Questions tagged [local-systems]

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 $.

Let $ X $ be a topological space. A local system (of abelian groups/modules/...) on $ X $ is a locally constant sheaf (of abelian groups/modules...) on $ X $. In other words, a sheaf $ \mathcal L $ is a local system if every point has an open neighborhood $ U $ such that $ \mathcal L | _ U $ is a constant sheaf.

If $ X $ is path-connected, a local system $ \mathcal L $ of abelian groups has the same fibre $ L $ at every point. To give such a local system is the same as to give a homomorphism $ \rho : \pi _ 1 ( X , x ) \to \operatorname {Aut} ( L ) $ and similarly for local systems of modules,... The map $ \pi _ 1 ( X , x ) \to \operatorname {End} ( L ) $ giving the local system $ \mathcal L $ is called the monodromy representation of $ \mathcal L $. This shows that (for $ X $ path-connected) a local system is precisely a sheaf whose pullback to the universal cover of $ X $ is a constant sheaf.

Another (stronger, nonequivalent) definition generalising the previous definition, and working for non-connected $ X $, is: a covariant functor $ \mathcal L : \Pi _ 1 ( X ) \to \operatorname {Mod} ( R ) $ from the fundamental groupoid of $ X $ to the category of modules over a commutative ring $ R $. Typically $ R = \mathbb Q ,\mathbb R ,\mathbb C $. What this is saying is that at every point $ x \in X $ we should assign a module $ M $ with a representations of $ \pi _ 1 ( X , x ) \to \operatorname {Aut} _ R ( M ) $ such that these representations are compatible with change of basepoint $ x \to y $ for the fundamental group.

Source: Wikipedia