Finite simplicial constructible sheaves have an actual derived incidence model

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.

The qualified cone estimate. Suppose in addition that the fibre augmentation $k\to H^0(F;k)$ is an isomorphism (for these finite models, $F$ nonempty and connected), and that $H^r(F;k)=0$ for $1\le r\le R$. Then $\tilde{C}=\mathrm{Cone}(k\to C^{\bullet }(F;k))$ has no cohomology in degrees $\le R$, including degree $-1$. For $A\in D^{\ge b}$, the Künneth formula gives $H^n(A\otimes \tilde{C})={⨁}_{p+r=n}{H}^p(A)\otimes H^r(\tilde{C})$ stalkwise; a nonzero term requires $p\ge b$ and $r\ge R+1$. The cone of the unit therefore lies in $D^{\ge b+R+1}$. Without the degree-zero condition the estimate is false, for example for a two-point fibre and $A=k$: the unit has cokernel $k$ in degree zero. The product comparison itself needs no connectedness condition and remains local to the supplied models.

Finite Hom and scalar extension. Full faithfulness identifies actual derived Hom with the finite diagram Hom model of [F10]. Its terms are finite products of ${\mathrm{Hom}}_k(V_{\sigma },W_{\tau })$ from finite-dimensional coefficient spaces. Tensor with a field extension commutes with these Hom spaces, finite sums, and the matrices and cohomology of the bounded model. This proves the Hom and scalar-extension clauses. The analytic injective replacement $J$ in step 5.2 need not have finite-dimensional terms; it is the equivalent diagram model that is finite.

The essential image. First, if $V$ is a bounded complex of diagrams, then each cohomology sheaf of $q^{-1}V$ is $q^{-1}(H^{j}V)$ by exactness of $q^{-1}$, and its restriction to an open simplex in $P$ is constant of finite rank, so such complexes lie in the stated image and have bounded finite-rank locally constant cohomology on each open simplex. Conversely, let $A$ be a bounded complex on $X$ whose cohomology sheaves are finite-rank locally constant on every open simplex. By step 3.1 each $H^j(A)$ is isomorphic to $q^{-1}{V}_j$ for a finite-dimensional diagram $V_j$, and $V_j=0$ outside a finite range of degrees. We build $A$ inductively from its cohomology. If $A$ has at most one nonzero cohomology sheaf, then $A\cong H^n(A)[-n]\cong q^{-1}(V_n[-n])$, using that $q^{-1}$ commutes with shifts up to canonical isomorphism because it is exact. Otherwise let $m$ be the least degree with $H^m(A)\mathrm{\ne }0$; the canonical truncation triangle of [F14] reads $H^m(A)[-m]\to A\to {\tau }^{\ge m+1}A\to H^m(A)[-m+1]$. By induction on the number of nonzero cohomology degrees, ${\tau }^{\ge m+1}A\cong q^{-1}W$ for a bounded diagram complex $W$, and $H^m(A)[-m]\cong q^{-1}(V_m[-m])$; by step 5.2 the connecting morphism $\delta :q^{-1}W\to q^{-1}(V_m[-m+1])$ is $q^{-1}({\delta }_0)$ for a unique morphism ${\delta }_0$ of the diagram derived category. Completing ${\delta }_0$ to a distinguished triangle $V_m[-m]\to Y\to W\to V_m[-m+1]$ of [F14] and applying the exact functor $q^{-1}$ yields a triangle $q^{-1}(V_m[-m])\to q^{-1}Y\to q^{-1}W\to q^{-1}(V_m[-m+1])$ whose connecting morphism is $\delta$; two distinguished triangles with isomorphic first two objects and the same connecting morphism are isomorphic, so $A\cong q^{-1}Y$ and $A$ is realized. Hence the essential image is exactly the full subcategory of complexes with bounded finite-rank locally constant cohomology on each open simplex.

Local internal-Hom models. For $x\in {\sigma }^{\circ }$, sufficiently small ambient Euclidean balls $U$ in $X$ lie in $\mathrm{St}(\sigma )$ and meet exactly the cofaces $Q_{\sigma }=\{\tau :\tau \ge \sigma \}$. For realized diagram complexes $A=q^{-1}V$, $B=q^{-1}W$, their restrictions are realized by the same restricted diagrams on $Q_{\sigma }$. The derived carrier-unit argument applies on $U$: for $\eta \le \tau$ in $Q_{\sigma }$, $U\cap \mathrm{St}(\eta )\cap \mathrm{\mid }\tau \mathrm{\mid }$ is empty or convex, so the restricted standard-injective resolution is acyclic on every restricted principal star. Consequently step 5.2 identifies ${\mathrm{Hom}}_{D(U)}(A{\mathrm{\mid }}_U,B{\mathrm{\mid }}_U[r])$ with the finite diagram Hom cohomology on $Q_{\sigma }$. Restriction to smaller such balls is the identity on these finite models. No convexity of an entire open star is required. By step 1.6 the stalk of internal Hom is therefore this same finite model. These identifications also prove local constancy along ${\sigma }^{\circ }$, rather than only equality of stalk dimensions: a class on one such ball maps canonically into the cohomology sheaf, and its restrictions to smaller balls about nearby points of ${\sigma }^{\circ }$ are the identical diagram class. This defines a map from the constant finite-model cohomology sheaf on $U\cap {\sigma }^{\circ }$ to the restricted internal-Hom cohomology sheaf, isomorphic on every stalk, hence an isomorphism.

Boundedness and field extension of internal Hom. The models ${\mathrm{Ext}}_{\mathrm{Diag}(Q_{\sigma })}^q(V{\mathrm{\mid }}_{Q_{\sigma }},W{\mathrm{\mid }}_{Q_{\sigma }})$ are finite-dimensional and vanish outside a range depending only on the internal amplitude of $V,W$ and the chain length of $Q_{\sigma }$, whose strict-chain length is at most that of $P$; since $P$ is finite and $Q_{\sigma }$ depends only on the simplex $\sigma$, the ranges are uniformly bounded over $\sigma$ and over points of ${\sigma }^{\circ }$, so $R\mathcal{H}om(A,B)$ is bounded with finite-rank locally constant cohomology. By [F10] the diagram Hom models commute with every field extension, and the identifications of step 6.4 use only the finite models, so extending scalars carries the internal-Hom models to the models of the extended objects. This proves the internal-Hom clause.

The ordinary equivalence is step 3.1; ordinary cohomology is step 3.2; compact supports are steps 2.2 and 4.3; full faithfulness, finite Hom and the essential image are steps 5.2, 6.2 and 6.3; internal derived Hom is steps 6.4 and 7.1; and the product comparison and qualified cone estimate are steps 5.1 and 6.1. AC is used through injective and comparison suppliers, whereas the finite bar construction itself is choice-free. Neither arbitrary analytic injectives nor arbitrary sheaf sections are asserted finite-dimensional.