Cocycle condition for equivariant sheaves

Question

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$ : $X_n := G^n \times X$, $s_0:X_0 \to X_1$ defined by $s_0(x)=(1,x)$ and $d_i:X_n \to X_{n-1}$ defined by

• $d_0(g_1,\dots,g_n,x) = (g_2,\dots,g_n,g_1^{-1}x)$,
• $d_i(g_1,\dots,g_n,x) = (g_1,\dots,g_ig_{i+1},\dots,g_n,x)$ if $0<i<n$,
• $d_n(g_1,\dots,g_n,x) = (g_1,\dots,g_{n-1},x)$.

Define a $G$-equivariant sheaf as a pair $(F,\theta)$, where $F$ is a quisi-coherent sheaf on $X$ and $\theta:d_1^*F \to d_0^*F$ is an isomorphism of sheaves satisfying the cocycle conditions: $d_2^*\theta \circ d_0^*\theta = d_1^*\theta$ and $s_0^*\theta = \mathrm{id}_F$.

What is the meaning of this simplicial scheme $X_\bullet$? Are there any relations with space $BG$ or group cohomology? Why this is a cocycle condition i.e. what is a cohomology theory?

Answer 1

• This simplicial scheme is a resolution of the homotopy quotient $X/G$ in a suitable higher category (the higher category presented by a suitable model structure on simplicial schemes; I don't know the story here but there should be a story here). An equivariant sheaf should be a sheaf on this quotient.
• The story for group cohomology is that the bar resolution is a resolution of the homotopy quotient $\text{pt}/G$ in the category of $\infty$-groupoids, which reproduces $BG$ (see also Borel construction).
• A cocycle condition is what happens when you try to describe morphisms in a higher category via a resolution of the source. Here the source is $X/G$ and the target is, at least morally speaking, the "classifying space of sheaves."
• Cohomology describes morphisms in higher categories; for example, in the case of Eilenberg-MacLane cohomology of spaces, the target is $B^n A$ for $A$ some abelian group. Presentations of cohomology theories describe these morphisms via resolutions. See the nLab for more details.