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Verdier Duality for Systems of Coefficients over Simplicial Sets
Author(s) -
Fimmel Torsten
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981900105
Subject(s) - mathematics , functor , sheaf , simplicial complex , pure mathematics , derived category , morphism , duality (order theory) , subcategory , surjective function , simplicial set , abelian category , topology (electrical circuits) , abelian group , homotopy category , combinatorics , homotopy
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 ∣.