Constructible Incidence And Cochain Models

The inverse image $f^{\ast }=f^{-1}$ is exact and $f^{\ast }⊣f_{\ast }$, and $f_{\ast }$ preserves injective objects. By [F4] there is a natural isomorphism $(f^{-1}G{)}_x\cong G_{f(x)}$ under which the stalk of $f^{-1}\phi$ is ${\phi }_{f(x)}$; since exactness is tested stalkwise, $f^{-1}$ preserves kernels, images and cokernels and is exact. The adjunction bijection of [F4] is $k$-linear, its two directions being given on sections by the canonical $k$-linear maps $G\to f_{\ast }{f}^{-1}G$ and $f^{-1}{f}_{\ast }F\to F$; hence $f^{\ast }$ is left adjoint to $f_{\ast }$ in ${\mathrm{Sh}}_k$. If $I$ is injective, then ${\mathrm{Hom}}_Y(-,f_{\ast }I)\cong {\mathrm{Hom}}_X(f^{\ast }(-),I)$ is a composite of the exact functor $f^{\ast }$ and the exact representable functor ${\mathrm{Hom}}_X(-,I)$, so $f_{\ast }I$ is injective.

The tensor product has the expected stalks, and tensoring is exact in each variable. Define $A{\otimes }_{k}B$ as the sheafification of the sectionwise tensor presheaf of [F5]. By [F5] sheafification preserves stalks, so $(A{\otimes }_{k}B{)}_x$ is the filtered colimit over the open neighbourhoods of $x$; since tensor products of vector spaces commute with filtered colimits, this is $A_x{\otimes }_k{B}_x$, and the stalk of $A\otimes \phi$ is $\mathrm{id}\otimes {\phi }_x$. For fixed $B$, the functor $A\mapsto A{\otimes }_{k}B$ is therefore stalkwise $A_x\mapsto A_x{\otimes }_k{B}_x$, which is exact because every $k$-vector space is flat, and exactness is stalkwise by [F4]; the same holds in the other variable. The restriction to opens is the evident one, so all these maps are compatible with restriction, and no choice is used in this step.

Injective $k$-sheaves are flasque. Let $I$ be injective and let $U\subseteq V$ be open. The canonical map $j_{U!}{k}_U\to j_{V!}{k}_V$ of [F6] is a monomorphism of $k$-sheaves: on stalks it is the identity $k\to k$ on $U$, the monomorphism $0\to k$ on $V\setminus U$, and $0\to 0$ elsewhere. By [F6], ${\mathrm{Hom}}_X(j_{U!}{k}_U,I)\cong {\mathrm{Hom}}_U(k_U,I{\mathrm{\mid }}_U)\cong I(U)$ and similarly with $V$. Injectivity of $I$ extends every morphism $j_{U!}{k}_U\to I$, that is, every section over $U$, along this monomorphism to a morphism $j_{V!}{k}_V\to I$, that is, a section over $V$ restricting to it. Hence the restriction $I(V)\to I(U)$ is surjective, and $I$ is flasque by [F8].

The adjunction $f^{\ast }⊣R{f}_{\ast }$ holds on bounded-below derived categories. Let $F$ be a bounded-below complex on $X$ and let $j:F\to I_F$ be a bounded-below termwise injective replacement, supplied under DC by [F3]; declare $R{f}_{\ast }F:=f_{\ast }{I}_F$, well defined up to canonical isomorphism because any two such replacements are homotopy equivalent by [F3] and $f_{\ast }$ preserves homotopy equivalences. Since $f_{\ast }$ preserves zero objects in each degree, $f_{\ast }{I}_F$ is bounded below, and it is termwise injective by step 1.1; hence it is K-injective by [F3], and for every bounded-below complex $G$ on $Y$ the localization identifies ${\mathrm{Hom}}_{D(Y)}(G,f_{\ast }{I}_F)\cong {\mathrm{Hom}}_{K(Y)}(G,f_{\ast }{I}_F)$. The levelwise adjunction of [F4] identifies this with ${\mathrm{Hom}}_{K(X)}(f^{\ast }G,I_F)$, and because $I_F$ is K-injective and quasi-isomorphic to $F$, this is ${\mathrm{Hom}}_{D(X)}(f^{\ast }G,F)$. All identifications are natural and $k$-linear, and the triangle identities are the triangle identities of the levelwise adjunction, so $f^{\ast }⊣R{f}_{\ast }$ on $D^{+}$.

