Abelian sheafification and quotient sheaves on a site

Fix Grothendieck universes $\mathcal{U}\in \mathcal{V}$ with $\omega \in \mathcal{V}$. A small site $\mathcal{C}$ here is a category whose object and morphism sets belong to $\mathcal{U}$, equipped with a Grothendieck topology $J$. A sieve on $T$ is a set of arrows into $T$ closed under precomposition. For $u:V\to T$, the pullback sieve $u^{\ast }R$ consists of arrows $v:W\to V$ with $uv\in R$. A Grothendieck topology specifies covering sieves on each object: the maximal sieve covers; pullback preserves covering sieves; and if $R$ covers $T$ and $u^{\ast }S$ covers the domain of every arrow $u$ in $R$, then the sieve $S$ covers $T$. No scheme condition is imposed on the objects or arrows. Presheaves take values in $\mathcal{V}$-small sets or abelian groups. A presheaf of sets is a sheaf when every compatible family of sections on a covering sieve has a unique amalgamation on its target.

An abelian presheaf is a contravariant functor from $\mathcal{C}$ to abelian groups; an abelian sheaf is one whose underlying presheaf of sets is a sheaf. Morphisms are natural transformations with homomorphisms as components. There is a functor $a_{\mathrm{A}\mathrm{b}}$ from abelian presheaves to abelian sheaves, left adjoint to inclusion. The category of abelian sheaves is abelian. For a morphism $f:A\to B$ of abelian sheaves, its cokernel is ${\mathrm{coker}}_{\mathrm{S}\mathrm{h}\mathrm{A}\mathrm{b}}(f)=a_{\mathrm{A}\mathrm{b}}(T\mapsto B(T)\mathrm{/}\mathrm{im}(f_T))\mathrm{.}$ In particular, the quotient is sheafified; its sections are not asserted to be the objectwise quotient.

For a presheaf of sets $P$, form the abelian presheaf $\mathbb{Z}[P{]}_{\mathrm{p}\mathrm{r}\mathrm{e}}$ by taking the free abelian group on $P(T)$ at every object $T$ and extending each restriction map to a homomorphism. Its associated abelian sheaf $a_{\mathrm{A}\mathrm{b}}\mathbb{Z}[P{]}_{\mathrm{p}\mathrm{r}\mathrm{e}}$ is functorial in $P$.

Scheme sites and sheafification

Free abelian group on a set

Abelian groups form an abelian category

Abelian category

Grothendieck universe closure convention

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Covariant functor, identity functor, composite functor, and contravariant functor

Natural transformation and its components

Adjunction by unit, counit, and the triangle identities

The kernel and image of a group homomorphism

The quotient group G/N and coset product (gN)(hN)=ghN

Reference 0 supplies associated abelian sheaves, the abelian-category assertion and the sheaf cokernel formula. Reference 1 supplies the free abelian sheaf construction and its functoriality. These are imported interfaces with the hypotheses and reading limits of the two references. No local proof of algebraic sheafification is supplied.

The general small site is defined inline above. Its sieve, topology and underlying set-sheaf conventions agree with those in Scheme sites and sheafification, without that definition's additional restriction to schemes. The universe convention is Grothendieck universe closure convention; smallness here means that both the object set and the morphism set belong to the indicated universe. Categories, functors, natural transformations and left adjoints have the meanings of Category, object, morphism, domain, codomain, identity, composition, and hom-collection, Covariant functor, identity functor, composite functor, and contravariant functor, Natural transformation and its components and Adjunction by unit, counit, and the triangle identities, respectively.

The values and free groups have the meanings supplied by Abelian groups form an abelian category and Free abelian group on a set. The image subgroup notation is that of The kernel and image of a group homomorphism, and the objectwise quotient uses The quotient group G/N and coset product (gN)(hN)=ghN. The category assertion has the AB1 and AB2 meaning of Abelian category.