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2 product -> limit and coproduct -> colimit

If I may be forgiven for climbing on my soap box ...

This kind of thing highlights one of the problems with these "one sided" approaches to the theory of smooth spaces. If one takes smooth functions out, then the resulting functor, Manifolds → Functions, does not preserve products limits (as Martin says). But if one takes smooth functions in then the resulting functor does not preserve coproductscolimits. The resolution is to take both smooth functions in and out, whereupon one does get a functor that preserves both coproducts colimits and productslimits. The resulting category is Hausdorff Frölicher spaces.

Taking functions both in and out also follows in the footsteps of the study of manifolds. I asked a bunch of topologists not long ago whether a chart was a map to a manifold or from it. It was about a 50-50 split! At the least, this shows that the direction of charts is ambiguous. At best, it shows that it's useful to have access to both directions.

So the answer to the question is as follows: the smooth functions $M \times N \to \mathbb{R}$ are those functions $f \colon M \times N \to \mathbb{R}$ with the property that $f \circ \gamma \in C^\infty(\mathbb{R},\mathbb{R})$ whenever $\gamma \colon \mathbb{R} \to M \times N$ is such that $(g,h) \circ \gamma \in C^\infty(\mathbb{R},\mathbb{R}^2)$ for all $g \in C^\infty(M)$ and $h \in C^\infty(N)$. This can probably be "sheafified" but I'm not a farmer.

Obligatory nlab link: for more on $C^\infty$-rings and the like: take a look at smooth algebra.

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If I may be forgiven for climbing on my soap box ...

This kind of thing highlights one of the problems with these "one sided" approaches to the theory of smooth spaces. If one takes smooth functions out, then the resulting functor, Manifolds → Functions, does not preserve products (as Martin says). But if one takes smooth functions in then the resulting functor does not preserve coproducts. The resolution is to take both smooth functions in and out, whereupon one does get a functor that preserves both coproducts and products. The resulting category is Hausdorff Frölicher spaces.

Taking functions both in and out also follows in the footsteps of the study of manifolds. I asked a bunch of topologists not long ago whether a chart was a map to a manifold or from it. It was about a 50-50 split! At the least, this shows that the direction of charts is ambiguous. At best, it shows that it's useful to have access to both directions.

So the answer to the question is as follows: the smooth functions $M \times N \to \mathbb{R}$ are those functions $f \colon M \times N \to \mathbb{R}$ with the property that $f \circ \gamma \in C^\infty(\mathbb{R},\mathbb{R})$ whenever $\gamma \colon \mathbb{R} \to M \times N$ is such that $(g,h) \circ \gamma \in C^\infty(\mathbb{R},\mathbb{R}^2)$ for all $g \in C^\infty(M)$ and $h \in C^\infty(N)$. This can probably be "sheafified" but I'm not a farmer.

Obligatory nlab link: for more on $C^\infty$-rings and the like: take a look at smooth algebra.