Flasque $k$-sheaves are acyclic for sections on every open set. Let $F$ be flasque and embed it into an injective $I$ using [F2], with quotient $Q$. Viewed as a short exact sequence of underlying abelian sheaves, the sequence $0\to F\to I\to Q\to 0$ has flasque kernel $F$ and flasque middle term $I$ by step 1.3, so [F8] gives that $I(U)\to Q(U)$ is surjective for every open $U$ and that $Q$ is flasque. Write $H^q(U,-)$ for the right derived functors of the $k$-linear sections functor at $U$ formed with the functorial datum of [F2]; no comparison with abelian sheaf cohomology is yet assumed. By [F7] the family $(H^q(U,-){)}_{q\ge 0}$ is a cohomological delta functor, so the displayed short exact sequence yields a long exact sequence; by [F7] again $H^q(U,I)=0$ for $q>0$ because $I$ is injective. Hence $H^1(U,F)=0$ and, for every $n\ge 1$, $H^{n+1}(U,F)\cong H^n(U,Q)$. Since $Q$ is again flasque, the same argument applies to $Q$, and induction on $n$ gives $H^n(U,F)=0$ for all $n\ge 1$ and every open $U$, uniformly over all flasque $k$-sheaves. To establish the comparison, resolve any $k$-sheaf $G$ by the functorial $k$-injective resolution of [F2]. Forgetting scalars and restricting to $U$ preserves exactness by the stalk criterion [F4]; each resolution term is flasque by step 1.3, so its underlying abelian sheaf on $U$ is acyclic by [F10]. The abelian resolution datum in Sheaf cohomology as right derived global sections is defined on all of $\mathrm{A}\mathrm{b}(U)$, hence includes every kernel in this resolution. The acyclic-resolution theorem of [F10] therefore identifies the cohomology of this section complex naturally with the published $H^q(U,G{\mathrm{\mid }}_U)$. The same section complex computes the $k$-linear derived functors, giving their natural comparison as underlying abelian groups. This uses flasqueness, not injectivity after forgetting scalars; for example ${\mathbb{F}}_p$ is injective as a vector space on a point but not as an abelian group.

Bounded complexes are K-flat. Filter a bounded complex $A$ by its finitely many descending stupid truncations; each successive quotient is one sheaf $A^p$ placed in degree $p$. For any acyclic complex $C$, tensoring this quotient with $C$ gives a shift of $A^p{\otimes }_{k}C$, acyclic by the exactness proved in step 1.2. The filtration sequences are split in each degree, so tensoring preserves them; their long exact cohomology sequences prove by finite induction that $\mathrm{Tot}(A{\otimes }_{k}C)$ is acyclic. This argument permits unbounded $C$, since only finitely many $A$-degrees occur in each total degree. For a quasi-isomorphism $B\to B^{\mathrm{\prime }}$ its cone is acyclic by [F9], and its tensor with $A$ is the cone of the tensor map, with the signed total differential. Hence tensor with $A$ preserves quasi-isomorphisms. Applying this in both variables shows that the signed total tensor of bounded representatives computes $A{\otimes }_k^{L}B$, independently of the representatives.

The sheaf tensor–Hom adjunction holds on sections over each open. For $k_X$-modules $H,A,G$ there is a natural bijection ${\mathrm{Hom}}_X(H{\otimes }_{k}A,G)\cong {\mathrm{Hom}}_X(H,\mathcal{H}om(A,G))$: on sections over an open $U$ it is the vector-space tensor–Hom transposition $g\mapsto (h\mapsto g(h\otimes -))$, with inverse $s\mapsto (h\otimes a\mapsto s(h)(a))$, and these sectionwise bijections are compatible with restriction to smaller opens and with the $k_X$-module structures because both constructions are built from restriction maps and scalar multiplication. Hence for fixed $A$ the functor $\mathcal{H}om(A,-)$ is right adjoint to the exact functor $-{\otimes }_{k}A$ of step 1.2; consequently $\mathcal{H}om(A,J)$ is injective whenever $J$ is injective, its representable functor being the composite of $-{\otimes }_{k}A$ and ${\mathrm{Hom}}_X(-,J)$. The bijection is natural in all three variables.

The internal derived Hom has the injective model. For bounded $A$ and a bounded-below injective replacement $B\to I_B$, put $R\mathcal{H}om(A,B)=\mathcal{H}o{m}^{\bullet }(A,I_B)$, with degree-$n$ term ${\prod }_p\mathcal{H}om(A^p,I_B^{p+n})$ and differential $d\phi =d_I\phi -(-1{)}^n\phi d_A$. The product is finite. Its terms are injective by step 2.4 and it is bounded below, hence K-injective by [F3]. Restriction of an injective sheaf to any open $U$ remains injective, because restriction is right adjoint to exact extension by zero [F6]. Thus the same model over $U$ computes $R{\mathrm{Hom}}_U(A{\mathrm{\mid }}_U,B{\mathrm{\mid }}_U)$ after taking sections. In particular the object represents the tensor–Hom functor $C\mapsto {\mathrm{Hom}}_D(C{\otimes }_k^{L}A,B)$, rather than $C\mapsto {\mathrm{Hom}}_D(C,B)$; it is a sheaf-valued internal Hom, not Hom of stalks.

