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\begin{document}

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\section*{global section}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{of_bundles}{Of bundles}\dotfill \pageref*{of_bundles} \linebreak
\noindent\hyperlink{of_sheaves_on_topological_spaces}{Of sheaves on topological spaces}\dotfill \pageref*{of_sheaves_on_topological_spaces} \linebreak
\noindent\hyperlink{GeneralTopos}{Of objects in a general Grothendieck topos}\dotfill \pageref*{GeneralTopos} \linebreak
\noindent\hyperlink{of_objects_in_an_topos}{Of objects in an $(\infty,1)$-topos}\dotfill \pageref*{of_objects_in_an_topos} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{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 object|terminal]] [[site]]. This definition generalizes to objects in a general [[topos]] and [[(∞,1)-topos]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{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.

\hypertarget{of_bundles}{}\subsubsection*{{Of bundles}}\label{of_bundles}

A \textbf{global [[section]]} of a [[bundle]] $E \overset{p}\to B$ is simply a [[section]] of $p$, that is a map $s\colon B \to E$ such that $p \circ s = \id_B$.

\begin{displaymath}
\itexarray{
    && E
    \\
    & {}^{\mathllap{s}}\nearrow & \downarrow^{\mathrlap{p}}
    \\
    B &\stackrel{id}{\to}& B
  }
  \,.
\end{displaymath}
The adjective `global' here is used to distinguish from a \textbf{[[local section]]}: a \emph{generalised} section over some subspace $i : U \hookrightarrow B$ which is a section of the map to $U$

\begin{displaymath}
i^* E := E|_U \to U
\end{displaymath}
from the [[pullback]]

\begin{displaymath}
\itexarray{
    i^* E := E|_U &\to& E
    \\
    \downarrow && \downarrow
    \\
    U &\stackrel{i}{\to}& B
  }
  \,.
\end{displaymath}
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 $E \overset{p}\to B$ in $\mathcal{C}$ \emph{is} simply a global point in the [[slice category]] $\mathcal{C}/B$.

One often writes

\begin{displaymath}
\Gamma_U(E) := Hom_{\mathcal{C}/U}(U , E|_U)
\end{displaymath}
for the \textbf{set of global sections} over $U$ (or $\Gamma(U,E)$ or similar).

\hypertarget{of_sheaves_on_topological_spaces}{}\subsubsection*{{Of sheaves on topological spaces}}\label{of_sheaves_on_topological_spaces}

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

\begin{displaymath}
A : U \mapsto \Gamma_U(E)
\end{displaymath}
for every $U \in Op(X)$. 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 $X$ is

\begin{displaymath}
\Gamma_X(A) = A(X) = Hom_{Sh(X)}(X, A)
  \,,
\end{displaymath}
where $X \in Sh(X)$ denotes the [[terminal object]] of the [[category of sheaves]] $Sh(X)$. Often this is written just using different notation

\begin{displaymath}
\Gamma_X(A) = Hom_{Sh(X)}(*,A)
\end{displaymath}
One notices that $\Gamma_X(-) : Sh(X) \to Set$ defined this way is the [[direct image]] functor on [[Grothendieck topos]]es that is induced from the canonical morphism $X \to *$ of [[topological space]]s (now ``$*$'' really denotes the [[point]] topological space!) and hence from the corresponding morphism of [[site]]s.

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

\hypertarget{GeneralTopos}{}\subsubsection*{{Of objects in a general Grothendieck topos}}\label{GeneralTopos}

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 [[Grothendieck topos|sheaf topos]]es over arbitrary [[site]]s.

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

\begin{displaymath}
\Gamma : \mathcal{T}  \stackrel{\leftarrow}{\to} Set : LConst
\end{displaymath}
called the \textbf{global sections} functor. It is given by the [[hom-set]] out of the [[terminal object]]

\begin{displaymath}
\Gamma(-) = Hom_{\mathcal{T}}({*}, -)
\end{displaymath}
and hence assigns to each object $A\in \mathcal{T}$ its set of [[global element]]s $\Gamma(A) = Hom_E(*,A)$.

The [[left adjoint]] $LConst : Set \to E$ of the global section functor is the canonical [[Set]]-[[copower|tensoring]] functor

\begin{displaymath}
\otimes : Set \times \mathcal{T} \to \mathcal{T}
\end{displaymath}
applied to the [[terminal object]]

\begin{displaymath}
const = (-)\otimes {*} : Set \to \mathcal{T}
\end{displaymath}
which sends a set $S$ to the [[coproduct]] of $|S|$ copies of the terminal object

\begin{displaymath}
S \otimes {*} = \coprod_{s \in S} {*}
  \,.
\end{displaymath}
This is called the \textbf{constant object} of $\mathcal{T}$ on the set $S$. Notably when $\mathcal{T}$ is a [[Grothendieck topos|sheaf topos]] this is the \textbf{[[constant sheaf]]} $LConst_S$ on $S$.

\begin{displaymath}
\mathcal{T} \stackrel{\stackrel{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set
  \,.
\end{displaymath}
If the topos $\mathcal{T}$ is a [[locally connected topos]] then the left adjoint functor $LConst$ is also a right adjoint, its left adjoint being the functor $\Pi_0 : \mathcal{T} \to Set$ that sends an object to its set of connected components.

\hypertarget{of_objects_in_an_topos}{}\subsubsection*{{Of objects in an $(\infty,1)$-topos}}\label{of_objects_in_an_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]] $\mathbf{H}$ is the [[hom-functor]]

\begin{displaymath}
\Gamma(-) := \mathbf{H}(*,-) : \;
  \mathbf{H} = Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(*) = \infty Grpd
\end{displaymath}
of morphisms out of the terminal object.

This is indeed again the terminal geometric morphism

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathbf{H}$ be an [[∞-stack]] [[(∞,1)-topos]]. Then the [[∞-groupoid]] $Geom(\mathbf{H}, \infty Grpd)$ of [[geometric morphism|geometric]] [[(∞,1)-functor]]s is [[contractible]].

So $\infty Grpd$ is the [[terminal object]] in the [[(∞,1)-category]] of [[(∞,1)-topos]]es and geometric morphisms.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT 6.3.4.1]]

\end{proof}
If the [[(∞,1)-topos]] is a [[locally contractible (∞,1)-topos]] then this is an [[essential geometric morphism]].

The composite [[(∞,1)-functor]] $\Gamma \circ LConst$ is the [[shape of an (∞,1)-topos|shape of]] $\mathbf{H}$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[local section]], [[flat section]]


\item [[abelian sheaf cohomology]]


\item [[ex-space]]


\item [[global element]]



\end{itemize}
[[!redirects global sections]]

[[!redirects global sections functor]] [[!redirects global section functor]] [[!redirects global sections functors]] [[!redirects global section functors]]

[[!redirects global section geometric morphism]] [[!redirects global section geometric morphisms]] [[!redirects global sections geometric morphism]] [[!redirects global sections geometric morphisms]]



\end{document}