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 $A{\mid }_{U_i}$ are constant sheaves.

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

Let $C=\mathrm{Core}(\mathrm{FinSet})\in$ Grpd be the core of the category FinSet of finite set, let ${\mathrm{const}}_C:\mathrm{Op}(X{)}^{\mathrm{op}}\to \mathrm{Grpd}$ 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 ${\overline{\mathrm{const}}}_C:\mathrm{Op}(X{)}^{\mathrm{op}}\to \mathrm{Grpd}$.

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

$$X\to {\overline{\mathrm{const}}}_C$$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 $U_{ij}=U_i\cap U_j$ an morphism $g_{ij}:F_i{\mid }_{I_{ij}}\stackrel{\simeq }{\to }{F}_j{\mid }_{U_{ij}}$, hence a bijection of finite sets;

• such that on triple overlaps we have $g_{ik}{\mid }_{U_{ijk}}=g_{jk}{\mid }_{U_{ijk}}\circ g_{ij}{\mid }_{U_{ijk}}$.

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

Definition

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

Without further assumption on $ℰ$ we have the following definition.

Definition

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

$$E\times U\simeq (\Delta F)\times U,$$E \times U \simeq (\Delta F) \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 $\Delta \mathrm{Aut}(X)$-principal bundle is called a Galois object .

If $ℰ$ is a locally connected topos there is another characterization of locally constant sheaves.

Definition

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

$$F(B^A)\to (F(B){)}^{F(A)}$$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 $\Delta :𝒮\to ℰ$ is a cartesian closed functor precisely if $ℰ$ is a locally connected topos.

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

Definition

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

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

Remark

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

Notice that

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

where the coproduct is over all cardinals smaller than $\kappa$ and where $B\mathrm{Aut}(F_i)$ 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 $ℰ$ a locally constant sheaf is the $\Delta \rho$-associated bundle to an $\Delta \mathrm{Aut}(F)$-principal bundle induced by the canonical permutation representation $\rho :B\mathrm{Aut}(F)\to \mathrm{Set}$ of the automorphism group $\mathrm{Aut}(F)$ on $F$.

Specifically for $g:*\to \Delta B\mathrm{Aut}(F)\simeq B\Delta \mathrm{Aut}(F)\hookrightarrow \Delta \mathrm{core}(\mathrm{set})$ 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

$$\begin{array}{ccccccc}U\times \Delta F& \to & P{\times }_{\Delta \mathrm{Aut}(F)(\Delta (F//\mathrm{Aut}(F)))}& \to & \Delta (F//\mathrm{Aut}(F))& \to & \Delta {\mathrm{Set}}^{\kappa }\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ U& \to & *& \stackrel{g}{\to }& B\Delta \mathrm{Aut}(F)& \hookrightarrow & \Delta \mathrm{core}({\mathrm{Set}}^{\kappa })\end{array},$$\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 $F//\mathrm{Aut}(F)$ is the action groupoid, the 2-colimit of $\rho B\mathrm{Aut}(F)\to \mathrm{Grpd}$.

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 / $\mathrm{\infty }$-stack is also called a local system.

References

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

• SGA 4, Exposé IX, 2.0 .

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)