Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds 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)
 A: Sikorski spaces, also known as differential modules, form the largest full subcategory of topological spaces equipped with subsheaves of real-valued continuous functions that satisfy 1. and are closed under composition with multi-variable smooth real-valued functions. Moreover, they are a full subcategory of locally ringed spaces (see below for proof).
According to Andrew Stacey's article Comparative smootheology in Theory and Applications of Categories, Vol. 25, 2011, No. 4, pp 64-117, these include as full subcategories so-called Smith spaces and Frolicher spaces. The latter form a cartesian closed category, and hence is a full subcategory of locally ringed spaces. In more detail, Frolicher spaces consists of sets equipped with the structure of smooth curves (functions form $\mathbb R$) and smooth functionals (functions to $\mathbb R$), satisfying the condition that each determines the other: smooth functionals are precisely the ones sending smooth curves to smooth functions on $\mathbb R$, and the smooth curves are precisely the ones sending smooth functionals to smooth functions on $\mathbb R$
The two full functors to Sikorski spaces in Stacey's article are obtained by giving the finest, resp. coarsest topology on the space rendering the curves, resp. functionals, continuous.

For the proof (as requested): since addition, multiplication, and constant functions of real numbers are smooth, the structure subsheaf of continuous real-valued functions of a Sikorski space has to be a sheaf of subrings (in fact of $\mathbb R$-algebras), so Sikorski spaces are ringed spaces. To see they are locally ringed, it suffices to show a germ of $f$ at a point $p$ is invertible if and only if it is not in the kernel of evaluation at $p$. But  $f(p)\neq0$ implies for any $\epsilon<|f(p|$ there is a smooth $g\colon\mathbb R\to\mathbb R$ such that $g|_U(x)=\frac1x$ for $U=(f(p)-\epsilon,f(p)+\epsilon)$. Consequently,  $g\circ f$ is the multiplicative inverse of $f$ on $f^{-1}(U)$.
That they're a full subcategory follows from the same argument as https://math.stackexchange.com/a/511604/400. Namely, a priori the components of a morphism between Sikorski spaces as locally ringed spaces consists are ring homomorphisms sending $s\colon U\to\mathbb R$ for $U\subseteq Y$ to some $s'\colon f^{-1}(U)\to\mathbb R$ so that for each point $p\in X$, the induced map on germs $s_{f(p)}\mapsto s'_p$ maps the maximal ideal of the stalk at $f(p)$ to the maximal idea of the stalk ot $p$. The induced map on the residue fields then sends $s(f(p))\mapsto s'(p)$. But the residue fields being isomorphic to $\mathbb R$, which has no ring endomorphisms ($a\leq b$ is equivalent to $b-a=c^2$, so implies $f(b)-f(a)=f(c)^2$, which is equivalent to $f(a)\leq f(b)$, so any endomorphism is order-preserving, hence the identity because $\mathbb Q$ is dense in $\mathbb R$), so we get $s(f(p))=s'(p)$.
Thus the components of every morphism of local rings between Sikorski spaces are given by $s\mapsto s\circ f$ for $f\colon X\to Y$ the underlying map on topological spaces, i.e. is a morphism of Sikroski spaces.
A: 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.
