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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 23rd 2012

started an entry on the Borel construction, indicating its relation to the nerve of the action groupoid.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeMar 23rd 2012
• (edited Mar 23rd 2012)

That is very useful and very ’timely’ as I was looking for a good reference for the Borel construction and orbifolds. :-)

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeMar 23rd 2012
I've done a very slight edit including the alternate name homotopy quotient but that leads to something in which homotopy quotient appears only very weakly.
• CommentRowNumber4.
• CommentAuthorMatanP
• CommentTimeMar 23rd 2012

As a homotopy colimit over the category associated to $G$

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 21st 2020
• (edited Oct 21st 2020)

This entry was lacking a decent reference. I have taken the liberty now of pointing to

Please feel invited to add you favorite classical textbook account on the Borel construction, instead (which is?)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 4th 2021
• (edited Jul 4th 2021)

added mentioning of the simplicial version and some lines (here) relating to the model structure on simplicial group actions

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

I have written out a proof (here) that the topological Borel construction of a well-pointed topological group action sits in the evident homotopy fiber sequence

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 17th 2021

I have added statement and proof (here) that the topological Borel construction of a free action is weakly equivalent – under some sufficient conditions – to the plain quotient.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 22nd 2021

have removed, in that Prop, the assumption that the fundamental group is abelian, and instead added the remark to the proof that the five-lemma still applies.

Of course it does. I have tried to make up for being silly here, previously, by expanding a little at five lemma on the case of homological categories.