The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this.
I hope the following question is appropriate for MathOverflow:
What does the classifying space of a topological monoid classify?
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2$\begingroup$ The classifying space of a topological monoid is equivalent to the classifying space of its group completion, so it certainly classifies principal bundles on the group completion. I don't know if you can make it more explicit than that. $\endgroup$– Denis NardinCommented Dec 13, 2018 at 9:45
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1$\begingroup$ @DenisNardin: What if the monoid is non-commutative? $\endgroup$– Thomas RotCommented Dec 13, 2018 at 11:04
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4$\begingroup$ Section 5 of Segal's Classifying spaces related to foliations shows that for discrete $M$ the space $BM$ still classifies principal $M$-bundles. In Moerdijk's Classifying spaces and classifying topoi there is a kind of answer for general topological monoids (Cor IV.4.5) if you specialize from topological categories (restricting to those with just one object), but it is not in terms of principal bundles. His version with principal bundles (Section IV.2) has again a discreteness condition. $\endgroup$– Lennart MeierCommented Dec 13, 2018 at 11:36
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2$\begingroup$ Related: mathoverflow.net/questions/23857/… $\endgroup$– Bruno StonekCommented Dec 14, 2018 at 10:25
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2$\begingroup$ @DenisNardin wrote: "The classifying space of a topological monoid is equivalent to the classifying space of its group completion." I don't think that's true; for example here is a 5-element discrete monoid whose classifying space is $S^2$: arxiv.org/abs/math/0202260. Its group completion must be some discrete group $G$, with classifying space the Eilenberg-Mac Lane space $K(G,1)$, but $S^2$ is not a $K(G,1)$. $\endgroup$– John BaezCommented Apr 6, 2020 at 17:22
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1 Answer
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Section 5 of Segal's Classifying spaces related to foliations shows that for discrete monoids $M$ the space $BM$ still classifies principal $M$-bundles (in a suitable sense). In Moerdijk's Classifying spaces and classifying topoi there is a kind of answer for general topological monoids (Cor IV.4.5); it is not in terms of principal bundles though, but rather in terms of linear orders. (He formulates it for topological categories, but topological monoids are just topological categories with one object.) His version with principal bundles (Section IV.2) has again a discreteness condition.
answered Dec 14, 2018 at 7:07