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Matteo Capucci (he/him) (Sep 21 2020 at 07:50):A diffeological space is a concrete sheaf on the site of open subsets of Euclidean spaces, that is, a diffeology on the set X is a 'coherent' choice of subsets of Set(U, X) for every open subset U of an Euclidean space. The idea, I understand, is that the diffeology at U tells me the smooth mappings from U to X.
My question is very simple: why smooth? What's the part of the definition which imposes a 'smooth-like' behavior on the chosen mappings? It looks like any adjective would be equally (un)motivated.
In fact, the cat of diffeological spaces hosts many other cats of spaces, first of all arbitrary topological spaces!
fosco (Sep 21 2020 at 08:11):Matteo Capucci said:
A diffeological space is a concrete sheaf on the site of open subsets of Euclidean spaces, that is, a diffeology on the set X is a 'coherent' choice of subsets of Set(U, X) for every open subset U of an Euclidean space. The idea, I understand, is that the diffeology at U tells me the smooth mappings from U to X.
My question is very simple: why smooth?
One source of smoothness is in that "coherent", no? Morphisms between and are smooth maps between subsets of .
[...] the cat of diffeological spaces hosts many other cats of spaces, first of all arbitrary topological spaces!
What definition are you using? It does not: https://ncatlab.org/nlab/show/adjunction+between+topological+spaces+and+diffeological+spaces
Matteo Capucci (he/him) (Sep 21 2020 at 08:37):fosco said:
One source of smoothness is in that "coherent", no? Morphisms between and are smooth maps between subsets of .
Alright, I skimmed through the def and naively thought the morphisms were inclusions. Now it makes sense.
Matteo Capucci (he/him) (Sep 21 2020 at 08:39):fosco said:
What definition are you using? It does not: https://ncatlab.org/nlab/show/adjunction+between+topological+spaces+and+diffeological+spaces
Ok, it's not an embedding, still there exists such a thing as a 'continuous diffeology', whose smooth functions = continuous functions and that bugs me a lot
Reid Barton (Sep 21 2020 at 10:57):It's similar to how any set can be given an indiscrete topology where the continuous functions = all functions.
Although the category of topological spaces isn't actually a (quasi)topos, we could pretend it is one, or replace it by the category of "continuological spaces": concrete sheaves on the site of open subsets of Euclidean space and continuous maps.
Then, there's a map of sites which is the non-full embedding of (opens in Euclidean spaces + smooth maps) into (opens in Euclidean spaces + continuous maps) and it induces an adjunction of categories of concrete sheaves, of which the right adjoint is this "continuous diffeology" functor.
In general, right adjoints defined like this have a non-geometric flavor, which I think is this unease that you're feeling--it's easy to describe what the maps into of something are, but "as a space" it's sort of mysterious.
By contrast, the left adjoint is nicer: if you have a nice diffeological space which you understand as a particular colimit of representables, then the left adjoint takes it to the corresponding colimit of representables in "continuological spaces", which will again be some nice object.
Matteo Capucci (he/him) (Sep 21 2020 at 11:35):I guess the point is that while we are used to talk about different topological structures on the same set we are not used to talk about different smooth structures. In particular, smooth structures are unique in all cases one meets unless you're a differential topologist and you're explicitly hunting for exotic ones. So smoothness feels more like a property than a structure to me.
Reid Barton (Sep 21 2020 at 11:48):That's an interesting observation. In fact, not only do we have multiple smooth structures here on the same set, they're related by an inequality. That is, the "identity map" is smooth in one direction but not in the other direction. And like you say, this is more familiar in topology than in the smooth setting.
Actually, it can happen for manifolds in pretty simple situations, too. For instance if you take a map like which is a smooth homeomorphism but whose inverse is not smooth, then you can transfer the smooth structure along it to get a new smooth structure on the same space with the identity smooth in one direction, and not the other. But somehow I'm not used to thinking about this example as exhibiting two smooth structures on the same space.
Morgan Rogers (he/him) (Sep 21 2020 at 11:53):Probably because it's an injective homeomorphism, so at the topological level the open sets you end up with are the same.
John Baez (Sep 21 2020 at 15:42):Matteo Capucci said:
In particular, smooth structures are unique in all cases one meets unless you're a differential topologist and you're explicitly hunting for exotic ones. So smoothness feels more like a property than a structure to me.
In the traditional approach to smoothness you start with a topological space and make it into a smooth manifold by equipping it with a maximal atlas. So being a smooth manifold is clearly an extra structure you put on a topological space, not a property.
And this structure is hardly ever truly unique. For example, if apply a non-smooth homeomorphism to the real line, the usual smooth structure gets mapped to a different smooth structure! It's isomorphic but not by the identity map on the real line. So one doesn't need to go to exotic smooth manifolds to see that smoothness is a structure: the real line has uncountably many different such structures.
Reid Barton (Sep 21 2020 at 16:50):Right, I mentioned this above with the example, but in connection with this example, I somehow find it easy/natural to visualize the map of the graph of projecting down to the -axis, but I have more trouble imagining it as the identity map from ( with the standard smooth structure) to ( with a cubic scrunching-up at the origin), even though it's the same map up to isomorphism. On the other hand, I don't have any trouble with ( with the discrete topology) to (the usual ) or (the usual ) to ( with the indiscrete topology), and it's not clear to me how to account for this difference; could just be that I'm not a differential geometer.