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global section

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The global sections of a bundle are simply its sections. When bundles are replaced by their sheaves of local sections, then forming global sections corresponds to the direct image operation on sheaves with respect to the morphism to the terminal site. This definition generalizes to objects in a general topos and (∞,1)-topos.

Definition

We start describing the more explicit notions of global sections of bundles and then work our way towards the more abstract notions in terms of topos theory.

Of bundles

A global section of a bundle EpBE \overset{p}\to B is simply a section of pp, that is a map s:BEs\colon B \to E such that ps=id Bp \circ s = \id_B.

E s p B id B. \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow^{\mathrlap{p}} \\ B &\stackrel{id}{\to}& B } \,.

The adjective ‘global’ here is used to distinguish from a local section: a generalised section over some subspace i:UBi : U \hookrightarrow B which is a section of the map to UU

i *E:=E UU i^* E := E|_U \to U

from the pullback

i *E:=E U E U i B. \array{ i^* E := E|_U &\to& E \\ \downarrow && \downarrow \\ U &\stackrel{i}{\to}& B } \,.

Compare the notion of global point, which is really the special case when BB is a terminal object (where the generalised section corresponds to a generalised element). On the other hand, a global section of EpBE \overset{p}\to B in 𝒞\mathcal{C} is simply a global point in the slice category 𝒞/B\mathcal{C}/B.

One often writes

Γ U(E):=Hom 𝒞/U(U,E U) \Gamma_U(E) := Hom_{\mathcal{C}/U}(U , E|_U)

for the set of global sections over UU (or Γ(U,E)\Gamma(U,E) or similar).

Of sheaves on topological spaces

Every sheaf ASh(X)=Sh(Op(X))A \in Sh(X) = Sh(Op(X)) on (the site that is given by the category of open subsets of) a topological space XX is the sheaf of local sections of its etale space bundle EXE \to X in that

A:UΓ U(E) A : U \mapsto \Gamma_U(E)

for every UOp(X)U \in Op(X). For this reasons one often speaks of the value of a sheaf on some object as a set of sections, even if the corresponding bundle is never mentioned and doesn’t really matter.

The set of global sections on XX is

Γ X(A)=A(X)=Hom Sh(X)(X,A), \Gamma_X(A) = A(X) = Hom_{Sh(X)}(X, A) \,,

where XSh(X)X \in Sh(X) denotes the terminal object of the category of sheaves Sh(X)Sh(X). Often this is written just using different notation

Γ X(A)=Hom Sh(X)(*,A) \Gamma_X(A) = Hom_{Sh(X)}(*,A)

One notices that Γ X():Sh(X)Set\Gamma_X(-) : Sh(X) \to Set defined this way is the direct image functor on Grothendieck toposes that is induced from the canonical morphism X*X \to * of topological spaces (now ”**” really denotes the point topological space!) and hence from the corresponding morphism of sites.

Again, this expression for global sections induces a relative version, e.g. for sheaves on SS-schemes, the direct image functor goes into the base scheme SS).

Of objects in a general Grothendieck topos

The definition of global sections of sheaves on topological spaces in terms of the direct image of the canonical morphism to the terminal site generalizes to sheaf toposes over arbitrary sites.

For every Grothendieck topos 𝒯\mathcal{T}, there is a geometric morphism

Γ:𝒯Set:LConst \Gamma : \mathcal{T} \stackrel{\leftarrow}{\to} Set : LConst

called the global sections functor. It is given by the hom-set out of the terminal object

Γ()=Hom 𝒯(*,) \Gamma(-) = Hom_{\mathcal{T}}({*}, -)

and hence assigns to each object A𝒯A\in \mathcal{T} its set of global elements Γ(A)=Hom E(*,A)\Gamma(A) = Hom_E(*,A).

The left adjoint LConst:SetELConst : Set \to E of the global section functor is the canonical Set-tensoring functor

:Set×𝒯𝒯 \otimes : Set \times \mathcal{T} \to \mathcal{T}

applied to the terminal object

const=()*:Set𝒯 const = (-)\otimes {*} : Set \to \mathcal{T}

which sends a set SS to the coproduct of S|S| copies of the terminal object

S*= sS*. S \otimes {*} = \coprod_{s \in S} {*} \,.

This is called the constant object of 𝒯\mathcal{T} on the set SS. Notably when 𝒯\mathcal{T} is a sheaf topos this is the constant sheaf LConst SLConst_S on SS.

𝒯ΓLConstSet. \mathcal{T} \stackrel{\stackrel{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set \,.

If the topos 𝒯\mathcal{T} is a locally connected topos then the left adjoint functor LConstLConst is also a right adjoint, its left adjoint being the functor Π 0:𝒯Set\Pi_0 : \mathcal{T} \to Set that sends an object to its set of connected components.

Of objects in an (,1)(\infty,1)-topos

The previous abstract definition generalizes straightforwardly to every context of higher category theory where the required notions of adjoint functor etc. are provided.

Notably in (∞,1)-category theory the global section functor on an ∞-stack (∞,1)-topos H\mathbf{H} is the hom-functor

Γ():=H(*,):H=Sh (,1)(C)Sh (,1)(*)=Grpd \Gamma(-) := \mathbf{H}(*,-) : \; \mathbf{H} = Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(*) = \infty Grpd

of morphisms out of the terminal object.

This is indeed again the terminal geometric morphism

Proposition

Let H\mathbf{H} be an ∞-stack (∞,1)-topos. Then the ∞-groupoid Geom(H,Grpd)Geom(\mathbf{H}, \infty Grpd) of geometric (∞,1)-functors is contractible.

So Grpd\infty Grpd is the terminal object in the (∞,1)-category of (∞,1)-toposes and geometric morphisms.

Proof

This is HTT 6.3.4.1

If the (∞,1)-topos is a locally contractible (∞,1)-topos then this is an essential geometric morphism.

The composite (∞,1)-functor ΓLConst\Gamma \circ LConst is the shape of H\mathbf{H}.

Revised on January 5, 2014 04:04:41 by heisenbug? (84.148.59.172)