nLab
matching family

Contents

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,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.

Properties

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).

Sheafification

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.

Co-representation by Cech groupoids

When presheaves of sets are regarded a presheaves of groupoids, the Cech 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

Last revised on June 13, 2018 at 10:54:33. See the history of this page for a list of all contributions to it.