Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

Question

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude.

The situation with topological groups is subtler. The fundamental group argument doesn't work, but the loop space could work. See this unanswered question.

What if I pass to groupoids? If two discrete groupoids have homotopic classifying spaces, are they equivalent as groupoids? I suspect that taking the fundamental groupoid should suffice, but I am not sure: could anything give me help or clue?

And more subtle again: what happens with topological groupoids? Here I'm afraid I don't have a clue, in fact.

Thank you in advance.

for discrete groupoids the answer is yes, for somewhat trivial reasons - just take the fundamental group of each of the path-connected component of the classifying space. No idea about topological groupoids. — user8268, Jan 04 '19 at 21:03
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same... — W. Rether, Jan 04 '19 at 21:06

JLA · Answer 1 · 2020-07-04T23:34:16.060

It’s not true for topological groupoids: the classifying spaces of a point and $\mathbb{R}$ are both the point, and these topological groupoids aren’t equivalent.

However, there is a notion of homotopy equivalence for groupoids, and the point and $\mathbb{R}$ are weakly homotopy equivalent through the inclusion of the point, which is a homomorphism of groups. That is, if there is a homomorphism between two groups which is a weak homotopy equivalence of the underlying spaces, then they will have the same classifying space. For discrete groups this happens if and only if they are isomorphic.

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

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