A premanifold is a locally ringed space locally isomorphic to an open subset of Euclidean space equipped with its sheaf of smooth functions. No assumption of paracompactness or the Hausdorff property.
What are the regular monomorphisms in this category? What are the regular epimorphisms? What are the universally regular epimorphisms - are they precisely the surjective submersions?
Does anything change for manifolds?