Thanks, Todd. For me to get started: Do you know just *any* (maybe more restrictive then necessary) category of spaces on which the statement holds?

In case it matters, I think Stephan might actually be wanting to know about the relation between traditional and category-theoretic compactness not so much in the category of topological spaces, but in just that of (topological / smooth/ ….) manifolds.

]]>Stephan, some weeks back I was ruminating on similar things (again), but came to no conclusion, and these matters still lie fallow for me. Maybe I’ll have another go at it.

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Todd(posted from n-category cafe): I don’t know if the story is any different for $X$ compactHausdorff, but it could be worth considering. =–

I am interested in this statement - i.e. in a (minimal) assumption on a category of topological spaces such that the notion of compact object in it reduces to that of a compact topological space. Have you already been considering this question?

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