# reference request for essential equivalence of top. groupoids

Let $G$ and $H$ be two topological groupoids. Recall that a morphism $F \colon H \to G$ is called an essential equivalence, if

• the map $t \circ \pi_1 \colon G_1 \times_{G_0} H_0 \to G_0$ is an open surjection, where $G_1 \times_{G_0} H_0$ is the pullback along the source map $s \colon G_1 \to G_0$.
• the following diagram is a pullback diagram: $$H_1 \to G_1 \\\\ \downarrow \qquad \downarrow \\\\ H_0 \times H_0 \to G_0 \times G_0$$ where the vertical arrow are given by $(s,t)$.

Basically this means that $F$ is an equivalence of categories (where the first condition implies essential surjectivity and the second it fullness), but it need not necessarily have a continuous inverse. Nevertheless, such a map $F$ induces a weak homotopy equivalence $BF \colon BH \to BG$.

I think, I know how to prove this. But for a paper I would rather have a reference in the literature for this result. Unfortunately, this seems to be so well-known that nobody bothers to write down a proof.

What is the best/first reference in the literature for the above statement?

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You should take Konrad's answer with a small grain of salt. Lemma 3.34 can be said to prove that an essential equivalence between Lie groupoids H and G defines an isomorphism of corresponding stacks. I presume the same proof works for topological groupoids. Then the result you want follows from a fact, the reference to which I don't know, that isomorphic stacks have homotopy equivalent classifying spaces (assuming that by $BG$ you mean the classifying space of a stack rather than the stack itself; the notation is ambiguous). –  Eugene Lerman Sep 23 '12 at 15:05
I would use an argument like this: From your proof, I can deduce that principal $G$-bundles over $S^n$ are in $1:1$-correspondence (induced by the functor) with principal $H$-bundles. From this we obtain a weak equivalence between $BG$ and $BH$ induced by $F$. –  Ulrich Pennig Sep 24 '12 at 9:58