There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will probably have the best chance of an answer. Roughly the free loop space is a space which is supposed to be the space of maps from the circle to a given space X. It is usually denoted LX. This is all fine and good for topological spaces. You get a nice mapping space LX by equipping the set of maps with the compactly generated compact open topology. It even satisfies the adjunction:
$ Map(Y, LX) = Map( Y \times S^1, X)$
However things get complicated when we want to work with manifolds. The first thing is that we want something that represents smooth maps from the circle into the manifold X.
There are basically two approaches to making such an object precise, and I want to know how they compare.
The first approach actually tries to construct an actual space of smooth maps. Here you start with the set of smooth maps LX, and with analytical muscle you give it the structure of an infinite dimensional Fréchet manifold. When X = G is a Lie group, this is an inifinite dimensional Lie group and is the thing whose (projective) representations make an appearence in conformal field theory.
The second approach is to study the loop space as a generalized smooth space. What is a generalized smooth space, you ask? Well there was a lot of discussion about it at the N-category Cafe, here and here. Roughly LX is thought of as a kind of sheaf via the formula:
$LX(Y) = Maps(Y, LX) = Maps(Y \times S^1, X)$
In some models it must be a concrete sheaf (i.e. it has a underlying set of points and every map from it to anything else which is concrete is a particular set map. The technical details appear in the Baez-Hoffnung paper in the second link). Clearly this model has its own desirable properties.
How does the manifold version of loop space compare to the sheaf theoretic version?
Presumably the Fréchet manifold model gives a sheaf (since we can just map into it). Does this agree with the sheaf defined by the adjunction formula? If they are not the same sheaf, they do seem to have the same points, right? And I think there is a comparison map from the manifold LX to the sheaf LX, which should be helpful. Can anyone explain their relationship? How similar/different are they?
Best Answer
They are the same.
If you want the full gory details, read "A Convenient Setting of Global Analysis" by Kriegl and Michor (available as a free PDF via the AMS bookstore). For a gentler approach, read my online seminar notes "The Differential Topology of Loop Spaces". Michor has also written a fair bit on manifolds of mappings, and I've written the odd article on them as well.
But basically, the manifold structure on $LM$ is the "right" structure so that the exponential law:
$$ C^\infty(N \times S^1,M) \cong C^\infty(N, LM) $$
holds.
I just (today, but before you posted this) started smooth loop space with the intention of copying over some of my seminar notes and other oddments into the nLab. If there's anything you are particularly interested in, let me know and that'll give me some incentives to get on with the project!
PS I notice you picked loops as an example. For the more general case of the exponential law, you should read the Kriegl and Michor book. Basically, it works fine where the object being exponentiated is a compact manifold. For non-compact manifolds you need to use a different structure which is a little weirder.