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Homotopy colimits and cohomology with local coefficients. (English) Zbl 1028.55012

Let \(X\) be a simplicial set and \(\Pi_1(X)\) its fundamental groupoid. Let \(A:\Pi_1(X)\to Ab\) be a local coefficient system on \(X\) and denote by \(K(A,n)\) the corresponding local coefficient Eilenberg-MacLane functor. Next, set \(L_{X}(A,n)= \text{hocolim}_{\Pi_{1}(X)}(K(A,n))\). The authors prove a representability result for local cohomology with coefficients in \(A\) analogous to the constant coefficient case, namely they prove that there is a natural isomorphism \[ H^{n}(X,A)\cong [X,L_{X}(A,n)], \] where \(H^{n}(\cdot)\) denotes cohomology with local coefficients \(A\) and \([\;]\) stands for homotopy classes of maps in the appropriate category.

MSC:

55U10 Simplicial sets and complexes in algebraic topology
55N25 Homology with local coefficients, equivariant cohomology
55N10 Singular homology and cohomology theory
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References:

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