(Presheaf)
A presheaf \(\mcf\) of abelian groups on a space \(X\) assigns a group \(\mcf(U)\) to each open \(U\) and a restriction \(\res{U}{V}: \mcf(U) \to \mcf(V)\) to each inclusion \(V \subseteq U\), functorially, with \(\mcf(\emptyset) = 0\). Equivalently, \(\mcf\) is a contravariant functor from the category of open subsets of \(X\), with inclusions as morphisms, to abelian groups. More generally, a presheaf with values in a category \(\mcc\) (sets, rings, modules) is a contravariant functor from open subsets of \(X\) to \(\mcc\), and sheaves with values in \(\mcc\) are defined by the same two axioms. Elements of \(\mcf(U)\) are sections of \(\mcf\) over \(U\), and elements of \(\mcf(X)\) are global sections.
(Sheaf)
A presheaf is a sheaf if for every open \(U\) and every open cover \(\ts{U_i}\) of \(U\):
identity: a section \(s \in \mcf(U)\) with \(\ro{s}{U_i} = 0\) for all \(i\) is \(0\);
gluing: sections \(s_i \in \mcf(U_i)\) agreeing on every overlap \(U_i \intersect U_j\) come from a section of \(\mcf(U)\).
A presheaf satisfying the identity axiom is a separated presheaf.
For a separated presheaf \(\mcf\), the map \(\mcf\to\mcf^+\) to its sheafification is injective on sections; for an arbitrary presheaf, its kernel on \(\mcf(U)\) consists of the sections whose restrictions to the members of some open cover of \(U\) vanish.
(The equalizer form)
Both axioms at once say that \[ \mcf(U) \to \prod_i \mcf(U_i) \rightrightarrows \prod_{i,j} \mcf(U_i \intersect U_j) \] is an equalizer, the two maps being restriction from \(U_i\) and from \(U_j\) to the overlap. Injectivity of the first map is identity; that its image is exactly the equalizer is gluing. This equalizer condition, with fibre products \(U_i\times_UU_j\) in place of intersections, is the definition of a sheaf on a site (D-GROTHTOP).
Backlinks
PR-IP6ZGDirect limits of sheaves, and the Noetherian hypothesis that removes the sheafificationusesFE-AM5Z8Presheaves that are not sheavesusesT-3VX80SheafificationusesD-GROTHTOPSites, sheaves on a site, and topoiusesD-0QSI0Stalks and germsuses