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New pages:
locally additive space: Something I’ve been musing on for a bit: inside all these “categories of smooth object” then we have the category of manifolds sitting as a nice subcategory, but that doesn’t give a very nice intrinsic definition of a “manifold”. By that I mean that suppose you knew a category of smooth spaces and took that as your starting point, could you figure out what manifolds were without knowing the answer in advance? “locally additive spaces” are an attempt to characterise manifolds intrinsically.
kinematic tangent space: Once out beyond the realm of finite dimensional manifolds, the various notions of tangent space start to diverge and so each acquires a name. kinematic refers to taking equivalence classes of curves. There’s a bit of an overlap here with some of the stuff on Frölicher spaces, but this applies to any (cartesian closed, cocomplete) category of smooth spaces.
Apart from a few little tweaks to do with wikilinks and entities, these were generated by my LaTeX-to-iTeX package. References and all.
Hmm, this “kinematic” is new terminology for the old notion of geometrical definition of a tangent space. In an elementary course of differential topology/geometry which I attended 23 years ago, we studied 3 definitions of tangent space, which were called
I have seen some of this (pretty intuitive) terminology in old books from time to time, but I can not point any off hand for sure (Hirsch ? Novikov ?). This is partly reflected at tangent bundle. There is also another definition by duality with cotangent space which itself can be defined in some situations via Kaehler differentials.
I got the name kinematic from Kriegl and Michor’s book, so I don’t claim originality. The point of needing the terminology is that all of those definitions of “tangent bundle” differ when one moves away from finite dimensional manifolds. The first two that you list correspond to “kinematic” and “operational” in KM, the third doesn’t make sense outside finite dimensional manifolds.
suppose you knew a category of smooth spaces and took that as your starting point, could you figure out what manifolds were without knowing the answer in advance?
See the section General abstract geometric definition at smooth manifold. When I wrote this I first made a dumb mistake which luckily was caught in discussion here. Now I think it should be okay, but there is plenty of room for improving the discussion.
I like the description at “general abstract geometric definition”. It feels as though there should be a way of tying this in effectively with the older description of generalized manifolds in terms of pseudogroups, as here. Thoughts?
Grothendieck wanted that Bourbaki has a much more general definition of a manifold which he envisioned, but the Bourbaki group has rejected his idea. Bourbaki also rejected Grothendieck’s idea to do 6-7 exposes in Bourbaki seminar on motives (imagine the size of this: historical FGA was 5 seminars and the write up had all the beautiful and deep Picard scheme, Quot scheme, descent theory, Hilbert scheme… treatments). Bourbaki said, it is too much to waste 6-7 seminars on somebody’s personal research.
Urs, that description is all very well but is orthogonal to the point of what I was writing about here. The description given in that link is constructive: you have a particular construction of “generalised smooth space” (sheaves on a site, or some fancier version) and then note that a particular type of sheaf coincides with the standard definition of a manifold (though I didn’t see anything about Hausdorff or paracompact in that, did I miss that?). Frankly, I am less than impressed! You assume too much and your eventual description is too weak to convince me that this is the “right” definition of a smooth manifold.
I’m starting (in this article) from the other side. We have a category of generalised smooth spaces, but we don’t know how it was made. So I can’t use anything about sheaves, because I don’t know that my category was built using sheaves.
Anyway, these aren’t meant to be “ground shaking” pages. They’re more little things I’ve been pondering and that I thought I’d get written down somewhere to try to clarify my thoughts. And in writing them down, I extended them a little: for example, I figured out how to show that the tangent space was a vector space: getting 2-dimensional stuff from 1-dimensional information.
Hi Andrew,
you write:
that description is all very well but is orthogonal to the point of what I was writing about here.
I didn’t mean to imply anything about your writing, by the way. I didn’t even have time to look at it until now. I just saw a question and replied with what I know about it. Only now do I realize that I replied to something that was probably meant as a purely rethorical question!?
You assume too much and your eventual description
Just so that we don’t misunderstand each other: this is not “my” description. This is the definition of “scheme” moved from the topos over to that that over . It’s stated explicitly for instance at the very end of Structured Spaces.
I am not sure how it “assumes too much”. There can’t be a way to characterize smooth manifolds without in some way mentioning their local model. You do this, too, on your page, in another way.
By the way, your definition reminds me of that of microlinear spaces.
Ah, okay. I over-reacted. Sorry.
I mean that it “assumes too much” because the model spaces (Euclidean spaces) are built in right from the start. If I started with, say, half-Euclidean spaces, presumably I’d get “manifolds with boundary”. So the eventual outcome is sensitive to the initial conditions. That’s what I meant. The approach I outlined doesn’t make any initial assumptions. The only “special” space is , which we use to define the tangent space. I then get the local models for free. So in the view at locally additive spaces, the fact that there are local models comes as a property not as part of the structure.
I guess what I’m trying to say is that to call something a manifold should mean that it behaves like a manifold, not that it happens to appear in our list of “known manifolds”. But what does “behaves like a manifold” mean? My proposal is that it be a space where the smooth and continuous tangent bundles agree.
Incidentally, my motivation for recording this is twofold, and neither of them is that I seriously want to rewrite differential topology text books! One is that when I visited Sheffield at the start of the summer then I gave a talk, and not having anything particular to talk about then I picked this as something that I’d been vaguely thinking about. The other is that it does simplify the construction of the manifold structure of mapping spaces considerably. We start by showing that the mapping space of a locally additive space is again a locally additive space, and then show that the tangent bundles work as expected, whereupon the manifold structure drops out without all the messy transition functions on charts. So rather than a local addition being a tool to construct the manifold structure, it shifts to central stage and conducts the whole orchestra.
Thanks for the link to microlinear spaces. That’s a neat example of why putting stuff here is a Good Thing: I wouldn’t have made that connection myself.
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