Motivation for equivariant sheaves?

Hello everyone;

i'm looking for a motivation for equivariant sheaves (see http://ncatlab.org/nlab/show/equivariant+sheaf) ~ Why are we interested in them?

More explicitely: Can I think of G-equivariant sheaves on a space X as a quotient of the category of sheaves (by some action? in a more general sense) by G?

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Equivariant sheaves are more like fixed points then a quotient. If $G$ acts on $X$ then we can pull-back a sheaf $\mathcal F$ on $X$ via an element $g \in G$, giving a $G$-action on the category of sheaves on $X$. To give a $G$-equivariant structure is then to give isomorphisms $g^*\mathcal F \cong \mathcal F$ for all $g\in G$, satisfying a suitable compatibility. This can be thought of as a categorification of the set-theoretic notion of a fixed point of the $G$-action on the sheaves on $X$. – Emerton Apr 7 '10 at 18:01
@Emerton: that is enough motivation for me! Maybe you should make that into an answer. – Reid Barton Apr 7 '10 at 18:34
To elaborate on Emerton's comment, one would like the (pseudo)functor assigning to a space its category of sheaves to take colimits to limits. However, depending on your notion of space and sheaf, this may not be the case for colimits given by taking the quotient by a group action. The category of G-equivariant sheaves is what we would like the category of sheaves on "X/G" to be, if we had a nice enough object X/G. – Reid Barton Apr 7 '10 at 18:46
maybe add equivariant-homotopy tag? – Bill Kronholm Apr 7 '10 at 19:59
I will make my comment into an answer in a little while, if nobody beats me to it. – Reid Barton Apr 7 '10 at 20:45

If you know that the sections of a vector bundle form a standard example of a sheaf, then the corresponding example of a $G$-equivariant sheaf on a space $X$ with $G$-action is a vector bundle $V$ over $X$ with a $G$-action compatible with the projection (i.e. making the projection $G$-equivariant, i.e. intertwining the actions). Such actions on vector bundles over homogeneous spaces were considered in representation theory by Borel, Weil, Bott and Kostant ("homogeneous vector bundles"; and later many generalizations to sheaves by Beilinson-Bernstein, Schmid, Miličić etc.). David Mumford introduced $G$-equivariant structures on sheaves under the name $G$-linearization for the purposes of geometric invariant theory.
While for a function on a $G$-space the appropriate notion is the $G$-invariance, for sheaves the invariance is useful only up to a coherent isomorphism, what spelled out yields the definition of the $G$-equivariant sheaf. This is an example of a categorification (recall that functions form a set and sheaves form a category). Using the Yoneda embedding one can indeed consider the $G$-equivariant objects as objects in some fibered category of objects on $X$ with an action on each hom-space (see the lectures by Vistoli).
While for a function to be invariant is a property, for a sheaf the $G$-equivariance entails the additional coherence data, so it is a structure.
Category of $G$-equivariant sheaves on $X$ is not a quotient of the category of usual sheaves on $X$, but rather equivalent to the category of sheaves on the geometric quotient $G/X$, in the case when the action of $G$ on $X$ is principal; or in general if we replace the geometric quotient by the appropriate stack $[G/X]$.