Bounded-below flasque complexes are acyclic for derived sections. Let $C^{\bullet }$ be a bounded-below complex of flasque $k$-sheaves, with $C^n=0$ for $n<b$, let $\phi :C^{\bullet }\to J^{\bullet }$ be a bounded-below termwise injective replacement supplied by [F3], and fix an open $U$. I first record a lemma: if $K^{\bullet }$ is an acyclic bounded-below complex whose every term is acyclic for the sections functor at $U$, then $\mathrm{\Gamma }(U,K^{\bullet })$ is acyclic. Indeed, put $Z^n=\ker (K^n\to K^{n+1})$; the complex is acyclic, so $0\to Z^n\to K^n\to Z^{n+1}\to 0$ is a short exact sequence for every $n$, and $Z^n=0$ for $n<b$. If $Z^n$ is acyclic for sections at $U$, then the long exact sequence of [F7] for this sequence, together with the acyclicity of $K^n$, shows that $Z^{n+1}$ is again acyclic for sections at $U$; inducting on $n$ upward from $n<b$, every $Z^n$ is acyclic. With all $Z^n$ acyclic, the same sequences give short exact sequences $0\to \mathrm{\Gamma }(U,Z^n)\to \mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,Z^{n+1})\to 0$ on sections, so the kernel of $\mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,K^{n+1})$ equals the image of $\mathrm{\Gamma }(U,K^{n-1})\to \mathrm{\Gamma }(U,K^n)$: the former is the kernel of $\mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,Z^{n+1})$, which is the image of $\mathrm{\Gamma }(U,Z^n)$ because $\mathrm{\Gamma }$ of a short exact sequence is left exact and the sequence is right exact at the quotient, and $\mathrm{\Gamma }(U,K^{n-1})\to \mathrm{\Gamma }(U,Z^n)$ is surjective, so the two images agree. Hence $H^n(\mathrm{\Gamma }(U,K^{\bullet }))=0$ for every $n$. Now apply this lemma to the cone $K^{\bullet }=\mathrm{Cone}(\phi )$ of the cochain map $\phi$ in the convention of Derived category of an abelian category. Its terms $J^n\oplus C^{n+1}$ are direct sums of an injective term, acyclic for sections at $U$ by [F7], and a flasque term, acyclic by step 2.2; a direct sum of two acyclic objects is acyclic by the long exact sequence [F7] applied to $0\to J^n\to J^n\oplus C^{n+1}\to C^{n+1}\to 0$. The cone is acyclic and bounded below, so the lemma gives that $\mathrm{\Gamma }(U,\mathrm{Cone}(\phi ))$ is acyclic. The canonical identification of complexes $\mathrm{\Gamma }(U,\mathrm{Cone}(\phi ))\cong \mathrm{Cone}(\mathrm{\Gamma }(U,\phi ))$ holds because sections over $U$ preserve finite direct sums and the differentials are those of $\phi$; by the cone criterion [F9] the map $\mathrm{\Gamma }(U,C^{\bullet })\to \mathrm{\Gamma }(U,J^{\bullet })$ is therefore a quasi-isomorphism for every open $U$. Since any two bounded-below injective replacements of $C^{\bullet }$ are homotopy equivalent by [F3], the ordinary section complex $\mathrm{\Gamma }(U,C^{\bullet })$ represents the derived sections $R\mathrm{\Gamma }(U,C^{\bullet })$ of the bounded-below flasque complex.

The internal model is independent of representatives. For a quasi-isomorphism $g:A\to A^{\mathrm{\prime }}$ of bounded complexes, put $C=\mathrm{Cone}(g)$. On every open $U$, the restriction $I{\mathrm{\mid }}_U$ of a bounded-below injective complex is K-injective by step 3.1 and [F3], so ${\mathrm{Hom}}_U^{\bullet }(C{\mathrm{\mid }}_U,I{\mathrm{\mid }}_U)$ is acyclic. These are the sections of $\mathcal{H}o{m}^{\bullet }(C,I)$; taking exact filtered stalk colimits shows that this sheaf complex is acyclic. The cone of $\mathcal{H}o{m}^{\bullet }(A^{\mathrm{\prime }},I)\to \mathcal{H}o{m}^{\bullet }(A,I)$ is isomorphic, with the usual Hom signs, to $\mathcal{H}o{m}^{\bullet }(C,I)[1]$, so the map is a quasi-isomorphism by [F9]. In the second variable, any homotopy equivalence between injective replacements is carried by $\mathcal{H}o{m}^{\bullet }(A,-)$ to a homotopy equivalence. The injective-model comparison of [F3] therefore makes the construction a natural bifunctor of the derived objects, compatible with restriction to opens.

