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\begin{document}

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\section*{matching family}

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

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_sheaves}{Characterization of sheaves}\dotfill \pageref*{characterization_of_sheaves} \linebreak
\noindent\hyperlink{sheafification}{Sheafification}\dotfill \pageref*{sheafification} \linebreak
\noindent\hyperlink{CoRepresentationByCechGroupoid}{Co-representation by Cech groupoids}\dotfill \pageref*{CoRepresentationByCechGroupoid} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{matching family} of elements is an explicit component-wise characterizaton of a morphism from a [[sieve]] into a general [[presheaf]].

Since such morphisms govern the [[sheaf]] property and the operation of [[sheafification]], these can be discussed in terms of matching families. As such, the set of matching families for a given [[covering]] and [[presheaf]] is the corresponding [[descent object]].

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

Let $(C,\tau)$ be a [[site]] and $P:C^{\mathrm{op}}\to\mathrm{Set}$ a [[presheaf]] on $C$. Let $S\in \tau(c)$ be a [[cover]]ing [[sieve]] on [[object]] $c\in C$ (in particular a subobject of the [[representable functor|representable]] presheaf $h_c$).

A \textbf{matching family} for $S$ of elements in $P$ is a rule assigning to each $f:d\to c$ in $S$ an element $x_f$ of $P(d)$ such that for all $g:e\to d$

\begin{displaymath}
P(g)(x_f) = x_{f\circ g}.
\end{displaymath}
Notice that $f\circ g\in S$ because $S$ is a sieve, so that the condition makes sense; furthermore the order of composition and the contravariant nature of $P$ agree. If we view the sieve $S$ as a subobject of the representable $h_c$, then a matching family $(x_f)_{f\in S}$ is precisely a [[natural transformation]] $x:S\to P$, $x: f\mapsto x_f$.

An \textbf{amalgamation} of the matching family $(x_f)_{f\in S}$ for $S$ is an element $x\in P(c)$ such that $P(f)(x) = x_f$ for all $f\in S$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{characterization_of_sheaves}{}\subsubsection*{{Characterization of sheaves}}\label{characterization_of_sheaves}

$P$ is a [[sheaf]] for the [[Grothendieck topology]] $\tau$ iff for all $c$, for all $S\in\tau(c)$ and every matching family $(x)_{f\in S}$ for $S$, there is a unique amalgamation. Equivalently $P$ is a sheaf if any natural transformation $x:S\to P$ has a unique extension to $h_C\to P$ (along inclusion $S\hookrightarrow h_c$); or to phrase it differently, $P$ is a sheaf (resp. [[separated presheaf]]) iff the precomposition with the inclusion $i_S : S\hookrightarrow h_C$ is an [[isomorphism]] (resp. [[monomorphism]]) $i_S:\mathrm{Nat}(h_C,P)\to \mathrm{Nat}(S,P)$.

Suppose now that $C$ has all [[pullback]]s. Let $R = (f_i:c_i\to c)_{i\in I}$ be any [[cover]] of $c$ (i.e., the smallest [[sieve]] containing $R$ is a [[covering]] sieve in $\tau$) and let $p_{ij}:c_i\times_c c_j\to c_i$, $q_{ij}:c_i\times_c c_j\to c_j$ be the two projections of the pullback of $f_j$ along $f_i$. A \textbf{matching family} for $R$ of elements in a presheaf $P$ is by definition a family $(x_i)_{i\in I}$ of elements $x_i\in P(c_i)$, such that for all $i,j\in I$, $P(p_{ij})(x_i) = P(q_{ij})(x_j)$.

\hypertarget{sheafification}{}\subsubsection*{{Sheafification}}\label{sheafification}

Let $\mathrm{Match}(R,P)$ be the set of matching families for $R$ of elements in $P$. Sieves over $c$ form a [[filtered category]], where partial ordering is by reverse inclusion ([[refinement]] of sieves). There is an endofunctor $()^+ : PShv(C,\tau)\to PShv(C,\tau)$ given by

\begin{displaymath}
P^+(c) := \mathrm{colim}_{R\in\tau(C)} \mathrm{Match}(R,P)
\end{displaymath}
In other words, elements in $P^+(c)$ are matching families $(x^R_f)_{f\in R}$ for all covering sieves modulo the equivalence given by agreement $x^R_f = x^{R'}_f$, for all $f\in R''$, where $R''\subset  R\cap R'$ is a common refinement of $R$ and $R'$. This is called the [[plus construction]].

Endofunctor $P\mapsto P^+$ extends to a presheaf on $C$ by $P^+(g:d\to c) : (x_f)_{f\in R}\mapsto (x_{g\circ h})_{h\in g^*R}$ where $g^* R = \{h:e\to d | e\in C, g\circ h\in R\}$ (recall that by the stability axiom of Grothendieck topologies, $g^*(d)\in \tau(d)$ is a covering sieve over $d$).

The presheaf $P^+$ comes equipped with a canonical natural transformation $\eta:P\to P^+$ which to an element $x\in P(c)$ assigns the equivalence class of the matching family $(P(f)(x))_{f\in Ob(C/c)}$ where the [[maximal sieve]] $Ob(C/c)$ is the class of objects of the slice category $C/c$.

$\eta$ is a [[monomorphism]] (resp. [[isomorphism]]) of presheaves iff the presheaf $P$ is a [[separated presheaf]] (resp. [[sheaf]]); moreover any morphism $P\to F$ of presheaves, where $F$ is a sheaf, factors uniquely through $\eta:P\to P^+$. For any presheaf $P$, $P^+$ is separated presheaf and if $P$ is already separated then $P^+$ is a sheaf. In particular, for any presheaf $P^{++}$ is a sheaf. A fortiori, $P^+(\eta)\circ\eta:P\to P^{++}$ realizes [[sheafification]].

\hypertarget{CoRepresentationByCechGroupoid}{}\subsubsection*{{Co-representation by Cech groupoids}}\label{CoRepresentationByCechGroupoid}

When [[presheaves]] of sets are regarded a [[presheaves of groupoids]], the [[Cech groupoid]] serves to co-represent matching families, hence serves as the \emph{[[codescent object]]} of the given covering and presheaf. See \href{Čech+groupoid#Codescent}{there} for more.

\hypertarget{references}{}\subsection*{{References}}\label{references}

A standard reference is

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}, Springer 1992. (chapter III)

\end{itemize}
[[!redirects matching families]]



\end{document}