Local coefficient system

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.

There are multiple equivalent definitions of local systems on a locally path connected topological space ${\displaystyle X}$. One of the most convenient but opaque definitions is that a local system is a functor

from the fundamental groupoid of ${\displaystyle X}$ to the category of modules over a commutative ring ${\displaystyle R}$. Typically ${\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }$. What this is saying is that at every point ${\displaystyle x\in X}$ we should assign a module ${\displaystyle M}$ with a representations of ${\displaystyle \pi _{1}(X,x)\to {\text{Aut}}_{R}(M)}$ such that these representations are compatible with change of basepoint ${\displaystyle x\to y}$ for the fundamental group.

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