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Given a (topological) group acting on $X$ a topological space continuously.

Then we have the category $Sh_G (X)$, it's the full subcategory of $Sh(X)$ consisting on the sheaves $\mathcal{L}$ such that $a^*\mathcal{L}\simeq \pi^* \mathcal{L}$, where $a:G\times X\rightarrow X$ is the action and $\pi$ the projection.

We can also consider the category $\widehat{Sh_G} (X)$ consisting on the sheaves $\mathcal{L}$ such that the restriction of $\mathcal{L}$ to any orbit is a constant sheaf.

Is it true that $Sh_G (X)$ and $\widehat{Sh_G} (X)$ are equal (or equivalent).

And if $G$ and $X$ are a algebraic group and variety over a field of positive characteristic respectively and replace sheaves with $\ell-$adic sheaves, or replace with the respective derived categories.

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    $\begingroup$ Actually, if you are interested in $G$-equivariant sheaves and $G$ is disconnected, then neither definition is correct: equivariance is an extra structure, and homomorphisms must preserve it, so the category of equivariant sheaves is not a full subcategory of the category of all sheaves. $\endgroup$
    – t3suji
    Commented Feb 9, 2017 at 14:54
  • $\begingroup$ These definitions are not even trying to do the same thing: the second one defines sheaves constructible wrt the stratification induced by $G$-orbits, which is a thing you might want to do sometimes but which is much more restrictive than considering arbitrary $G$-equivariant sheaves. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 9, 2017 at 19:06
  • $\begingroup$ And what if the group acting is unipotent and connected, then can I say that if I have a $G-$ equivariant sheaf (in the correct definition) then his restriction to any orbit is a constant sheaf, if isn't, then what can I say about the restriction to any orbit. (And what if my sheaf belongs to the bounded derived category of constructible complexes of $\ell-$adic sheaves) $\endgroup$
    – João Dias
    Commented Feb 21, 2017 at 10:44

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