TITLE: Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
QUESTION [8 upvotes]: This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following properties:

$\mathrm{GDiff}$ is the category of topological spaces with additional structure
$\mathrm{Diff}$ (the category of smooth manifolds) is a full subcategory of $\mathrm{GDiff}$.
$\mathrm{GDiff}$ is Cartesian-closed.
The natural functor  $\mathrm{GDiff} \to \mathrm{LocallyRingedSpace}$ is a full embedding.

Question: can such a category exist, or are these conditions inconsistent?
If the conditions are inconsistent, then you can deviate somewhat from property 2 (in fact, for the most part I just want to be able to talk about infinite-dimensional manifolds such as spaces of smooth mappings). On the other hand, any additional nice categorical properties are welcome as long as you keep properties 1 and 3.
I think that thanks to the 3rd property,  $\mathrm{GDiff}$ is as close as possible to the classical differential topology, since the most important concepts of differential topology are defined in terms of locally ringed spaces. In particular, there is a natural canonical functor $\mathrm{Tangent} \colon  \mathrm{GDiff}_{*} \to \mathrm{Vect}$ extending resp. functor for  $\mathrm{GDiff}$ (here $\mathrm{Vect}$ is the category of vector spaces over different fields, morphisms in it are pairs from a morphism of scalar fields and a morphism of vector spaces). When, as far as I understand, for diffeological spaces there is no standard definition of a tangent space at the moment and different definitions are being explored now (this is related to my other question about the existence of a full embedding of diffological spaces into locally ringed spaces)

REPLY [4 votes]: Since you said that you are interested in infinite-dimensional manifolds such as mapping spaces, I wouldn't give up to use diffeological spaces. Diffeological spaces satisfy your requirements 0,1,2, but probably not 3.
The reason why I write "probably" is that 3 is not satisfied for the naive way of mapping diffeological spaces to locally ringed spaces (see the comment by Simon Henry at Is the category of diffeological spaces a full subcategory of locally ringed spaces?); however, there might be another way which we currently don't know.
The naive way is equip a diffeological space $X$ with the D-topology and with the sheaf that associated to a D-open subset the set of smooth functions on $X$. This defines a functor that is faithful but in general not full.
My point is that the naive functor is full when restricted to a very general class of manifolds such as mapping spaces. This is because

the D-topology coincides with the usual topology on such mapping spaces (often called the Fréchet topology or Whitney $C^{\infty}$-topology). This follows from Lemma 4.13 and Theorem 5.14 in these Lecture Notes of Christoph Wockel: https://www.math.uni-hamburg.de/home/wockel/teaching/data/HigherStructures2013/hs.pdf

the smooth functions (in the diffeological sense, when the mapping space is equipped with the functional diffeology) are precisely the smooth functions (in the manifold sense); this is Theorem 7.6 (c) in above notes of Wockel.


Thus, viewing a mapping space as a diffeological space, and then using the naive construction of a locally ringed space, gives the same result as viewing the mapping space (the manifold) as a locally ringed space in the usual way - no information will be lost.