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Recall that a morphism $f:C \to D$ in a category $\mathscr{C}$ is representable if for all maps $g:E \to D$ in $\mathscr{C},$ the pullback $C \times_{D} E$ exists.

Let now $\mathscr{C}$ be the category of smooth manifolds. Then any submersion is representable. Is the converse true? I have heard from various people that the converse is true, but the only reference I have found is David Metzler's preprint on the arXiv:

Topological and Smooth Stacks

However, the proof he gives there is not complete, for it assumes implicitly that if $M \times_N L$ is a pullback of manifolds, then the induced map $$M \times_N L \to M \times L$$ is a a smooth embedding. I do not see how this is automatic.

I do have a sketch of a proof that this map must be a topological embedding (using diffeological spaces) but is it necessarily an immersion? I would like to argue this using curves, however, this is difficult without knowledge of how to differentiate them in the pullback.

Does anyone have either have a proof or a counterexample for this statement?

flag	asked Aug 2 2011 at 17:27 David Carchedi 7,727●10●27

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I hadn't heard that usage of "representable" before. Where does it come from? I note that it clashes with the standard usage of "representable" in the case of Set-valued morphisms in CAT. – Tom Leinster Aug 3 2011 at 1:22

@Tom: I am sort of making up the usage. More correctly, I am borrowing it by considering the morphism as actually being a morphism of presheaves via Yoneda. – David Carchedi Aug 3 2011 at 4:44

Ah, I see where you're coming from. Thanks. – Tom Leinster Aug 3 2011 at 9:40

Indeed, I do not see why the proof of Lemma 71 in Metzler's preprint works. Let's look at an example: suppose $M=N=L=\mathbb{R}$ and the maps M->N and L->N are x^2 and y^3, respectively. Isn't the pullback just a line? More explicitly: suppose f and g are differentiable functions such that f^2=g^3. Clearly, f/g is continuous. But actually, it seems that it is differentiable. If so, the pullback does exist, but the induced map $M\times_N L\to M\times L$ is not a smooth embedding. – t3suji Aug 3 2011 at 14:30

1+, nice question. It asks for a categorical characterization of submersions in the category of smooth manifolds. – Martin Brandenburg Aug 5 2011 at 9:20

Is every representable map a submersion?

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