The tensor–Hom adjunction and its coherence identities hold on the derived category. For a bounded complex $C$ and a bounded-below complex $I$ of injectives, the currying map $\mathcal{H}o{m}^{\bullet }(A{\otimes }_{k}C,I)\to \mathcal{H}o{m}^{\bullet }(C,\mathcal{H}o{m}^{\bullet }(A,I))$, which sends a morphism $\phi$ to $c\mapsto (a\mapsto (-1{)}^{\mathrm{\mid }c\mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }}\phi (a\otimes c))$ on pure tensor sections and extends by linearity and gluing, is an isomorphism of complexes: over each open it is the vector-space tensor–Hom transposition of step 2.4, bijective on sections, and the Koszul sign makes both sides carry the single differential rule $d\psi =d_I\psi -(-1{)}^n\psi d_{\text{source}}$ of Derived hom in the bounded setting, so it commutes with differentials and with restriction. Taking $I$ to be an injective replacement of the bounded-below complex $B$ and using that $C$ is K-flat and $I$, $\mathcal{H}o{m}^{\bullet }(A,I)$ are K-injective, the $H^0$ transposition identifies ${\mathrm{Hom}}_D(C{\otimes }_k^{L}A,B)\cong {\mathrm{Hom}}_D(C,R\mathcal{H}om(A,B))$ naturally in the bounded $C,A$ and bounded-below $B$; this is the tensor–Hom adjunction, with unit and counit the canonical currying maps. The associativity isomorphism $(A{\otimes }_{k}B){\otimes }_{k}C\cong A{\otimes }_k(B{\otimes }_{k}C)$, the symmetry isomorphism $A{\otimes }_{k}B\cong B{\otimes }_{k}A$ with its Koszul sign, the evaluation morphism $R\mathcal{H}om(A,B){\otimes }_k^{L}A\to B$, and the currying bijection all hold on pure tensor sections, where they are the usual vector-space identities, and extend by linearity, gluing and the comparison maps of step 4.1; being defined on sections they are natural and compatible with restriction to opens, hence internal. The triangle identities are those of step 2.4 at chain level.

The claims are now assembled. Inverse image is exact and $f^{\ast }⊣R{f}_{\ast }$ on $D^{+}$ is step 2.1; the derived tensor of bounded complexes is the signed total tensor by step 2.3; the internal derived Hom is the injective model of step 3.1, well defined by step 4.1; flasque sheaves are acyclic for sections on every open set by step 2.2, and bounded-below flasque complexes compute their derived sections by step 3.2; the tensor–Hom adjunction and the dg associativity, symmetry, evaluation and currying identities are step 5.1. AC is assumed and consumed exactly as follows: AC yields DC by [F1], DC is used by the injective replacement suppliers of [F3] and the delta-functor and injective-acyclicity statements of [F7], the functorial injective resolutions of [F2] use the Grothendieck embedding theorem under AC, and the flasque section lifting of [F8] uses Zorn's lemma under AC; the stalk, tensor, Hom and adjunction verifications are choice-free. No bounded-constructibility claim for the internal Hom is made: that is supplied later on this page by the geometric finite conical and incidence models, which are not prerequisites of this lemma.

The carrier map. The principal upsets $P_{\ge \sigma }=\{\tau \in P:\tau \supseteq \sigma \}$ form a basis of the Alexandrov topology: a subset is open exactly when it is upward closed, and an upward-closed set is the union of the principal upsets of its elements. Since $D$ is a subcomplex, every face of a simplex in $D$ lies in $D$, so contraposition makes the set $P$ of nonempty simplices outside $D$ upward closed under taking cofaces; hence $q^{-1}(P_{\ge \sigma })=\{x\in X:\sigma \subseteq q(x)\}=\mathrm{St}(\sigma )$, because a point of $\mathrm{\mid }K\mathrm{\mid }$ whose carrier contains $\sigma$ cannot lie in $\mathrm{\mid }D\mathrm{\mid }$ and every coface of a simplex of $P$ again lies in $P$. The star $\mathrm{St}(\sigma )={\bigcup }_{\tau \supseteq \sigma }{\tau }^{\circ }$ is open in $\mathrm{\mid }K\mathrm{\mid }$, since its trace on each closed simplex $\mathrm{\mid }\eta \mathrm{\mid }$ is $\{x\in \mathrm{\mid }\eta \mathrm{\mid }:\sigma \subseteq \mathrm{supp}(x)\}$, which is relatively open when $\sigma \subseteq \eta$ and empty otherwise; being contained in $X$ it is also open in $X$. As the principal upsets form a basis, $q$ is continuous. Each ${\sigma }^{\circ }$ is convex in the affine simplex spanned by $\sigma$, hence connected, locally path-connected and simply connected, and each $\mathrm{St}(\sigma )$ is open and nonempty.

