nLab
local system

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Cohomology

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 equivalently 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

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

on page 58 :

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

Textbook account and relation to twisted de Rham cohomology:

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

See also at function field analogy.