nLab
local system

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cohomology

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Idea

A local system – which is short for local system of coefficients for cohomology – is a system of coefficients for twisted cohomology.

Often this is presented or taken to be presented by a locally constant sheaf. Then cohomology with coefficients in a local system is the corresponding sheaf cohomology.

More generally, we say a local system is a locally constant stack, … and eventually a locally constant ∞-stack.

Under suitable conditions (if we have Galois theory) local systems on XX correspond to functors out of the fundamental groupoid of XX, or more generally to (∞,1)-functors out of the fundamental ∞-groupoid. These in turn are equivalently flat connections (this relation is known as the Riemann-Hilbert correspondence) or generally flat ∞-connections.

Definitions

A notion of cohomology exists intrinsically within any (∞,1)-topos. We discuss local systems first in this generality and then look at special cases, such as local systems as ordinary sheaves.

General

For H\mathbf{H} an (∞,1)-sheaf (∞,1)-topos, write

(LConstΓ):HΓLConstGrpd (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

for the terminal (∞,1)-geometric morphism, where Γ\Gamma is the global section (∞,1)-functor and LConstLConst the constant ∞-stack-functor.

Write 𝒮:=core(FinGrpd)\mathcal{S} := core(Fin \infty Grpd) \in ∞Grpd for the core ∞-groupoid of the (∞,1)-category of finite \infty-groupoids. (We can drop the finiteness condition by making use of a higher universe.) This is canonically a pointed object *𝒮* \to \mathcal{S}, with points the terminal groupoid.

Definition

For XHX \in \mathbf{H} an object, a local system or locally constant ∞-stack on XX is a morphism

˜:XLConst𝒮 \tilde \nabla \colon X \longrightarrow LConst \mathcal{S}

in H\mathbf{H} or equivalently the object in the over-(∞,1)-topos

(PX)H/X (P \to X) \in \mathbf{H}/X

that is classified by ˜\tilde \nabla under the (∞,1)-Grothendieck construction

P LConst𝒵 X ˜ LConst𝒮 \array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} }

In other words, local systems are locally constant ∞-stacks or equivalenlty their classifying cocycles for cohomology with constant coefficients.

(See principal ∞-bundle for discussion of how cocycles ˜:XLConst𝒮\tilde \nabla : X \to LConst \mathcal{S} classify morphisms PXP \to X.)

Remark

If H\mathbf{H} happens to be a locally ∞-connected (∞,1)-topos in that there is the further left adjoint (∞,1)-functor Π\Pi

(ΠLConstΓ):HGrpd (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd

we call Π(X)\Pi(X) the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos. In this case, by the adjunction hom-equivalence we have

H(X,LConst𝒮)Func(Π(X),𝒮). \mathbf{H}(X, LConst \mathcal{S}) \simeq Func(\Pi(X), \mathcal{S}) \,.

This means that local systems are naturally identified with representations (\infty-permutation representations, as it were) of the fundamental ∞-groupoid Π(X)\Pi(X):

Maps(X,LConst𝒮)Maps(Π(X),𝒮). Maps(X, LConst \mathcal{S}) \simeq Maps(\Pi(X), \mathcal{S}) \,.

This is essentially the basic statement around which Galois theory revolves.

The (∞,1)-sheaf (∞,1)-topos over a locally contractible space is locally \infty-connected, and many authors identify local systems on such a topological space with representations of its fundamental groupoid.

Definition

Given a local system ˜:XLConst𝒮\tilde \nabla : X \to LConst \mathcal{S}, the cohomology of XX with this local system of coefficients is the intrinsic cohomology of the over-(∞,1)-topos H/X\mathbf{H}/X:

H(X,˜):=H /X(X,P ˜), H(X,\tilde \nabla) := \mathbf{H}_{/X}(X, P_{\tilde \nabla}) \,,

where P ˜P_{\tilde\nabla} is the homotopy fiber of ˜\tilde \nabla.

Remark

Unwinding the definitions and using the universality of the (∞,1)-pullback, one sees that a cocycle cH(X,˜)c \in \mathbf{H}(X,\tilde \nabla) is a diagram

X c * LConst𝒮 \array{ X &&\stackrel{c}{\to}&& * \\ & \searrow &\swArrow& \swarrow \\ && LConst \mathcal{S} }

in H\mathbf{H}. This is precisely a section of the locally constant ∞-stack ˜\tilde \nabla.

Sheaf-theoretic case

Local systems can also be considered in abelian contexts. One finds the following version of a local system

Definition

A linear local system is a locally constant sheaf on a topological space XX (or manifold, analytic manifold, or algebraic variety) whose stalk is a finite-dimensional vector space.

Regarded as a sheaf FF with values in abelian groups, such a linear local system serves as the coefficient for abelian sheaf cohomology. As discussed there, this is in degree nn nothing but the intrinsic cohomology of the \infty-topos with coefficients in the Eilenberg-MacLane object B nF\mathbf{B}^n F.

Lemma

On a connected topological space this is the same as a sheaf of sections of a finite-dimensional vector bundle equipped with flat connection on a bundle; and it also corresponds to the representations of the fundamental group π 1(X,x 0)\pi_1(X,x_0) in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent D XD_X-modules and local systems on XX.

References

An early version of the definition of local system appears in

  • Norman Steenrod: Homology with local coefficients, Annals 44 (1943) pp. 610 - 627,

This is before the formal notion of sheaf was published by Jean Leray. (Wikipedia’s entry on Sheaf theory is interesting for its historical perspective on this.)

A definition appears as an exercise in

  • Edwin Spanier, 1966, Algebraic Topology , McGraw Hill. (republished by Springer, 1982).

on page 58 :

A local system on a space XX is a covariant functor from the fundamental groupoid of XX to some category.

A blog exposition of some aspects of linear local system is developed here:

A clear-sighted description of locally constant (n1)(n-1)-stacks / nn-local systems as sections of constant nn-stacks is in

for locally constant stacks on topological spaces. The above formulation is pretty much the evident generalization of this to general (∞,1)-toposes.

Discussion of Galois representations as encoding local systems in arithmetic geometry includes

  • Tom Lovering, Étale cohomology and Galois Representations, 2012 (pdf)

See also at function field analogy.

Revised on September 24, 2014 15:38:04 by Urs Schreiber (185.26.182.25)