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).