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?



What does the classifying space of a topological monoid classify?
Section 5 of Segal's [Classifying spaces related to foliations](https://core.ac.uk/download/pdf/82283884.pdf) 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. 