Exact realization. A diagram $V:P\to k\text{-}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}$ defines a sheaf $U\mapsto {\lim }_{\sigma \in U}{V}_{\sigma }$: compatible families glue uniquely componentwise. The least neighbourhood $P_{\ge \sigma }$ has initial object $\sigma$, so its sections and its stalk are $V_{\sigma }$. Conversely, a sheaf on $P$ has these values on the principal opens; restrictions give the diagram, and the sheaf axiom recovers its sections on every open. Thus sheaves on $P$ and diagrams are equivalent. Define $\widehat{V}=q^{-1}V$. Its stalk at $x$ is $V_{q(x)}$, so realization is exact. On ${\sigma }^{\circ }$, the carrier factors through the point $\sigma$ of $P$; composition of inverse images identifies its restriction with the constant sheaf of value $V_{\sigma }$. This constructs the constant-sheaf identification, rather than inferring it from an abstract equality of stalk values.

Triviality and generization. First, a finite-rank local system on a convex simplex interior is constant. By [F18], subdivide a path into finitely many intervals lying in trivializing opens and compose their stalk identifications. Refining the subdivision does not change the composite. On overlaps, transition matrices are locally constant, so a further subdivision proves independence of the trivializing opens. A homotopy between paths has compact square domain; subdivide that square finely enough that each small square maps into one trivializing open. Transport around each small square is the identity, and cancellation of its shared edges shows that transport along the two boundary paths agrees. Convexity provides such a homotopy between any two paths with fixed endpoints. Transport from a fixed point therefore defines a sheaf map from its constant stalk sheaf to the local system: near each point it is the identification from one trivialization. It is an isomorphism by the stalk criterion. Put $V_{\sigma }=F_{b_{\sigma }}$. For $\sigma \le \tau$, represent a germ at $b_{\sigma }$ on a sufficiently small ambient Euclidean ball; its intersection with ${\tau }^{\circ }$ is nonempty and convex. Extend its restriction uniquely along the constant sheaf on ${\tau }^{\circ }$ and evaluate at $b_{\tau }$. Equal germ representatives agree on a smaller such ball, so this map is well defined and linear. The same construction based at any point of ${\sigma }^{\circ }$ is locally constant after the canonical transport identifications: lift finitely many basis germs to ambient sections and shrink so that their values on each of the finitely many cofaces are fixed. Connectedness of ${\sigma }^{\circ }$ makes each incidence matrix constant throughout it. For $\sigma \le \eta \le \tau$, choose a point of ${\eta }^{\circ }$ in the ball representing the original germ and restrict the same section near that point into ${\tau }^{\circ }$; uniqueness of extension and the just-proved independence of the point on ${\eta }^{\circ }$ give the transitivity equation. Identities are immediate. This reconstructs a diagram functorially from $F$.

The standard injectives. For $\tau \in P$ and a finite-dimensional $k$-vector space $W$ let $I_{\tau }(W)$ be the diagram with value $W$ at every $\sigma \subseteq \tau$ and $0$ elsewhere, the structure maps being identities where both values are $W$. A diagram map $V\to I_{\tau }(W)$ is exactly a $k$-linear map $V_{\tau }\to W$, since its $\tau$-component determines all components and conversely any such linear map extends by the zero maps; therefore $I_{\tau }(W)$ is injective because $\mathrm{Hom}(-,W)$ is exact. Every diagram $V$ embeds into the finite direct sum ${⨁}_{\tau \in P}{I}_{\tau }(V_{\tau })$ by the identity component at $\tau$, a nonzero vector of $V_{\tau }$ being detected by that component. Hence the finite diagram category has enough injectives and explicit finite injective embeddings, and by [F10] every bounded complex of finite-dimensional diagrams has a bounded injective model whose terms are finite sums of these standard injectives with the stated finite-dimensional ranges.

Representables. The diagram $P_{\sigma }$ has value $k$ on cofaces of $\sigma$ and zero elsewhere, and $\mathrm{Hom}(P_{\sigma },V)=V_{\sigma }$; hence it is projective and these diagrams generate. On $\mathrm{St}(\sigma )$ its realization is canonically constant, since all its diagram maps there are identities. The extension-by-zero adjunction gives $j_{\sigma !}{k}_{\mathrm{St}(\sigma )}\to q^{-1}{P}_{\sigma }$, identity on that star; both sides have zero stalks elsewhere, so this actual morphism is an isomorphism. The finite projective bar construction of [F10] resolves every bounded finite-dimensional diagram complex by a bounded complex of finite sums of these representables.

