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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?

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    $\begingroup$ 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. $\endgroup$
    – Rob Arthan
    Jun 6, 2021 at 23:01
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    $\begingroup$ @RobArthan Actually, $B$ stands for base. The bundle is denoted by $E$, and the fibre by $F$, hence the "fibre sequence" $F \to E \to B$. $\endgroup$
    – Zhen Lin
    Jun 6, 2021 at 23:20
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    $\begingroup$ @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.) $\endgroup$
    – Rob Arthan
    Jun 6, 2021 at 23:39
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    $\begingroup$ $E$ stands for espace (total). $\endgroup$
    – Zhen Lin
    Jun 6, 2021 at 23:48
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    $\begingroup$ 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. $\endgroup$ Jun 7, 2021 at 19:18

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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.

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