TITLE: Etymology of $\mathbf{B}G$ for category of one object for $G$?
QUESTION [7 upvotes]: Emily Riehl's book "Category theory in Context" uses the notation $\mathbf{B}G$ for the one object category whose objects is a single object $*$ and whose arrows correspond to group elements. Why is the notation $\mathbf{B}G$ used? Where does $\mathbf{B}$ come from?

REPLY [8 votes]: It is traditional to use the notation $F \to E \to B$ for the spaces involved in a fibre bundle. Here, I believe $F$ stands for "fibre", $E$ stands for the French term "espace total" ("total space" in English), and $B$ stands for "base space" (not "bundle" - my comment was wrong). For any group $G$ there is an important fibre bundle (actually a principal $G$-bundle) $F \to EG \to BG$, where $F = G$, $EG$ is contractible and $BG$ is called the classifying space of $G$, which, in a certain sense, represents all principal $G$-bundles. The construction of $BG$ from $G$ can be generalised to categories other than the category of topological spaces, see this article on delooping. The notation $BG$ seems to have been adopted from the special case of the classifying space in topology.
Acknowledgments to Zhen Lin for valuable corrections.