Internal derived Hom. For bounded representatives $A,B$, choose a bounded-below injective replacement $B\to J$. By [F16], $\mathcal{H}o{m}^{\bullet }(A,J)$ represents $R\mathcal{H}om(A,B)$ and its derived sections on $U$ represent $R\mathrm{Hom}(A{\mathrm{\mid }}_U,B{\mathrm{\mid }}_U)$. The reason ordinary sections suffice in this model is that $\mathcal{H}om(A^i,J^j)$ is injective: tensor with a $k$-sheaf is exact stalkwise and its right adjoint is sheaf Hom. There are finitely many $A$-degrees in each Hom degree. Restriction preserves injectives by its exact left adjoint $j_{!}$. Taking the exact filtered colimit of the section complexes therefore gives . Boundedness will follow from the local finite models.

Sections on a star. The generization maps of step 1.3 are locally constant along each face: trivialize the finite-rank restriction near a point of that face, lift its finitely many basis germs to ambient sections, and shrink one neighbourhood where their restrictions to each of the finitely many incident cofaces are constant. Their coface matrices are then constant on that face neighbourhood. Since each face interior is connected, these local matrices define the same transported map throughout it; restrictions through nested faces compose as in step 1.3. For $\sigma \in P$, a section on $\mathrm{St}(\sigma )$ with zero germ at $b_{\sigma }$ vanishes on a nonempty subset of every incident open simplex, hence on that whole simplex by [F6], so it is zero. Conversely, start with a germ $a\in F_{b_{\sigma }}$ and define its value along every ${\tau }^{\circ }$ with $\tau \supseteq \sigma$ by the generization map followed by the unique constant transport on ${\tau }^{\circ }$. At a point of ${\eta }^{\circ }$, lift that assigned germ to an ambient local section. Its values on every incident ${\tau }^{\circ }$ are the prescribed ones, by the local constant matrices and their composition rule; shrinking finitely many times ensures agreement for all cofaces. Thus the assigned family of germs is locally represented by ambient sections. These sections agree stalkwise on their overlaps and glue by the sheaf axiom to a section on the whole star. This proves $F(\mathrm{St}(\sigma ))\cong F_{b_{\sigma }}$. The simplex interiors were not treated as open subsets of the star.

Compact-support reduction. Extend $V$ by zero to the full face poset $P_K$, obtaining $\tilde{V}$; no arrow goes from a retained face to a deleted coface because $D$ is a subcomplex. The restriction of $q_K^{-1}\tilde{V}$ to $X$ is canonically $q^{-1}V$. The open extension-by-zero adjunction gives $j_{!}{q}^{-1}V\to q_K^{-1}\tilde{V}$; it is the identity on $X$, and both sides have zero stalks on $\mathrm{\mid }D\mathrm{\mid }$, so it is an isomorphism. The space $X$ is locally compact Hausdorff: its finite Euclidean realization is compact Hausdorff and $X$ is open. Thus [F15] and compactness give $R{\mathrm{\Gamma }}_c(X,q^{-1}V)\simeq R\mathrm{\Gamma }(\mathrm{\mid }K\mathrm{\mid },q_K^{-1}\tilde{V})$.

The product pair. Let $V=\mathrm{\mid }{K}_V\mathrm{\mid }\setminus \mathrm{\mid }{D}_V\mathrm{\mid }$ and $F=\mathrm{\mid }{K}_F\mathrm{\mid }\setminus \mathrm{\mid }{D}_F\mathrm{\mid }$ be realizations of finite pairs with face posets $P_V,P_F$ as in the statement, let $\mathrm{pr}:V\times F\to V$ be the projection, and let $B$ be a bounded complex of finite-dimensional diagrams on $P_V$, realized as $A:=q_V^{-1}B$, with the product realized along $q_{V\times F}:V\times F\to P_V\times P_F$. The product $\mathrm{\mid }{K}_V\mathrm{\mid }\times \mathrm{\mid }{K}_F\mathrm{\mid }$ has the finite regular product-cell decomposition whose closed cells are the convex polytopes $\mathrm{\mid }\sigma \mathrm{\mid }\times \mathrm{\mid }\eta \mathrm{\mid }$; the star, diagram and acyclicity arguments apply verbatim to these regular cells, using their convex faces, so no product of simplices is asserted to be a simplex, and $V\times F$ is the complement of the subcomplex $D$ consisting of the cells lying in $\mathrm{\mid }{D}_V\mathrm{\mid }\times \mathrm{\mid }{K}_F\mathrm{\mid }$ or in $\mathrm{\mid }{K}_V\mathrm{\mid }\times \mathrm{\mid }{D}_F\mathrm{\mid }$; a product cell $\sigma \times \eta$ lies outside $D$ exactly when $\sigma \mathrm{\notin }{D}_V$ and $\eta \mathrm{\notin }{D}_F$, so its poset is exactly $P_V\times P_F$, and ${\mathrm{pr}}^{\ast }A=q_{V\times F}^{-1}(B⊠k)$, where $B⊠k$ denotes the product diagram with value $B_{\sigma }$ at $(\sigma ,\eta )$ and the identity in the $\eta$-variable.

