Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 26th 2020
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexSuppose 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.

   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.