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?

Classifying spaces related to foliationsshows that for discrete $M$ the space $BM$ still classifies principal $M$-bundles. In Moerdijk'sClassifying spaces and classifying topoithere 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 Meier Dec 13 '18 at 11:36