The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of locales of the form Y→X.
Is the same statement true for smooth manifolds instead of locales and submersions instead of continuous maps?
More precisely, consider the site of smooth manifolds equipped with its standard Grothendieck topology of surjective submersions.
Is the category of internal smooth manifolds in the Grothendieck topos of sheaves on a smooth manifold M equivalent to the category whose objects are surjective submersions of the form N→M and morphisms are commuting triangles?
Definitions tend to branch when we internalize them, so an answer to this question should include the correct definition of a smooth manifold in an arbitrary Grothendieck topos.
According to nLab, internal dualizable modules over the locale of real numbers in the category of sheaves on a smooth manifold M are precisely finite-dimensional vector bundles over M. This can be seen as a further motivation for the above question.
