nLab locally constant sheaf

Idea

A locally constant sheaf $A$ over a topological space is a sheaf of sections of a covering space of $X$: there is a cover of $X$ by open subsets $\{U_i\}$ such that the restrictions $A|_{U_i}$ are constant sheaves.

More elegantly said: locally constant sheaves are the sections of constant stacks:

Let $C = Core(FinSet)\in$ Grpd be the core of the category FinSet of finite set, let $const_C : Op(X)^{op} \to Grpd$ the presheaf constant on $C$, i.e. the functor on the opposite category of the category of open subsets of $X$ that sends everything to (the identity on) $C$. Then the constant stack on $C$ is the stackification $\bar const_C : Op(X)^{op} \to Grpd$.

Write then $X$ for the space $X$ regarded as a sheaf or trivial covering space over itself, i.e. the terminal object $X$ in sheaves and hence in stacks over $X$. Then by definition of stackification morphisms

$$X \to \bar const_C$$

are represented by

• an open cover $\{U_i\}$ of $X$;

• over each $U_i$ a choice $F_i \in C$ of object in $C$, hence a finite set in $C$;

• over each double overlap $U_{i j} = U_i \cap U_j$ an morphism $g_{i j} : F_i|_{U_{i j}} \stackrel{\simeq}{\to} F_j|_{U_{i j}}$, hence a bijection of finite sets;

• such that on triple overlaps we have $g_{i k}|_{U_{i j k}}= g_{j k}|_{U_{i j k}}\circ g_{i j}|_{U_{i j k}}$.

Such data clearly is the local data for a covering space over $X$ with typical fiber any of the $F_i$.

Definition

Let $(\Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \mathcal{S}$ be the global section geometric morphism of a topos $\mathcal{E}$ over base $\mathcal{S}$.

Without further assumption on $\mathcal{E}$ we have the following definition.

Definition

For $U \to *$ an epimorphism in $\mathcal{E}$, an object $E \in \mathcal{E}$ is called locally constant and split by $U$ if in the over category $\mathcal{E}/U$ we have an isomorphism

$$E \times U \simeq (\Delta S) \times U \,,$$

for some $S \in$ Set.

An object which is locally constant and $U$-split for some $U$ is called locally constant.

A locally constant object $E$ which is in addition an $\Delta Aut(X)$-principal bundle is called a Galois object .

If $\mathcal{E}$ is a locally connected topos there is another characterization of locally constant sheaves.

Definition

For $C$ and $C$ cartesian closed categories, a functor $F \,\colon\, C \to D$ that preserves products is called a cartesian closed functor if the canonical natural transformation

$$F(B^A) \to (F(B))^{F(A)}$$

(which is the adjunct of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an isomorphism.

From the discussion at locally connected topos we have that

Proposition

The constant sheaf-functor $\Delta \,\colon\, \mathcal{S} \to \mathcal{E}$ is a cartesian closed functor precisely if $\mathcal{E}$ is a locally connected topos.

In this case the above definition is equivalent to the following one.

Definition

Let $\mathcal{E} = Sh(C)$ be a locally connected topos. Let $p \,\colon\, core(Set^\kappa_*) \to core(Set^\kappa)$ be the core of the generalized universal bundle for sets of cardinality less than some $\kappa$.

A locally constant $\kappa$-bounded object in $\mathcal{E}$ is the pullback of $\Delta(p)$ along a morphism $* \to core(Set^\kappa)$ in the (2,1)-topos $Sh_{(2,1)}(C)$.

Remark

This says that locally constant sheaves are the sections of the constant stack on the groupoid $core(Set^\kappa)$.

Notice that

$$core(Set \kappa) \simeq \coprod_i \mathbf{B}Aut(F_i) \,,$$

where the coproduct is over all cardinals smaller than $\kappa$ and where $\mathbf{B}Aut(F_i)$ denotes the delooping groupoid of the automorphism group of the set $F_i$: the symmetric group on $F_i$.

This means that on each connected component of $\mathcal{E}$ a locally constant sheaf is the $\Delta \rho$-associated bundle to an $\Delta Aut(F)$-principal bundle induced by the canonical permutation representation $\rho : \mathbf{B} Aut(F) \to Set$ of the automorphism group $Aut(F)$ on $F$.

Specifically for $g : * \to \Delta \mathbf{B} Aut(F) \simeq \mathbf{B} \Delta Aut(F) \hookrightarrow \Delta core(set)$ the classifying morphism of a locally constant sheaf and for $U \to *$ an epimorphism on which it trivializes, we have a pasting diagram of pullbacks

$$\array{ U \times \Delta F &\to& P \times_{\Delta Aut(F) (\Delta (F // Aut(F)))} &\to& \Delta(F // Aut(F)) &\to& \Delta Set^\kappa \\ \downarrow && \downarrow && \downarrow && \downarrow \\ U &\to& * &\stackrel{g}{\to}& \mathbf{B} \Delta Aut(F) &\hookrightarrow& \Delta core(Set^\kappa) } \,,$$

where $F//Aut(F)$ is the action groupoid, the 2-colimit of $\rho \mathbf{B}Aut(F) \to Grpd$.

Applications

• Locally constant sheaves are sheaves of sections of covering spaces.

• When used as coefficient objects in cohomology they are also called local systems.

• The action of the fundamental groupoid $\Pi_1(X)$ on the fibers of a local system give rise to the notion of monodromy.

• This may be used to define homotopy groups of general objects in a topos, and the fundamental group of a topos.

• This is the content of Galois theory.

Pattern

In sufficiently highly locally connected cases, we have:

• A locally constant function is a section of a constant sheaf;

• a locally constant sheaf is a section of a constant stack;

• a locally constant stack is a section of (… and so on…)

• a locally constant ∞-stack is a section of a constant ∞-stack.

A locally constant sheaf / $\infty$-stack is also called a local system.

Related concepts

• constructible sheaf

References

The definition of locally constant sheaf originates in the notion of covering projection

• SGA 4, Exposé IX, 2.0 .

Lecture notes are in

• James Milne, section 6 of Lectures on Étale Cohomology

The topos-theoretic definition is reproduced in the context of a discussion of the notion of Galois topos as definition 5.1.1 in

• Eduardo Dubuc, On the representation theory of Galois and Atomic Topoi (arXiv:0208222)

and definition 2.2 in

• Eduardo Dubuc, The fundamental progroupoid of a general topos (arXiv:0706.1771)

or as definition 1 in

• Michael Barr, Radu Diaconescu, On locally simply connected toposes and their fundamental groups (NUMDAM)

Discussion of the notions of locally constant sheaves is at

• Michael Shulman, Locally constant sheaves (blog discussion)