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minor in 5.2
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marcbezem committed Oct 3, 2024
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Expand Up @@ -176,7 +176,7 @@ \section{Group actions ($G$-sets)}
\[
\Hom(H,G)(x,y)\jdeq
%\Copy_{\mkgroup}((\BH_\div,x) \ptdto (\BG_\div,y)).
\Copy_{\mkgroup}(\sum_{f:\BH_\div\to \BG_\div}(y\eqto f(x))).
\Copy_{\mkgroup}\bigl(\sum_{f:\BH_\div\to \BG_\div}(y\eqto f(x))\bigr).
\]
Thus the type $\Hom(H,G)$ may also be considered to be a $(H\times G)$-set
\[
Expand Down Expand Up @@ -408,7 +408,9 @@ \subsection{Actions in a type}
Oftentimes it is interesting not to have an action on a set, but on an element in any given type (not necessarily the type of sets). For instance, a group can act on another, giving rise to the notion of the semidirect product in \cref{sec:Semidirect-products}. We will return these more general types of actions many times.

\begin{definition}\label{action}
If $G$ is any (possibly higher) group and $A$ is any type of objects,
If $G$ is any group\footnote{%
Even an $\infty$-group in the sense of \cref{sec:inftygps}.}
and $A$ is any type of objects,
then we define an \emph{action} by $G$ in %the world of elements of
$A$ as a function
\[
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