Realized injectives are acyclic on stars. Put $Z_{\tau }=\mathrm{\mid }\tau \mathrm{\mid }\cap X$, closed in $X$. The restriction of $q^{-1}{I}_{\tau }(W)$ to $Z_{\tau }$ is canonically constant: the restricted diagram has value $W$ and identity maps everywhere. Its stalks vanish off $Z_{\tau }$, so the actual closed-inclusion unit of [F8] identifies it with $i_{\tau \ast }{W}_{Z_{\tau }}$. Consequently its cohomology on an open $U$ is the constant-sheaf cohomology of $U\cap Z_{\tau }$. For $U=\mathrm{St}(\sigma )$ this intersection is empty unless $\sigma \le \tau$; otherwise it is the convex set in $\mathrm{\mid }\tau \mathrm{\mid }$ where all coordinates indexed by $\sigma$ are positive, and cannot meet $D$. It has cohomology $W$ in degree zero and zero above by [F7]. Globally $Z_{\tau }$ contracts to $b_{\tau }$: at positive time the segment from a retained point to $b_{\tau }$ lies in ${\tau }^{\circ }$. It has a contractible-open neighbourhood basis, being open in the convex simplex $\mathrm{\mid }\tau \mathrm{\mid }$. Hence its constant-sheaf cohomology is also $W$ in degree zero and zero above. This proves ordinary global and star acyclicity of realized standard injectives and their finite sums. It does not claim that they are injective or c-soft among all sheaves.

The ordinary equivalence. The constructions of steps 1.2 and 1.3 are inverse equivalences between finite-dimensional covariant $k$-diagrams on $P$ and sheaves on $X$ whose restriction to every open simplex is locally constant of finite rank. On the diagram side, the reconstruction of $\widehat{V}$ has $V_{\sigma }(\widehat{V})=(\widehat{V}{)}_{b_{\sigma }}\cong V_{\sigma }$ by step 1.2, and the generization maps of step 1.3 reproduce the maps of $V$, since the canonical sections over the stars have the prescribed values. On the sheaf side, the canonical morphism $\widehat{V(F)}\to F$ is adjunct to the family $V_{\sigma }\cong F(\mathrm{St}(\sigma ))$ of inverse isomorphisms from step 2.1 and is an isomorphism on stalks by steps 1.2, 1.3 and 2.1, hence an isomorphism by [F2]. Both functors are exact: realization is exact by step 1.2, and reconstruction is exact because a sequence of such sheaves is exact exactly when it is exact at the stalks, which by step 1.3 are the values of the reconstructed diagrams and the generization maps are functorial, so a diagram sequence is exact exactly when the corresponding sheaf sequence is.

Ordinary cohomology models. Dualizing the finite projective bar resolution of [F10] on the opposite poset gives a bounded injective model $V\to I$ with terms finite sums of standard injectives. Realization is exact, and step 2.4 makes these terms acyclic for global sections and for every retained star. The acyclic-resolution comparison [F13] consequently gives $R\mathrm{\Gamma }(\mathrm{St}(\sigma ),q^{-1}V)\simeq I_{\sigma }\simeq V_{\sigma }$ and $R\mathrm{\Gamma }(X,q^{-1}V)\simeq {\lim }_{\sigma \in P}{I}_{\sigma }$. Here its bounded-complex version follows by resolving the finitely many source degrees and filtering the total complex; the sheaf-level comparisons of [F13] identify each graded piece, hence the total. All syzygies lie in the whole field-sheaf category covered by [F11], and AC supplies DC. This last complex is a bounded finite-dimensional cochain model. Equivalently the bar model of [F10] has incidence terms ${\prod }_{{\sigma }_0<\cdots <{\sigma }_p}{V}_{{\sigma }_p}$ and its coefficient and face differential. Its finite matrix construction commutes with every field extension. If $P$ is empty then $X$ is empty, both categories and all these models are zero; no nonempty-poset supplier is applied in that case.

