Local coefficients

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In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod.

In sheaf theory terms, a constant sheaf has locally constant functions as its sections. Consider instead a sheaf F, such that locally on X it is a constant sheaf. That means that in the neighbourhood of any x in X, it is isomorphic to a constant sheaf. Then F may be used as a system of local coefficients on X.

Larger classes of sheaves are useful: for example the idea of a constructible sheaf in algebraic geometry. These turn out, approximately, to be local coefficients away from a singular set.