible with the symmetric monoidal structure, which is the case for the above example. I am somewhat hesitant to require the category to be preabelian or even abelian, but I would not mind an answer that uses one of these stronger conditions.

For the category of smooth manifolds I expect the relevant additive symmetric monoidal category to be a subcategory of the category of complete locally convex nuclear Hausdorff topological vector spaces, possibly equipped with some additional structures, with the monoidal structure probably coming from the nuclear tensor product.

I am also considering the option (communicated to me by Jeff Egger) that an involutive structure might be crucial in this setting, similarly to the case of C*-algebras and von Neumann algebras. Thus I am willing to replace monoids and categories with involutive monoids and involutive categories in the above description.

Is the category of smooth manifolds equivalent to the opposite category of the category of commutative (involutive?) monoids of some additive (involutive?) symmetric monoidal category?

I am also interested in the case of commutative C*-algebras and von Neumann algebras, as well as in the noncommutative case. Restricting to compact smooth manifolds is also possible.

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    $\begingroup$ How wedded are you to considering manifolds, as opposed to manifold-like structures? Part of the reason I'm asking is that the category of manifolds is not well-behaved (as a category), and a lot of technical work is easier to carry out in slightly expanded categories. There is in particular a notion of $C^\infty$-ring, which seems close to your concerns... $\endgroup$ – Todd Trimble Sep 30 '11 at 19:52
  • $\begingroup$ @Todd: I am not sure whether the absence of (co)limits and/or the failure of the category of smooth manifods to be cartesian closed is an issue, but I will be very happy to see a solution of this problem for any expanded category of smooth spaces. $\endgroup$ – Dmitri Pavlov Sep 30 '11 at 22:05
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    $\begingroup$ Dmitri, those are just some of the defects (I wasn't even thinking of failure of cartesian closure here). Anyway, interesting question! $\endgroup$ – Todd Trimble Oct 1 '11 at 0:11
  • $\begingroup$ I think that one could take CBorn_C, the category of complete, convex bornological vector spaces over the complex numbers with the projective tensor product. It is a quasi-abelian closed symmetric monoidal category. The category of manifolds is opposite to a full subcategory of the commutative monoids over CBorn_C. I have not written down all the details but something like this should be true. This provides an approach that uses bornological algebras and not C^{\infty} rings. We have been using this approach in analytic geometry, see for instance arxiv.org/abs/1502.01401 $\endgroup$ – Oren Ben-Bassat Apr 21 '15 at 21:16
  • $\begingroup$ @OrenBen-Bassat: This is interesting, but if you allow full subcategories, then you might as well take real vector spaces (smooth manifolds embed fully faithfully in the opposite category of real algebras). $\endgroup$ – Dmitri Pavlov Apr 22 '15 at 17:18

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