Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds

Question

Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:

Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.

I understand that this category $\text{Man}$ is not well behaved in more than one sense or do not have enough objects, for it to be

• closed under pullback,
• to have mapping space, an appropriate smooth structure on $\text{Map}(X,Y)$ for manifolds $X$ and $Y$.

Then, people added more spaces to the category of manifolds, in an attempt to make sure the resulting category has (some) of the nice properties which the category $\text{Man}$ did not had. Some examples are

1. Chen spaces (On the proof of "Mapping space is a Chen space"),
2. Differentiable spaces (I saw the first in the paper, section $2.7$) which are sheaves over the category $\text{Man}$ that are Differentiable stacks over the category $\text{Man}$ (recall that, any manifold is a sheaf over the category $\text{Man}$ that are Differentiable stacks over the category $\text{Man}$).
3. Frölicher spaces. These are introduced to have a Cartesian closed category (please correct me if I have misunderstood something).

Question : Are there any (What are the) results that hold in these generalised spaces whose counterparts does not hold true in the set up of smooth manifolds?

There is one result (Lemma $2.35$ in above paper) I am aware of that holds true for Differentiable spaces but there is no appropriate counterpart for smooth manifolds.

Sub questions :

1. It looks like diffeological spaces are introduced not to “enrich” (not sure if it is correct word) the category of manifolds, but actually to study sheaves on the category of manifolds. Is that correct? I am not sure to what extent this question is making sense, so feel free to ask for more clarification or ignore it.
2. I also observe similarity with the notion of “Algebriac spaces”. Those were also (roughly) defined (similar to Differentiable spaces) as sheaves of particular kind (over some appropriate site). I think there are more than a handful of results that holds true in Algebriac spaces but not in the category $\text{Sch}/S$. You can also add them, but I am not sure if I can appreciate them enough.

Answer 1

There are many such results.

Consider some smooth manifolds M and N. The internal hom Hom(M,N) is a sheaf on smooth manifolds. We can compute its tangent bundle, and it turns out that the tangent space at some point f in Hom(M,N), i.e., f:M→N is a smooth map, equals the vector space of smooth sections of the vector bundle f*TN. This is the expected result, but the setting of sheaves allows us to make it completely rigorous and precise with minimal technicalities.

Now take M=N and consider the open subobject of Hom(M,M) consisting of diffeomorphisms. This is a group object (i.e., an infinite-dimensional Lie group) and its Lie algebra is precisely the Lie algebra of vector fields on M.

Differential k-forms form a sheaf Ω^k on smooth manifolds. In particular, morphisms Hom(M,N)→Ω^k are differential k-forms on the infinite-dimensional space of smooth maps M→N. We also immediately obtain the de Rham complex on Hom(M,N) in the same manner, and it satisfies the expected properties.

Liekwise, we have a sheaf of groupoids B_∇(G) of principal G-bundles with connection. Maps Hom(M,N)→B_∇(G) are principal G-bundles with connection over the infinite-dimensional space of smooth maps M→N.

Hopkins and Freed compute the de Rham complex of B_∇(G), and it turns out to be the vector space of invariant polynomials on the Lie algebra of G.

This means, for instance, that you can immediately start computing Chern–Weil forms of principal G-bundles with connection on Hom(M,N), for example.

Now, we can also take G to be any group object in sheaves, such as, for example, the group Diff(M) of diffeomorphisms of M considered above. This immediately allows us to consider principal G-bundles with connection for such groups.

Other objects that can be encoded in this setting include the (higher) sheaves of bundle (n-1)-gerbes with connection and structure abelian Lie group A, denoted by B_∇^n(A). Morphisms M→B_∇^n(A) are precisely bundle (n-1)-gerbes with connection over M.

Now you can talk about bundle (n-1)-gerbes with connection over Hom(M,N). The Cheeger–Simons differential refinement of the Chern character in this language is a morphism B_∇(G)→B_∇^n(A), etc.

So in particular, not only de Rham cohomology, but also differential cohomology make sense in this framework.