In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.
Definition Let X be a topological space. A local system (of abelian groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X. In other words, a sheaf L {\displaystyle {\mathcal {L}}} is a local system if every point has an open neighborhood U {\displaystyle U} such that the restricted sheaf L | U {\displaystyle {\mathcal {L}}|_{U}} is isomorphic to the sheafification of some constant presheaf.
Equivalent definitions
Path-connected spaces If X is path-connected, a local system L {\displaystyle {\mathcal {L}}} of abelian groups has the same stalk L {\displaystyle L} at every point. There is a bijective correspondence between local systems on X and group homomorphisms
ρ : π 1 ( X , x ) → Aut ( L ) {\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)}
and similarly for local systems of modules. The map π 1 ( X , x ) → Aut ( L ) {\displaystyle \pi _{1}(X,x)\to {\text{Aut}}(L)} giving the local system L {\displaystyle {\mathcal {L}}} is called the monodromy representation of L {\displaystyle {\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. This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} (equivalently, Z [ π 1 ( X , x ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X,x)]} -modules).
Stronger definition on non-connected spaces A stronger nonequivalent definition that works for non-connected X is the following: a local system is a covariant functor
L : Π 1 ( X ) → Mod ( R ) {\displaystyle {\mathcal {L}}\colon \Pi _{1}(X)\to {\textbf {Mod}}(R)}
from the fundamental groupoid of X {\displaystyle X} to the category of modules over a commutative ring R {\displaystyle R} , where typically R = Q , R , C {\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} } . This is equivalently the data of an assignment to every point x ∈ X {\displaystyle x\in X} a module M {\displaystyle M} along with a group representation ρ x : π 1 ( X , x ) → Aut R ( M ) {\displaystyle \rho _{x}:\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)} such that the various ρ x {\displaystyle \rho _{x}} are compatible with change of basepoint x → y {\displaystyle x\to y} and the induced map π 1 ( X , x ) → π 1 ( X , y ) {\displaystyle \pi _{1}(X,x)\to \pi _{1}(X,y)} on fundamental groups.
Examples Constant sheaves such as Q _ X {\displaystyle {\underline {\mathbb {Q} }}_{X}} . This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:
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