Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds

Question

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:

0. $\mathrm{GDiff}$ is the category of topological spaces with additional structure
1. $\mathrm{Diff}$ (the category of smooth manifolds) is a full subcategory of $\mathrm{GDiff}$.
2. $\mathrm{GDiff}$ is Cartesian-closed.
3. 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)

Answer 1

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

1. 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

2. 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.

Answer 2

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.

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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.