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. As such, the set of matching families for a given covering and presheaf is the corresponding descent object.

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 $x_f$ of $P(d)$ 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 $f\in S$.

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 $p_{ij}:c_i\times_c c_j\to c_i$, $q_{ij}:c_i\times_c c_j\to c_j$ 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 $i,j\in I$, $P(p_{ij})(x_i) = P(q_{ij})(x_j)$.

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

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

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.

Co-representation by Čech groupoids

When presheaves of sets are regarded a presheaves of groupoids, the Čech groupoid serves to co-represent matching families, hence serves as the codescent object of the given covering and presheaf. See there for more.

References

A standard reference is

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