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\begin{document}

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\section*{equivariant object}

Given a [[fibered category]] $\pi=\pi_F: F\to C$, a [[presheaf]] of [[group]]s $\hat{G}:C^{\mathrm{op}}\to\mathrm{Grp}$ (possibly [[representable functor|representable]]), an object $X$ in $C$, and an [[action]] $\hat\nu: \hat{G}\times h_X\to h_X$ (generalizing the action $G\times X\to X$ in the representable case), one says that the object $\rho$ in the fiber $F_X = \pi^{-1}(\mathrm{id}_X)$ is \textbf{$\hat{G}$-equivariant} if $\hat{G}\circ\pi$ is equipped with an action on $h_\rho$, i.e., it is equipped with a natural transformation $\sigma:(\hat{G}\circ\pi)\times h_\rho : F^{\mathrm{op}}\to\mathrm{Set}$ such that for any $\xi\in F$, $\sigma_\xi:\hat{G}(\pi(\xi))\times\mathrm{Hom}_F(\xi,\rho)\to\mathrm{Hom}_F(\xi,\rho)$ satisfies the action axiom of a group $\hat{G}(\pi(\xi))$ on a set $\mathrm{Hom}_F(\xi,\rho)$, and such that $\pi(\sigma)=\hat\nu$. The $\hat{G}$-equivariant objects naturally form a fibered category $\pi^{\hat{G}}:F^{\hat{G}}\to C$ of equivariant objects.

Alternatively, if $F$ and $C$ are [[complete category|complete]] and $\hat{G}$ is represented by $G$ in $C$, the equivariant fiber over a $G$-object $X$ is simply the fiber over the [[simplicial object]] $[ X/G ]$ (the simplicial [[Borel construction]]), that is the category of Cartesian functors $(\sigma^F,\sigma^C):\mathrm{Id}_{\Delta^{\mathrm{op}}}\to\pi_F$ from the opposite of the category of [[simplex|simplices]] $\Delta^{\mathrm{op}}$ considered as a fibered category $\mathrm{Id}_{\Delta^{\mathrm{op}}}:\Delta^{\mathrm{op}}\to\Delta^{\mathrm{op}}$ to the fibered category $\pi_F:F\to C$, such that the bottom component is $\sigma_C = [X/G]:\Delta^{\mathrm{op}}\to C$.

Equivariant objects in fibered categories generalize [[equivariant sheaf|equivariant sheaves]] which in turn generalize (sheaves of sections of) $G$-[[equivariant bundle]]s. The description of $G$-equivariant sheaves via cartesian sections over the simplicial Borel construction translates into the statement that they form the category equivalent to the subcategory of the category of [[Deligne sheaf|Deligne sheaves]] over the simplicial Borel construction all of whose structure morphisms are isomorphisms. Mumford has expressed $G$-equivariant sheaves over a $G$-space $(X,\nu)$ as sheaves $\rho$ equipped with an isomorphism $\theta:\nu^*(\rho)\to p^*(\rho)$ where $p:G\times X\to X$ is the projection and $\nu:G\times X\to X$ is the action, and such that $\theta$ satisfies a certain cocycle condition, which is an identity of sheaves over $G\times G\times X$.

For $G$-equivariant sheaves over a \emph{topological} $G$-space $(X,\nu)$ (as opposed to a $G$-object $(X,\nu)$ in a Grothendieck [[site]]), one can extend the canonical equivalence between \'e{}tale spaces over $X$ and sheaves over $X$ to a canonical equivalence between \'e{}tale $G$-spaces over $(X,\nu)$ and $G$-equivariant sheaves over $(X,\nu)$.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item P. Deligne, Th\'e{}orie de Hodge : III. Publications Math\'e{}matiques de l'IH\'E{}S, 44 (1974), p. 5--77 (\href{http://www.numdam.org/item?id=PMIHES_1974__44__5_0}{numdam})
\item S. MacLane, I. Moerdijk, Sheaves in geometry and logic, Springer 1992.
\item D. Mumford, Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer 1965. 2nd edition with J. Fogarty, Springer 1982.
\item Z. \v{S}koda, Some equivariant constructions in noncommutative algebraic geometry, accepted to Georgian Math. J., \href{http://arxiv.org/abs/0811.4770}{arXiv:0811.4770}.
\item A. Vistoli, Grothendieck topologies, fibered categories and descent theory. Fundamental algebraic geometry, 1--104, Math. Surveys Monogr., 123, AMS 2005, \href{http://front.math.ucdavis.edu/0412.5512}{arXiv:math.AG/0412512}.

\end{itemize}


\end{document}