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rvk
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Homotopy quotients, fixed points and stalks of simplicial (pre)sheaves

Let $\mathcal{F}$ be a simplicial (pre)sheaf on some site $\mathcal{C}$ (assume the site has enough stalks; if you like also assume every representable functor on $\mathcal{C}$ is a sheaf). Suppose $\mathcal{G}$ is a (pre)sheaf of groups acting on $\mathcal{F}$. Then we can form a homotopy quotient $\mathcal{F}_{h\mathcal{G}}$ and a homotopy fixed point sheaf $\mathcal{F}^{h\mathcal{G}}$ via a section wise prescription (this is my naive 'on the spot' construction - I don't have a reference and would like one if it exists for the 'correct' construction):

$\mathcal{F}_{h\mathcal{G}}(U) = \mathcal{F}(U)_{h\mathcal{G}} := hocolim_{\mathcal{G}(U)} F(U)$,

$\mathcal{F}^{h\mathcal{G}}(U) = \mathcal{F}(U)^{h\mathcal{G}} := holim_{\mathcal{G}(U)} F(U)$,

for $U \in \mathcal{C}$.

If $\mathcal{F}_x$ is a stalk, then we can also form:

$(\mathcal{F}_x)_{h\mathcal{G}} := hocolim_{\mathcal{G}_x}\mathcal{F}_x$,

$(\mathcal{F}_x)^{h\mathcal{G}} := holim_{\mathcal{G}_x}\mathcal{F}_x$.

Question 1: Is $(\mathcal{F}_x)_{h\mathcal{G}} \simeq (\mathcal{F}_{h\mathcal{G}})_x$?

Question 2: Is $(\mathcal{F}_x)^{h\mathcal{G}} \simeq (\mathcal{F}^{h\mathcal{G}})_x$?

In both cases I am worried about commuting colimits (albeit filtrant) with homotopy limits/colimits.

Naively, 1) seems to be ok to me: using the explicit Borel model for the homotopy colimit, it's just the diagonal for a bisimplicial complex and stalks are more or less by definition taken level wise. Regardless, I worry I am missing some subtlety, and am quite lost with the dual homotopy fixed point model in terms of a totalization of a cosimplicial set.

Question 3: Is there a decent reference for these constructions for a neophyte for simplicial (pre)sheaves (I am not partcularly versed with the subtleties of simplicial methods - my training is in representation theory)? I have Jardine's 'Local Homotopy Theory' which is a godsend for a lot of stuff, but seems to not quite have much along these lines.

Added later: In my questions I am implicitly assuming that stalks are given by a filtrant colimit. This is unnecessary for Question 1, but probably is required for Question 2 to have any hope of having an affirmative answer - along with the group being finite (see Dmitri's answer and the comments underneath it).

rvk
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