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An immersion of smooth manifolds is a smooth map whose Jacobian has full rank at each point in the source manifold.

Is there a notion of ``immersion'' for geometric morphisms of topoi which conservatively generalizes the usual notion of immersion for smooth manifolds (i.e. such that a map between smooth manifolds is an immersion if and only if the geometric morphism it induces on sheaf topoi is an immersion)?

Many properties about continuous maps of nice spaces have been conservatively generalized to properties of geometric morphisms of topoi (open, closed, proper, topological embedding, . . .), and I'm looking for a similar generalization for immersions of smooth manifolds. This may require an implicit generalization of the notion of smoothness to topoi, but my hope is that such a generalization has been done and is known (to people more knowledgeable about topoi than I).

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If by "sheaf topos" you mean the topos of sheaves on the manifold, then surely not – this topos doesn't know anything about the smooth structure of the manifold. But perhaps you could say something if you consider a different site... – Zhen Lin Dec 29 '12 at 17:57
I guess that part of what I'm asking is what extra structure is present on the sheaf topos which does capture the smooth structure? Certainly there is extra structure which does this, as the topos of O_X modules (where O_X here denotes the sheaf of smooth functions) is a category of diagrams within the sheaf topos. – Jesse Wolfson Dec 29 '12 at 19:08
You don't need the whole category of modules: there's a notion of locally ringed topos that neatly extends the notion of locally ringed space. However a morphism of locally ringed toposes is more than just a geometric morphism, just as a morphism of locally ringed spaces is more than just a continuous map. – Zhen Lin Dec 30 '12 at 0:38
This sounds promising. Is there a concise notion of smoothness for locally ringed topoi and a good notion of immersion between smooth topoi? – Jesse Wolfson Dec 30 '12 at 2:52
A locally ringed space is a manifold precisely if it is (second countable Hausdorff) locally isomorphic to Euclidean space with the usual sheaf of smooth functions. It completely and wholly captures the smooth structure, but somehow this is just a very sophisticated way of saying the same thing, rather than imparting any new insight... – Zhen Lin Dec 30 '12 at 12:12

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