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Suppose A: Cart^op → Ab is a sheaf of abelian groups on the cartesian site.
Denote by A[0]: Cart^op → Ch the 1-sheaf of unbounded chain complexes that sends S∈Cart to A(S)[0].
Under what conditions is A[0] an ∞-sheaf?
This is true if A is a constant sheaf or A is the representable sheaf of an abelian Lie group, or A is a sheaf of real vector spaces. However, I do not know a general criterion.
As a special case, for the vanishing of the first cohomology group the map A(U)⊕A(V)→A(U∩V) must be surjective.
I do not know any examples of A where this map is not surjective.
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