Start with a group $G$ that acts on a set $X$, and a second group $H$. We want to consider functions $\varphi: G \times X \to H$ such that $\varphi(g g', x) = \varphi(g, g'x) \varphi(g', x)$ for all $g$, $g'$, and $x$. Note that if $X$ is singleton (or if $G$ acts trivially), then $\varphi$ is essentially just an ordinary group homomorphism. Is there a name for a function like this?
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4$\begingroup$ If I'm correct it's widely called "cocycle", maybe have a look for instance at arxiv.org/abs/math/0303236 (Fisher-Margulis). The keyword "cocycle rigidity" often points to this meaning of cocycle. $\endgroup$– YCorCommented May 28, 2020 at 14:54
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4$\begingroup$ Specifically, to fit in the usual set-up for these things, I'd call it a co-cycle for $G$ with values in $\operatorname{Func}(X, H)$, where the latter is viewed as a $G$-set in the obvious way. (More generally, whenever you find yourself saying "twisted …", cohomology is probably a good place to look to describe your objects.) $\endgroup$– LSpiceCommented May 28, 2020 at 14:56
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$\begingroup$ If H = Aut(Y) then your phi can also be described as an action G -> Aut(X x Y) which is compatible with the action G -> Aut(X) relative to the projection X x Y -> X. $\endgroup$– Joel SjögrenCommented Sep 2, 2023 at 20:07
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$\begingroup$ Am I wrong to assume that this use of the word "cocycle" in a different from the sense of group cohomology cocycle? Such a cocycle can be described as a group homomorphism G -> G |x M (a section in fact), which looks a little similar to G x X -> H but not very. $\endgroup$– Joel SjögrenCommented Sep 2, 2023 at 20:13
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$\begingroup$ (I'm asking because I've also heard the word "twisted homomorphism" used in the context of group cohomology cocycles - specifically here: kurims.kyoto-u.ac.jp/~motizuki/… .) $\endgroup$– Joel SjögrenCommented Sep 2, 2023 at 20:21
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