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What is the definition of equivariant $l$-adic or ($\mathbb{Z}_l$-) sheaves?

Suppose $G$ acts on $X$, I could pick a $G$-equivariant etale sheaf of $\mathbb{Z}/l^n$ module on $X$ for each $n$, and posing certain compatibility conditions as the non-equivariant case. On the other hand, I could pick a $l$-adic sheaf on $X$ and fix an isomorphism from two different ways of pullbacks of $G \times X \rightarrow X$.

Which one is correct? Same question for equivariant derived category of $l$-adic or ($\mathbb{Z}_l$-) sheaves.

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  • $\begingroup$ I don't remember the specific details for that specific case. But just as a general remark, if one wants to use $\infinity$-categorical language, then I believe the two options should give the same answer, because both are taking some limits of categories, so the operations will commute (please correct me if I am wrong). $\endgroup$
    – Sasha
    Commented Feb 2, 2019 at 8:49
  • $\begingroup$ @AknazarKazhymurat Any reference for this? $\endgroup$
    – userabc
    Commented Feb 2, 2019 at 20:37
  • $\begingroup$ @Sasha I don't know anything about infinity category. The fact that equivariant derived category is not the same as derived category of equivariant sheaves makes me feel doubtful for changing order of the adjectives. $\endgroup$
    – userabc
    Commented Feb 3, 2019 at 0:02
  • $\begingroup$ @userabc There the problem is because you change order with the adjective "derived". In my comment, everything is derived from the beggining! That is the "correct" way of doing things. But for this, you need to consider not triangulated categories but $\infty$-categories, because one can take limits of $\infty$-categories. So, roughly, cat of $\ell$-adic sheaves will be the limit of cats of $\mathbb{Z}/ \ell^n \mathbb{Z}$-sheaves, and equivariant cat. will be limit of cats of sheaves on $G \times X$ , $G \times G \times X$, etc. Well, the comment box is not the best place to try to unfold this $\endgroup$
    – Sasha
    Commented Feb 3, 2019 at 1:42

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