matching family

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.

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 $g:e\to d$

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

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 $f\in S$.

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

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

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

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.

A standard reference is

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

Revised on December 17, 2011 14:23:24
by Urs Schreiber
(89.204.137.170)