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As we know that a topological space can be viewed as a groupoid with only identity morphisms, is there a kind of equivalence which covers the homotopy equivalence between spaces? I mean that if two spaces are homotopy equivalent to each other as topological spaces then the corresponding groupoid are equivalent.

flag	asked Feb 7 at 0:16 Bei Liu 1

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How can a topological space be viewed as a groupoid with only identity morphisms? Isn't a groupoid with only identity morphisms basically just a set? – Noah S Feb 7 at 2:08

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Noah, by the title of the question, I guess he's thinking of internal groupoids in topological spaces. This question is definitely not well posed. – Fernando Muro Feb 7 at 7:20

@Noah S @Fernando Muro I am very sorry I didn't pose my question clearly. I mean if we have a topological space X then we can construct a groupoid G with G_1=X, G_0=X and the source and target maps are both identity map. My question is that if there is an relation between groupoids which would correspond to homotopy equivalence between topological spaces under this construction. – Bei Liu Feb 8 at 0:34

@Beren: Since you didn't confirm Fernando's guess that you meant internal groupoids in topological spaces, I'm inclined to assume that you didn't mean that and instead meant literally what you wrote, namely just plain groupoids. But then the answer to your question is trivially negative, since your groupoid completely ignores the topology of $X$ and involves $X$ only as a set. If you meant something nontrivial instead, please tell us what you meant. For now, I'm voting to close as "not a real question". – Andreas Blass Feb 8 at 14:16

@Andreas Blass In the above construction I mean the groupoid $G$ is a topological groupoid and $G_1$ and $G_0$ are both topological spaces with the same topology as $X$ and the source and target maps are continuous maps. – Bei Liu Feb 8 at 16:11

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Homotopy equivalence of topological groupoid

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