Scheme sites and sheafification

PDF p. 4: Nisnevich sheaves and cdh cohomology

Fix Grothendieck universes U in V. A scheme site here is a U-small category C whose objects are schemes and whose morphisms are scheme morphisms, equipped with a Grothendieck topology J. A sieve on X is a set of arrows into X closed under precomposition; a topology selects covering sieves, including the maximal sieve, stable under pullback and satisfying transitivity. A presheaf of sets is a contravariant functor C → Set_V. It is a sheaf when, for every covering sieve, every compatible family of sections indexed by its arrows has a unique amalgamation on the target.

The sheafification of a presheaf F is the sheaf aF = (F+)+, equipped with the composite F → F+ → (F+)+. Here F+(X) consists of matching families on covering sieves of X, with two families identified when they agree on a common covering subsieve; restriction uses pullback of sieves. Equivalently, a is left adjoint to the inclusion of sheaves into presheaves, with the displayed composite as unit.

A big site over a base S has a specified category of S-schemes as its objects; an S-scheme is a scheme equipped with a morphism to S, and its morphisms commute with those structure maps. A small site of a fixed S-scheme X instead uses a specified category of schemes over X, such as a suitable category of étale X-schemes. The chosen categories and covering topologies must be stated separately: the words small and big alone do not determine them. A complex of abelian sheaves, or a derived presheaf, requires its own abelian or derived-category conventions; set-valued sheafification alone does not assert a complex-valued descent or localization theorem.

Schemes and morphisms over a base

Small sites and the plus construction

The size hypotheses make the sets of sieves, matching families and equivalence classes legitimate V-small sets. The plus construction is a presheaf and its unit is natural by The plus construction is a presheaf. It is separated by The plus presheaf is separated, so its second application is a sheaf by The plus construction turns a separated presheaf into a sheaf. For every sheaf G, applying Universal mapping property of the plus construction successively to F and F+ gives natural bijections Hom((F+)+,G) ≅ Hom(F+,G) ≅ Hom(F,G). This proves the stated adjunction and identifies its unit. The definitions of the scheme categories require the separately specified topology and size bound; they do not claim that every possible category of schemes is U-small.