An acyclic product resolution. In the setting of step 2.3, choose bounded diagram-injective models $B\to I$ and $k\to J$ on $P_V$ and $P_F$. The signed total $I⊠J$ realizes a resolution of ${\mathrm{pr}}^{\ast }A$: over a field tensor is exact, and the finite double-complex filtration proves the augmentation is a quasi-isomorphism. Each elementary term is the closed pushforward of the constant coefficient $W\otimes U$ on $Z_{\tau }\times Z_{\nu }$. At a point of $V$, take small ambient Euclidean balls meeting only cofaces of its carrier. Their intersections with $Z_{\tau }$ are empty or convex; $Z_{\nu }$ is contractible by step 2.4. The constant-polyhedral comparison gives no higher cohomology on the corresponding support intersection in the inverse image of the ball. Taking stalk colimits in an injective direct-image model shows that these terms are ${\mathrm{pr}}_{\ast }$-acyclic. Their ordinary direct images are $i_{\tau \ast }(W\otimes U{)}_{Z_{\tau }}$: on any open base set a locally constant function on its product with the nonempty connected $Z_{\nu }$ is constant in the fibre, and evaluation at one fibre point identifies it with a locally constant function on the base support. Thus termwise pushforward of this resolution is canonically $q_V^{-1}I{\otimes }_k{C}^{\bullet }(F;k)$, where $C^{\bullet }(F;k)={\lim }_{\nu \in P_F}{J}_{\nu }$ is the finite model for $R\mathrm{\Gamma }(F,k)$ from step 3.2. This keeps every internal degree of the bounded complex $B$; no degree-zero assumption on $B$ is made. For empty $F$, both the pushforward and $C^{\bullet }(F;k)$ are zero.

The derived carrier unit. By step 2.1, the actual unit $V\to q_{\ast }{q}^{-1}V$ is an isomorphism on every principal open: its map to the germ at $b_{\sigma }$ is the identity of $V_{\sigma }$. Evaluation on a principal open is exact in the sheaf category on $P$, by its diagram description in step 1.2. Consequently, for an injective replacement $q^{-1}V\to J$, the $\sigma$-component of $R^j{q}_{\ast }{q}^{-1}V$ is $H^j(\mathrm{St}(\sigma ),q^{-1}V)$, zero for $j>0$ by step 3.2. Thus the derived unit is an isomorphism for a diagram sheaf. Finite truncation triangles extend the same conclusion to every bounded diagram complex. This argument concerns principal opens in $P$; it makes no assertion that open stars form a neighbourhood basis in $X$.

The compact-support cochain model. Apply the argument of steps 2.4 and 3.2 to the pair $(K,\mathrm{\varnothing })$, whose face poset is $P_K$ and whose realization is the compact space $\mathrm{\mid }K\mathrm{\mid }$: for a bounded complex $\tilde{V}$ of finite-dimensional diagrams on $P_K$, choose a bounded injective resolution $\tilde{V}\to J^{\bullet }$ with terms finite sums of standard injectives [F10], realize it, use the acyclicity of realized standard injectives for $\mathrm{\Gamma }(\mathrm{\mid }K\mathrm{\mid },-)$ from step 2.4, and apply the acyclic-resolution theorem [F13]; the outcome is that $R\mathrm{\Gamma }(\mathrm{\mid }K\mathrm{\mid },q_K^{-1}\tilde{V})$ is computed by the finite-dimensional bounded complex , with the range bound of [F10] for the poset $P_K$ and its coefficient range. By step 2.2 this complex is a bounded finite cochain model for the compact-support cohomology $R{\mathrm{\Gamma }}_c(X,q^{-1}V)$, and as in step 3.2 every map in the construction is $k$-linear between finite-dimensional spaces, so the model and hence the cohomology commute with every coefficient field extension by [F10].

Product comparison and unit. The acyclic resolution of step 4.1 identifies $R{\mathrm{pr}}_{\ast }{\mathrm{pr}}^{\ast }A$ with $q_V^{-1}I{\otimes }_k{C}^{\bullet }(F;k)$. The augmentation $B\to I$ and exact tensor over $k$ identify the latter with $A{\otimes }_k{C}^{\bullet }(F;k)$, giving the claimed natural isomorphism. Naturality and restriction compatibility follow from the diagram maps, the termwise direct-image identification and the canonical acyclic-resolution comparisons; they do not rely on testing all-sheaf stalks on whole stars. The adjunction unit is represented by $B\to I$ tensored with the fibre augmentation $k\to C^{\bullet }(F;k)$. It is therefore $A\otimes (k\to C^{\bullet }(F;k))$ under the comparison. After full faithfulness and the essential-image argument below, this applies to every $A$ in the stated constructible subcategory; in the Statement this realized base object is denoted $B$.

Derived full faithfulness. For bounded diagram complexes $U,W$, take a finite projective bar model $P\to U$ and an injective replacement $q^{-1}W\to J$. By [F11], [F12] and the inverse/direct-image adjunction, the Hom complex from $q^{-1}P$ to $J$ equals the Hom complex from $P$ to $q_{\ast }J$. The derived carrier unit of step 4.2 identifies $q_{\ast }J$ with $W$. Since the bounded projective model $P$ is K-projective, taking degree-zero cohomology identifies ${\mathrm{Hom}}_{D(X)}(q^{-1}U,q^{-1}W)$ with ${\mathrm{Hom}}_{D(P)}(U,W)$, naturally and compatibly with realization. Shifts give the same comparison in every degree, and [F14] identifies the bounded localizations with their full subcategories. Thus realization is fully faithful.