Timeline for How are infinite-dimensional manifolds most commonly treated?
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| Apr 16, 2016 at 21:54 | comment | added | Dmitri Pavlov | @SaalHardali: Colimits of diffeological spaces are computed as colimits of smooth sets followed by the concretization functor (ncatlab.org/nlab/show/concrete+sheaf). The concretization functor destroys meaningful geometric information. In those examples where there is a difference between these two colimits, it is the colimit in smooth sets that has a more geometric flavor. The infinite jet bundle of a fiber bundle is a concrete sheaf (smooth set), hence it doesn't make any difference in which category to consider it. | |
| Apr 16, 2016 at 21:24 | comment | added | Saal Hardali | @DmitriPavlov This is a humble question: There must be some (co-)limits which are not preserved by the embedding of diffeological spaces into smooth sets. And if so I'd be inclined to believe that limits/colimits in the diffeological category are closer to "geometry". Does this qualify as an argument against working with general smooth sets or is it a techical point which doesn't affect anything in practice? For a pressing example: does the infinite jet bundle of a fibre bundle commute with this embedding? | |
| Feb 21, 2013 at 11:19 | comment | added | Dmitri Pavlov | @Urs: Doesn't Corollary 3.14 only talk about finite-dimensional vector spaces as domains? | |
| Feb 14, 2013 at 12:01 | comment | added | Urs Schreiber | Maybe a better reference for a proof, and of a more general statement, is cor. 3.14 in Kriegl-Michor ncatlab.org/nlab/show/The+convenient+setting+of+global+analysis which shows that a map between locally convex topological vector spaces is smooth precisely already if it sends smooth curves to smooth curves, hence it it is diffeological. | |
| Feb 10, 2013 at 0:14 | comment | added | Dmitri Pavlov | @André: Yes. For Fréchet manifolds, see Urs' answer below, where he cites a reference for this fact, namely a paper by Losik, which cites another paper by Losik (ams.org/mathscinet-getitem?mr=1213569), where this fact is actually proven. The latter paper also points out that for Banach manifolds a similar fact has been established by Hain, which can be seen from the following review of Hain's paper: ams.org/mathscinet-getitem?mr=539632. | |
| Feb 10, 2013 at 0:06 | comment | added | Urs Schreiber | The embedding of Frechet manifolds into diffeological spaces is stated in M. V. Losik, "Fréchet manifolds as diffeological spaces", Soviet. Math. 5 (1992) and reviewed in section 3 of M. V. Losik, "Categorical Differential Geometry", Cah. Topol. Géom. Différ. Catég., 35(4):274–290, 1994. (mat.univie.ac.at/~esiprpr/shadow/esi038.html) There is further discussion in appendix A.1 of Konrad Waldorf, "Transgression to Loop Spaces and its Inverse I" arxiv.org/abs/0911.3212 | |
| Feb 9, 2013 at 23:20 | comment | added | André Henriques | Banach and Fréchet manifolds are full subcategories? Are you sure? | |
| Feb 9, 2013 at 19:23 | comment | added | Urs Schreiber | I was waiting for you to say that. In an unpublished version of my previous comment I had a side remark saying "and if we say 'diffeological space' we may also say 'smooth space' ". But either way "smooth set" also redirects to the entry. Personally, I think it is a bad move to think that "space" means "oo-groupoid" by default, because 'topological spaces' represent oo-groupoids. The thing is that its really the homotopy type, not the space, that is the oo-groupoid. All around us 'space' refers to geometric spaces. I think it is a bad move to identify spaces with their homotopy types. | |
| Feb 9, 2013 at 18:59 | comment | added | Dmitri Pavlov | @Urs: I guess I should add that the term ‘smooth space’ makes me somewhat uncomfortable, because ‘space’ often means ‘∞-groupoid’ and ‘smooth space’ can be misunderstood as ‘smooth ∞-groupoid’. ‘Smooth set’ avoids this ambiguity. | |
| Feb 9, 2013 at 17:56 | comment | added | Dmitri Pavlov | @Urs: I replaced ‘manifold-like objects’ with ‘smooth sets’ and ‘smooth objects’. | |
| Feb 9, 2013 at 17:55 | history | edited | Dmitri Pavlov | CC BY-SA 3.0 |
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| Feb 9, 2013 at 17:25 | comment | added | Urs Schreiber | Dmitry, while I of course agree that (oo-)sheaves on the site of smooth manifolds is the right context for doing (higher) differential geometry, it is not good to say, as you do, that these are categories of "manifold-like objects" -- unless the term "manifold" loses all its intended meaning. The objects in these sheaf toposes have the only common property that they are characterized by smooth probes, but only very few of them are characterized as locally equivalent to a given model space V. Those that are are the V-manifolds (the "differential geometric V-schemes"). | |
| Feb 9, 2013 at 16:45 | comment | added | Todd Trimble | Also important are sheaves of sets on slightly larger sites which include smooth manifolds, such as the category of f.p. $C^\infty$ rings with appropriate topologies, as considered in synthetic differential geometry. This allows one to consider objects of nilpotent infinitesimals as representing jet-space functors; a comprehensive treatment is given in the book by Moerdijk and Reyes. See also the pioneering work of Urs Schreiber and collaborators throughout the nLab on the importance of cohesive sites for higher geometry, as in differential cohomology in $(\infty, 1)$-toposes. | |
| Feb 9, 2013 at 14:36 | history | answered | Dmitri Pavlov | CC BY-SA 3.0 |