Contents

cohomology

# Contents

## 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 $X$ correspond to functors out of the fundamental groupoid of $X$, 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 $\mathbf{H}$ an (∞,1)-sheaf (∞,1)-topos, write

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

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

Write $\mathcal{S} := core(Fin \infty Grpd) \in$ ∞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 $* \to \mathcal{S}$, with points the terminal groupoid.

###### Definition

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

$\tilde \nabla \colon X \longrightarrow LConst \mathcal{S}$

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

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

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

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

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 $\tilde \nabla : X \to LConst \mathcal{S}$ classify morphisms $P \to X$.)

###### Remark

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

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

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

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

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

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

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

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

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

###### Remark

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

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

in $\mathbf{H}$. 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 $X$ (or manifold, analytic manifold, or algebraic variety) whose stalk is a finite-dimensional vector space.

Regarded as a sheaf $F$ 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 $n$ nothing but the intrinsic cohomology of the $\infty$-topos with coefficients in the Eilenberg-MacLane object $\mathbf{B}^n F$.

###### 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 $\pi_1(X,x_0)$ in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent $D_X$-modules and local systems on $X$.

• inner local system

• simplicial local system: within Sullivan’s (1977) theory of Infinitesimal computations in topology, he refers to ‘local systems’ several times. This seems to be simplicial in nature. This entry explores some of the uses of that notion based on Halperin’s lecture notes on minimal models

• D. Sullivan, Infinitesimal computations in topology (pdf)

## 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 $X$ is a covariant functor from the fundamental groupoid of $X$ to some category.

Then the first major account and relation to twisted de Rham cohomology:

• Pierre Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes Math. 163, Springer (1970) $[$publications.ias:355$]$

Textbook accounts:

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

A clear-sighted description of locally constant $(n-1)$-stacks / $n$-local systems as sections of constant $n$-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)