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?

anotherway of defining loop spaces. He's put the details at the nLab page: ncatlab.org/nlab/show/smooth+loop+space I don't understand it all yet but looks interesting! – Loop Space Jan 22 '10 at 21:28