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,τ)(C,\tau) be a site and P:C opSetP:C^{\mathrm{op}}\to\mathrm{Set} a presheaf on CC. Let Sτ(c)S\in \tau(c) be a covering sieve on object cCc\in C (in particular a subobject of the representable presheaf h ch_c).

A matching family for SS of elements in PP is a rule assigning to each f:dcf:d\to c in SS an element x fx_f of P(d)P(d) such that for all g:edg:e\to d

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

Notice that fgSf\circ g\in S because SS is a sieve, so that the condition makes sense; furthermore the order of composition and the contravariant nature of PP agree. If we view the sieve SS as a subobject of the representable h ch_c, then a matching family (x f) fS(x_f)_{f\in S} is precisely a natural transformation x:SPx:S\to P, x:fx fx: f\mapsto x_f.

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


Characterization of sheaves

PP is a sheaf for the Grothendieck topology τ\tau iff for all cc, for all Sτ(c)S\in\tau(c) and every matching family (x) fS(x)_{f\in S} for SS, there is a unique amalgamation. Equivalently PP is a sheaf if any natural transformation x:SPx:S\to P has a unique extension to h CPh_C\to P (along inclusion Sh cS\hookrightarrow h_c); or to phrase it differently, PP is a sheaf (resp. separated presheaf) iff the precomposition with the inclusion i S:Sh Ci_S : S\hookrightarrow h_C is an isomorphism (resp. monomorphism) i S:Nat(h C,P)Nat(S,P)i_S:\mathrm{Nat}(h_C,P)\to \mathrm{Nat}(S,P).

Suppose now that CC has all pullbacks. Let R=(f i:c ic) iIR = (f_i:c_i\to c)_{i\in I} be any cover of cc (i.e., the smallest sieve containing RR is a covering sieve in τ\tau) and let p ij:c i× cc jc ip_{ij}:c_i\times_c c_j\to c_i, q ij:c i× cc jc jq_{ij}:c_i\times_c c_j\to c_j be the two projections of the pullback of f jf_j along f if_i. A matching family for RR of elements in a presheaf PP is by definition a family (x i) iI(x_i)_{i\in I} of elements x iP(c i)x_i\in P(c_i), such that for all i,jIi,j\in I, P(p ij)(x i)=P(q ij)(x j)P(p_{ij})(x_i) = P(q_{ij})(x_j).


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

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

In other words, elements in P +(c)P^+(c) are matching families (x f R) fR(x^R_f)_{f\in R} for all covering sieves modulo the equivalence given by agreement x f R=x f Rx^R_f = x^{R'}_f, for all fRf\in R'', where RRRR''\subset R\cap R' is a common refinement of RR and RR'. This is called the plus construction.

Endofunctor PP +P\mapsto P^+ extends to a presheaf on CC by P +(g:dc):(x f) fR(x gh) hg *RP^+(g:d\to c) : (x_f)_{f\in R}\mapsto (x_{g\circ h})_{h\in g^*R} where g *R={h:ed|eC,ghR}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)τ(d)g^*(d)\in \tau(d) is a covering sieve over dd).

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

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


A standard reference is

Last revised on January 3, 2016 at 19:16:35. See the history of this page for a list of all contributions to it.