Categories of manifolds (possibly with extra structure) tend not to have all colimits. Other questions have addressed when colimits of manifolds exist.

I'd like to know what we can say in general about those colimits which do happen to exist. In particular: given a colimit which does exist in some category $C$ of manifolds, I'd like to know whether it necessarily coincides with the underlying colimit in $\mathsf{Top}$ (and hence also $\mathsf{Set}$).

Hopefully, there is a general principle that I'm missing which answers this for many sensible categories of manifolds. But for the sake of a sharp question: does the forgetful functor $C \to \mathsf{Top}$ preserve colimits, for

- $C = \mathsf{Man}$, the category of topological manifolds and continuous functions [*]?
- $C = \mathsf{Diff}$, the category of smooth manifolds and smooth functions?
- $C = \mathsf{Rm}$, the category of smooth Riemannian manifolds and smooth local isometries?

Or maybe the functor $\mathsf{Man} \to \mathsf{Top}$ fails to preserve colimits, while forgetful $\mathsf{Diff} \to \mathsf{Man}$ or $\mathsf{Rm} \to \mathsf{Diff}$ does preserve colimits?

[*] Since being a topological manifold is a property of, rather than a structure on, a topological space, (1) isn't so much a forgetful functor as an inclusion.

I don't see any obvious right adjoints to these forgetful functors, although I'd be happy to learn otherwise.

A comment under this question suggests that there are examples of colimits which exist in $\mathsf{Man}$ (or maybe $\mathsf{Diff}$ in my notation?) but which don't agree with the underlying colimits in $\mathsf{Top}$. So a negative answer to (1) or (2) might spell out such an example.

I know also that there are various categories of 'generalized smooth spaces' (Frölicher spaces, diffeological spaces, Chen spaces, ...) with $\mathsf{Diff}$ as a subcategory. If the inclusion functor is known to preserve colimits, and colimits of the generalized smooth spaces are understood, then this could be useful to address (2). But I currently know nothing about such generalized smooth spaces, so a very helpful answer along these lines might indicate e.g. which versions of generalized smooth space are best able to answer (2), and whether any of these generalizations have been extended to the Riemannian case for (3).

Edit:

It seems that the category $\mathsf{Haus}$ of Hausdorff topological spaces and continuous maps is important to this story.

With this in mind, a better formulation of my question is as follows: we have a sequence of forgetful/inclusion functors $$ \mathsf{Rm} \rightarrow \mathsf{Diff} \rightarrow \mathsf{Man} \hookrightarrow \mathsf{Haus} \hookrightarrow \mathsf{Top}. $$ Which, if any, of these functors preserve all colimits?

The comment below by Martin Brandenburg already shows that the functors with target $\mathsf{Top}$ do not preserve all colimits.

creatingcolimits. This is methodological advice not an answer: Coproducts no problem. Next think about filtered colimits of inclusions - what kind? probably open ones. Then quotients of equivalence relations - what kind? probably closed ones. Those are likely to be the colimits you want. There could be many more scary ones, so try out some topological counterexamples. $\endgroup$1more comment