# Etymology of $\mathbf{B}G$ for category of one object for $G$?

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?

• I think it is related to the notation $BG$ for the classifying space of a group, where $B$, I believe, stands for "bundle". See ncatlab.org/nlab/show/… and the linked topics there. Jun 6, 2021 at 23:01
• @RobArthan Actually, $B$ stands for base. The bundle is denoted by $E$, and the fibre by $F$, hence the "fibre sequence" $F \to E \to B$. Jun 6, 2021 at 23:20
• @ZhenLin: P.S., I wouldn't call $E$ the "bundle": it's the total space of the bundle - for me, the bundle is really the map from $E$ to $B$. Do you know if I am right that it $E$stands for "ensemble"? (I don't have the right books to hand.) Jun 6, 2021 at 23:39
• $E$ stands for espace (total). Jun 6, 2021 at 23:48
• I belong to the (rare?) species of category theorists who call the one-object category corresponding to a group (or monoid) $G$ just by $G$. Several advantages: 1) Less notation, 2) the notation reflects that monoids are really just special cases of categories, 3) the notation $BG$ is already used for a topological space, not a category, and it's always better to not overload notations. Jun 7, 2021 at 19:18

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.