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I got confused by the page (Are these two definitions of $EG$ equivalent?) about the following.

Doesn't this also define an action on BG?

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    $\begingroup$ Strictly speaking, $BG$ does not have a $G$-action, right? It is, somehow, the quotient of $EG$ by $G$. There is a projection $EG \to BG$, and this projection forgets the action because it drops the $G$-action component of $EG$. $\endgroup$
    – MathsIsFun
    Jul 12, 2022 at 14:12

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A $G$-simplicial set $X$ is cofibrant if, and only if, each $X_n$ has a free $G$-action. The only sensible $G$-action on $BG$ is the trivial $G$-action. But the trivial action is not free: every simplex is a fixed point! Thus, $BG$ is not cofibrant as a $G$-simplicial set.

Your confusion might arise from trying to see the object $BG$, whose natural category is $\mathsf{sSet}$, inside of $G-\mathsf{sSet}$. A different question is whether $BG$ is cofibrant in $\mathsf{sSet}$, and this is indeed the case: all simplicial sets are cofibrant.

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