topos theory

# Contents

## Idea

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 explit 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 $E\stackrel{p}{\to }B$ is simply a section of $p$, that is a map $s:B\to E$ such that $p\circ s={id}_{B}$.

$\begin{array}{ccc}& & E\\ & {}^{s}↗& {↓}^{p}\\ B& \stackrel{\mathrm{Id}}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $i:U↪B$ which is a section of the map to $U$

${i}^{*}E:=E{\mid }_{U}\to U$i^* E := E|_U \to U

from the pullback

$\begin{array}{ccc}{i}^{*}E:=E{\mid }_{U}& \to & E\\ ↓& & ↓\\ U& \stackrel{i}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $B$ is a terminal object (where the generalised section corresponds to a generalised element). On the other hand, a global section of $E\stackrel{p}{\to }B$ in $𝒞$ is simply a global point in the slice category $𝒞/B$.

One often writes

${\Gamma }_{U}\left(E\right):={\mathrm{Hom}}_{𝒞/U}\left(U,E{\mid }_{U}\right)$\Gamma_U(E) := Hom_{\mathcal{C}/U}(U , E|_U)

for the set of global sections over $U$ (or $\Gamma \left(U,E\right)$ or similar).

### Of sheaves on topological spaces

Every sheaf $A\in \mathrm{Sh}\left(X\right)=\mathrm{Sh}\left(\mathrm{Op}\left(X\right)\right)$ on (the site that is given by the category of open subsets of) a topological space $X$ is the sheaf of local sections of its etale space bundle $E\to X$ in that

$A:U↦{\Gamma }_{U}\left(E\right)$A : U \mapsto \Gamma_U(E)

for every $U\in \mathrm{Op}\left(X\right)$. 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 $X$ is

${\Gamma }_{X}\left(A\right)=A\left(X\right)={\mathrm{Hom}}_{\mathrm{Sh}\left(X\right)}\left(X,A\right)\phantom{\rule{thinmathspace}{0ex}},$\Gamma_X(A) = A(X) = Hom_{Sh(X)}(X, A) \,,

where $X\in \mathrm{Sh}\left(X\right)$ denotes the terminal object of the category of sheaves $\mathrm{Sh}\left(X\right)$. Often this is written just using different notation

${\Gamma }_{X}\left(A\right)={\mathrm{Hom}}_{\mathrm{Sh}\left(X\right)}\left(*,A\right)$\Gamma_X(A) = Hom_{Sh(X)}(*,A)

One notices that ${\Gamma }_{X}\left(-\right):\mathrm{Sh}\left(X\right)\to \mathrm{Set}$ defined this way is the direct image functor on Grothendieck toposes that is induced from the canonical morphism $X\to *$ 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 $S$-schemes, the direct image functor goes into the base scheme $S$).

### 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 $𝒯$, there is a geometric morphism

$\Gamma :𝒯\stackrel{←}{\to }\mathrm{Set}:\mathrm{LConst}$\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

$\Gamma \left(-\right)={\mathrm{Hom}}_{𝒯}\left(*,-\right)$\Gamma(-) = Hom_{\mathcal{T}}({*}, -)

and hence assigns to each object $A\in 𝒯$ its set of global elements $\Gamma \left(A\right)={\mathrm{Hom}}_{E}\left(*,A\right)$.

The left adjoint $\mathrm{LConst}:\mathrm{Set}\to E$ of the global section functor is the canonical Set-tensoring functor

$\otimes :\mathrm{Set}×𝒯\to 𝒯$\otimes : Set \times \mathcal{T} \to \mathcal{T}

applied to the terminal object

$\mathrm{const}=\left(-\right)\otimes *:\mathrm{Set}\to 𝒯$const = (-)\otimes {*} : Set \to \mathcal{T}

which sends a set $S$ to the coproduct of $\mid S\mid$ copies of the terminal object

$S\otimes *=\coprod _{s\in S}*\phantom{\rule{thinmathspace}{0ex}}.$S \otimes {*} = \coprod_{s \in S} {*} \,.

This is called the constant object of $𝒯$ on the set $S$. Notably when $𝒯$ is a sheaf topos this is the constant sheaf ${\mathrm{LConst}}_{S}$ on $S$.

$𝒯\stackrel{\stackrel{\mathrm{LConst}}{←}}{\stackrel{\Gamma }{\to }}\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{T} \stackrel{\stackrel{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set \,.

If the topos $𝒯$ is a locally connected topos then the left adjoint functor $\mathrm{LConst}$ is also a right adjoint, its left adjoint being the functor ${\Pi }_{0}:𝒯\to \mathrm{Set}$ that sends an object to its set of connected components.

### Of objects in an $\left(\infty ,1\right)$-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 $H$ is the hom-functor

$\Gamma \left(-\right):=H\left(*,-\right):\phantom{\rule{thickmathspace}{0ex}}H={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)\to {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(*\right)=\infty \mathrm{Grpd}$\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 $H$ be an ∞-stack (∞,1)-topos. Then the ∞-groupoid $\mathrm{Geom}\left(H,\infty \mathrm{Grpd}\right)$ of geometric (∞,1)-functors is contractible.

So $\infty \mathrm{Grpd}$ 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 $\Gamma \circ \mathrm{LConst}$ is the shape of $H$.

Revised on August 14, 2013 13:17:06 by Urs Schreiber (24.131.18.91)