Can nonisomorphic groupoids have homotopy equivalent classifying spaces?
$begingroup$
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.
algebraic-topology fundamental-groups simplicial-stuff groupoids
asked Jan 4 at 20:49
W. RetherW. Rether
748417
$endgroup$
add a comment |
$begingroup$
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.
algebraic-topology fundamental-groups simplicial-stuff groupoids
asked Jan 4 at 20:49
W. RetherW. Rether
748417
$endgroup$
3
$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
1
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08
add a comment |
$begingroup$
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.
algebraic-topology fundamental-groups simplicial-stuff groupoids
asked Jan 4 at 20:49
W. RetherW. Rether
748417
$endgroup$
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.
algebraic-topology fundamental-groups simplicial-stuff groupoids
algebraic-topology fundamental-groups simplicial-stuff groupoids
asked Jan 4 at 20:49
W. RetherW. Rether
748417
asked Jan 4 at 20:49
W. RetherW. Rether
748417
edited Jan 4 at 20:57
W. Rether
asked Jan 4 at 20:49
W. RetherW. Rether
748417
asked Jan 4 at 20:49
W. RetherW. Rether
748417
asked Jan 4 at 20:49
W. RetherW. Rether
748417
748417
3
$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
1
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08
add a comment |
3
$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
1
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08
3
3
$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
1
1
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08
add a comment |
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$begingroup$
the groupoids don't have to be isomorphic, only equivalent
$endgroup$
– user8268
Jan 4 at 20:52
$begingroup$
Of course, thanks. Fixed.
$endgroup$
– W. Rether
Jan 4 at 20:57
$begingroup$
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.
$endgroup$
– user8268
Jan 4 at 21:03
$begingroup$
Good point! Just for curiosity: do you think that the fundamental groupoid could work as well? The idea is more or less the same...
$endgroup$
– W. Rether
Jan 4 at 21:06
1
$begingroup$
yes, it's basically the same thing
$endgroup$
– user8268
Jan 4 at 21:08