nLab
locally constant sheaf

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

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

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

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

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

Xconst¯ C X \to \bar const_C

are represented by

  • an open cover {U i}\{U_i\} of XX;

  • over each U iU_i a choice F iCF_i \in C of object in CC, hence a finite set in CC;

  • over each double overlap U ij=U iU jU_{i j} = U_i \cap U_j an morphism g ij:F i| I ijF j| U ijg_{i j} : F_i|_{I_{i j}} \stackrel{\simeq}{\to} F_j|_{U_{i j}}, hence a bijection of finite sets;

  • such that on triple overlaps we have g ik| U ijk=g jk| U ijkg ij| U ijkg_{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 XX with typical fiber any of the F iF_i.

Definition

Let (ΔΓ):ΓΔ𝒮(\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 𝒮\mathcal{S}.

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

Definition

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

E×U(ΔF)×U, E \times U \simeq (\Delta F) \times U \,,

for some SS \in Set.

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

A locally constant object EE which is in addition an ΔAut(X)\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 CC and CC cartesian closed categories, a functor F:CDF : C \to D that preserves products is called a cartesian closed functor if the canonical natural transformation

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

(which is the adjunct of F(A)×F(B A)F(A×B A)F(B)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 Δ:𝒮\Delta : \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 =Sh(C)\mathcal{E} = Sh(C) be a locally connected topos. Let p:core(Set * κ)core(Set κ)p : core(Set^\kappa_*) \to core(Set^\kappa) be the core of the generalized universal bundle for sets of cardinality less than some κ\kappa.

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

Remark

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

Notice that

core(Setκ) iBAut(F i), core(Set \kappa) \simeq \coprod_i \mathbf{B}Aut(F_i) \,,

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

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

Specifically for g:*ΔBAut(F)BΔAut(F)Δcore(set)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*U \to * an epimorphism on which it trivializes, we have a pasting diagram of pullbacks

U×ΔF P× ΔAut(F)(Δ(F//Aut(F))) Δ(F//Aut(F)) ΔSet κ U * g BΔAut(F) Δcore(Set κ), \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)F//Aut(F) is the action groupoid, the 2-colimit of ρBAut(F)Grpd\rho \mathbf{B}Aut(F) \to Grpd.

Applications

Pattern

In sufficiently highly locally connected cases, we have:

A locally constant sheaf / \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 .

Lecture notes are in

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

and definition 2.2 in

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

Revised on November 22, 2013 04:09:34 by Urs Schreiber (82.169.114.243)