# What does the classifying space of a topological monoid classify?

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?

• 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. – Denis Nardin Dec 13 '18 at 9:45
• @DenisNardin: What if the monoid is non-commutative? – Thomas Rot Dec 13 '18 at 11:04
• If the monoid does not have a calculus of fractions, the group completion is kind of horrifying to compute, but it still exists and it is a topological group (note that it might well, nay it will, be infinite dimensional even if $M$ is discrete, but such is life). – Denis Nardin Dec 13 '18 at 11:08
• 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. – Lennart Meier Dec 13 '18 at 11:36
• – Bruno Stonek Dec 14 '18 at 10:25

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.