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 varieties) 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 studied now (this is related to my other question about the existence of a full embedding of diffological spaces into locally ringed spaces)

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)

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 varieties) 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 studied now (this is related to my other question about the existence of a complete embedding of diffological spaces into locally ringed spaces)

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 varieties) 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 studied now (this is related to my other question about the existence of a full embedding of diffological spaces into locally ringed spaces)