Timeline for Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 24, 2015 at 9:28 | comment | added | Oren Ben-Bassat | Yeah, but this embedding does not take product of manifold to the ordinary tensor product. If you put extra structure on the vector spaces, you can use a completed tensor product and then the product of manifolds and the coproduct of algebras will match up. | |
| Apr 22, 2015 at 17:18 | comment | added | Dmitri Pavlov | @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). | |
| Apr 21, 2015 at 21:16 | comment | added | Oren Ben-Bassat | 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 | |
| Oct 1, 2011 at 0:11 | comment | added | Todd Trimble | Dmitri, those are just some of the defects (I wasn't even thinking of failure of cartesian closure here). Anyway, interesting question! | |
| Sep 30, 2011 at 22:05 | comment | added | Dmitri Pavlov | @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. | |
| Sep 30, 2011 at 19:52 | comment | added | Todd Trimble | 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... | |
| Sep 30, 2011 at 16:43 | history | asked | Dmitri Pavlov | CC BY-SA 3.0 |