4
$\begingroup$

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?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

For example, this is proved as:

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ 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). $\endgroup$
    – Eugene Lerman
    Commented Sep 23, 2012 at 15:05
  • 1
    $\begingroup$ (continued) A good place to look for info on topological stacks are papers of Behrang Noohi (maths.qmul.ac.uk/~noohi) $\endgroup$
    – Eugene Lerman
    Commented Sep 23, 2012 at 15:07
  • 1
    $\begingroup$ 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 $F$. $\endgroup$
    – Ulrich Pennig
    Commented Sep 24, 2012 at 9:58

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .