Local simplicial sheaf and additive chain model interfaces

Assume the Axiom of Choice and fixed compatible Grothendieck universes $\mathcal{U}\in \mathcal{V}$ with $\omega \in \mathcal{U}$. Let $\mathcal{C}$ be any $\mathcal{U}$-small category with a Grothendieck topology, in the general small-site convention. Presheaves take values in $\mathcal{V}$-small sets. A simplicial presheaf is a contravariant functor from $\mathcal{C}$ to simplicial sets; a simplicial sheaf has a sheaf in every degree. An abelian presheaf is a contravariant functor to abelian groups, and it is an abelian sheaf when its underlying presheaf of sets is a sheaf. Write $a$ for associated abelian sheafification and $\mathrm{Ab}(\mathcal{C})$ for abelian sheaves. This category is abelian, and $a$ is exact and left adjoint to inclusion. For a presheaf of sets $S$, write ${\mathbb{Z}}_{\mathrm{s}\mathrm{h}}[S]=a(U\mapsto \mathbb{Z}[S(U)])$.

Here is the precise local weak-equivalence convention. For a simplicial set $T$, use its component set ${\pi }_0(T)$ and the ordinary groups ${\pi }_i(T,t)={\pi }_i(\mathrm{\mid }T\mathrm{\mid },t)$ of its geometric realization for a vertex $t$ and $i\ge 1$. Let ${\tilde{\pi }}_{0}S$ be the associated sheaf of $V\mapsto {\pi }_0(S(V))$. For $U\in \mathcal{C}$, the slice $\mathcal{C}\mathrm{/}U$ has objects $v:V\to U$ and morphisms the commuting triangles; a sieve on $v$ covers precisely when its underlying sieve on $V$ covers in $\mathcal{C}$. Given $x\in S_0(U)$, let ${\tilde{\pi }}_i(S{\mathrm{\mid }}_U,x)$ be the associated group sheaf on this slice of $v\mapsto {\pi }_i(S(V),v^{\ast }x)$. A map $f:S\to T$ is a local weak equivalence exactly when ${\tilde{\pi }}_{0}S\to {\tilde{\pi }}_{0}T$ is an isomorphism and, for every $U$, $x\in S_0(U)$ and $i\ge 1$, the map ${\tilde{\pi }}_i(S{\mathrm{\mid }}_U,x)\to {\tilde{\pi }}_i(T{\mathrm{\mid }}_U,f(x))$ is an isomorphism. This definition requires no enough-points hypothesis.

Simplicial presheaves and simplicial sheaves have the local injective simplicial model structures: weak equivalences are the local weak equivalences just defined, cofibrations are monomorphisms, and fibrations have the right lifting property against trivial cofibrations. Sheafification and inclusion give a Quillen equivalence. The simplicial function complex $\mathrm{Map}(X,Y)$ has $q$-simplices the maps from $X\otimes \mathrm{\Delta }[q]$ to $Y$; for presheaves the tensor is the sectionwise product, and for sheaves it is its degreewise associated sheaf. Fibrations are sectionwise Kan fibrations, and any two fibrant models of one object are sectionwise simplicially homotopy equivalent.

Simplicial abelian presheaves and sheaves have simplicial model structures whose fibrations and trivial fibrations are created by the forgetful functor $u$ to the corresponding simplicial presheaves or sheaves. Their weak equivalences are precisely the normalized-chain homology-sheaf isomorphisms. Free abelianization, with sheafification when appropriate, is simplicial left Quillen; additive sheafification and inclusion give a Quillen equivalence. The additive function complex has $q$-simplices the additive maps $A\otimes \mathbb{Z}[\mathrm{\Delta }[q]]\to B$. Here the tensor in simplicial degree $r$ is the ordinary abelian-group tensor product $A_r(V){\otimes }_{\mathbb{Z}}\mathbb{Z}[\mathrm{\Delta }[q{]}_r]$ on each section $V$, followed by abelian sheafification for sheaves; simplex operators act on both factors.

Normalization $N$ and its inverse $\mathrm{\Gamma }$ transport this additive model to nonnegative chain complexes. In degree $r$, $N(A{)}_r={\bigcap }_{j<r}\ker d_j$ with differential $(-1{)}^r{d}_r$; the inverse is the ordinary Dold–Kan inverse applied sectionwise, with the sheaf-local assertion supplied by the imported additive model. We also use its reindexing to cochain degrees at most zero, by $E_r=E^{-r}$ with the same differential. Cohomology is $H^j(E)=\ker (d^j)\mathrm{/}\mathrm{im}(d^{j-1})$ in the abelian category of sheaves; quasi-isomorphism means isomorphism on every such cohomology sheaf.

The mapping-corner axiom in these simplicial models says that a cofibration $i:A\to B$ and fibration $p:X\to Y$ induce a Kan fibration $\mathrm{Map}(B,X)\to \mathrm{Map}(A,X){\times }_{\mathrm{Map}(A,Y)}\mathrm{Map}(B,Y)$, trivial if either $i$ or $p$ is a weak equivalence. A derived function complex is $\mathrm{Map}(QX,RY)$, using a cofibrant replacement $QX\to X$ and a fibrant replacement $Y\to RY$; the simplicial model axioms give its invariance under replacements and weak equivalences.

Small sites and the plus construction

Abelian category

Cochain complex in an abelian category

Cohomology object of a cochain complex

Model categories and Quillen adjunctions

Simplicial sets, homotopies and trivial Kan fibrations

Simplicial horns and Kan fibrations

Free abelian group on a set

The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums

Dold-Kan equivalence for simplicial modules with explicit inverse

The Axiom of Choice

Reference 0 imports the local simplicial models, their function complexes and mapping-corner axiom, sheaf comparison and fibrant evaluation. Reference 1 imports their additive counterparts and normalization transport. Reference 4 supplies the ordinary homotopy-group conventions and the local weak-equivalence definition with its component sheaf and all based slice-site homotopy sheaves. These are unlocalized models; no ${\mathbb{A}}^1$ localization is asserted.

The general site and presheaf conventions are supplied by Small sites and the plus construction, which allows arbitrary small categories. The slice and based homotopy-sheaf constructions are specified in the Statement and reference 4. Model and Quillen terminology is supplied by Model categories and Quillen adjunctions, simplices and simplicial homotopies by Simplicial sets, homotopies and trivial Kan fibrations, and Kan fibrations by Simplicial horns and Kan fibrations. The mapping-corner axiom is stated explicitly because the model-category definition alone does not state it.

References 2 and 3 supply algebraic sheafification and the abelian category in the sense of Abelian category. Exactness combines left exactness with preservation of cokernels by the left adjoint. Hence sheafifying a presheaf chain complex commutes with its homology. Free groups have the meaning of Free abelian group on a set; the ordinary tensor product is The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums with $R=\mathbb{Z}$. Cochains and cohomology have the meanings of Cochain complex in an abelian category and Cohomology object of a cochain complex, applied to this abelian category, without any restriction to scheme sites. The explicit ordinary normalization and inverse are supplied by Dold-Kan equivalence for simplicial modules with explicit inverse; reference 1, rather than that fixed-ring theorem alone, supplies their sheaf-local model application. Choice, as in The Axiom of Choice, is allowed in the imported model factorizations and selections of lifts. This is an external interface; no local proof of these existence theorems is claimed.