nLab
matching family

Idea

A 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.

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 covering sieve on object $c\in C$ (in particular a subobject of the representable presheaf $h_c$).

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

$$P(g)(x_f)=x_{f\circ g}.$$P(g)(x_f) = x_{f\circ g}.

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 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 $f\in S$.

Properties

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 pullbacks. 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 $p_{\mathrm{ij}}:c_i{\times }_c{c}_j\to c_i$, $q_{\mathrm{ij}}:c_i{\times }_c{c}_j\to c_j$ be the two projections of the pullback of $f_j$ along $f_i$. A 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 $i,j\in I$, $P(p_{\mathrm{ij}})(x_i)=P(q_{\mathrm{ij}})(x_j)$.

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 $({)}^{+}:\mathrm{PShv}(C,\tau )\to \mathrm{PShv}(C,\tau )$ given by

$$P^{+}(c):={\mathrm{colim}}_{R\in \tau (C)}\mathrm{Match}(R,P)$$P^+(c) := \mathrm{colim}_{R\in\tau(C)} \mathrm{Match}(R,P)

In other words, elements in $P^{+}(c)$ are matching families $(x_f^R{)}_{f\in R}$ for all covering sieves modulo the equivalence given by agreement $x_f^R=x_f^{R\prime }$, for all $f\in R″$, where $R″\subset R\cap R\prime$ is a common refinement of $R$ and $R\prime$. 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 $g^{*}R=\{h:e\to d\mid e\in C,g\circ h\in R\}$ (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 $(P(f)(x){)}_{f\in \mathrm{Ob}(C/c)}$ where the maximal sieve $\mathrm{Ob}(C/c)$ 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.

References

A standard reference is

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer 1992. (chapter III)