Verdier Duality for Systems of Coefficients over Simplicial Sets

Let X. be a simplicial set. A cohomological system of coefficients F. on X. with values in an abelian category A is given by an object F σ ∈ A for every σ ∈ X. and by a compatible set of morphisms F α*σ → F σ for every map of ordered sets α : m → n , m , n = 0, … and σ ∈ X n . We denote by SH ( X. ) the category of cohomological systems F. of coefficients on X, such that the maps F α * σ → F σ, are isomorphisms if α is surjective. This category is equivalent to a certain subcategory of the category of sheaves SH ( X ) on the geometric realization X = ∣ X. ∣ of the simplicial set X . The present paper gives a construction of a duality theory for SH ( X. ) which is analogue to the topological Verdier duality for SH ( X ). It consists of a construction of a right adjoint to the cohomology with compact support functor and a resulting duality theory given by a dualizing complex. These constructions are much more explicit than in the topological case. We compare the simplicial and topological constructions via the embedding SH ( X. ) → SH ( X ). In particular, we get an explicit description of the dualizing sheaf complex for topological spaces X = ∣ X ∣.