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edited Mar 1, 2014 at 18:43

This simplicial scheme is a resolutionresolution 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 (e.g. a simplicial resolution) of the objectsource. 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 (e.g. via cocycles) describe these morphisms, again, via resolutions of the source. See the nLab for more details.

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 (e.g. a simplicial resolution) of the object. 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 (e.g. via cocycles) describe these morphisms, again, via resolutions of the source. See the nLab for more details.

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.

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created Mar 1, 2014 at 18:38

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 (e.g. a simplicial resolution) of the object. 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 (e.g. via cocycles) describe these morphisms, again, via resolutions of the source. See the nLab for more details.