Analytic Constructible Sheaf Foundations

(Setup.) A sheaf of $k$-vector spaces on $X$ is an ${\mathcal{O}}_X$-module for the ringed space $(X,k_X)$ and conversely, and the stalk $k$ is a local ring, so $(X,k_X)$ is a locally ringed space with ${\mathrm{Sh}}_k(X)=\mathrm{M}\mathrm{o}\mathrm{d}(k_X)$; this category is abelian, and $D^b({\mathrm{Sh}}_k(X))$ is taken with the cochain conventions of Derived category of an abelian category, every Hom-set assertion being under its standing smallness hypothesis. In the second part, $D_{\mathfrak{S}}^b(X;k)$ is the full subcategory of complexes whose cohomology sheaves are bounded and $\mathfrak{S}$-constructible.

(Extensions of local systems.) Let $0\to K\to E\stackrel{\pi }{\to }Q\to 0$ be exact on any space $Y$, with $K,Q$ locally constant of finite rank. Near $y\in Y$ trivialize both, and let $e_1,\dots ,e_q$ be the constant basis sections of $Q$. By [F6], their germs lift to $E_y$; represent these finitely many lifts by sections $t_i$ and shrink to one neighbourhood $N^{\mathrm{\prime }}$ where $\pi (t_i)=e_i$. For any open $W\subseteq N^{\mathrm{\prime }}$ and locally constant coefficients $a_i:W\to k$, define $s({\sum }_i{a}_i{e}_i)={\sum }_i{a}_i{t}_i$ by gluing the ordinary scalar multiples on opens where all $a_i$ are constant. Locality makes this well defined and compatible with restriction, and $\pi s=\mathrm{id}$. Identifying $K$ with the kernel of $\pi$, the maps $(u,v)\mapsto u+s(v)$ and $e\mapsto (e-s\pi (e),\pi (e))$ are inverse sheaf morphisms between $K{\mathrm{\mid }}_{N^{\mathrm{\prime }}}\oplus Q{\mathrm{\mid }}_{N^{\mathrm{\prime }}}$ and $E{\mathrm{\mid }}_{N^{\mathrm{\prime }}}$. Therefore $E$ is locally constant of finite rank. This uses finitely many lifts and no connected-neighbourhood hypothesis.

(Injective models.) By [F1] applied to $(X,k_X)$ every $k$-sheaf receives a specific injective resolution, and the category has enough injectives. AC supplies DC by [F3]. For each bounded-below complex, [F11] now supplies a bounded-below termwise injective replacement; the successive embeddings may be taken from the fixed embedding functor of [F1]. Thus the replacement hypothesis of [F2] is met for complexes as well as sheaves, and $K^{+}(\mathrm{Inj}{\mathrm{Sh}}_k(X))\to D^{+}({\mathrm{Sh}}_k(X))$ is an equivalence. By [F12], the canonical functors from $D^b$ and $D^{+}$ to $D$ are fully faithful, with essential images the bounded-cohomology and bounded-below-cohomology objects, respectively. Their commutation with $D^b\to D^{+}$ therefore makes that functor fully faithful, with essential image precisely the bounded-cohomology objects of $D^{+}$. Since replacement preserves cohomology, restricting the injective-model equivalence to complexes with bounded cohomology models $D^b({\mathrm{Sh}}_k(X))$. Hom groups between bounded complexes are computed by homotopy classes of maps between their bounded-below injective replacements, even when the replacements have unbounded terms.

(Shifts and isomorphisms.) Fix $\mathfrak{S}$ and let $A\in D_{\mathfrak{S}}^b(X;k)$. By [F7], $H^j(A[m]){\mathrm{\mid }}_S=H^{j+m}(A){\mathrm{\mid }}_S$, so the cohomological support is reindexed and every restricted cohomology sheaf remains locally constant of finite rank. Isomorphic derived objects have isomorphic cohomology sheaves: the cohomology functors on complexes send quasi-isomorphisms to isomorphisms and therefore descend through localization. Hence the subcategory is closed under all shifts and isomorphisms.

(Cones on one stratum.) Let $f:A\to B$ be a morphism in $D_{\mathfrak{S}}^b(X;k)$ and fix a stratum $S$. By [F6], inverse image to $S$ is exact, since its stalk maps are the original stalk maps. It preserves finite sums, differentials and shifts, hence cone complexes by [F7], and it preserves quasi-isomorphisms. Applying it to a bounded roof representing $f$ consequently gives a derived morphism and its cone triangle on $S$; no chain-map representative $A\to B$ is assumed. The long exact sequence [F4] yields $0\to \mathrm{coker}(f^j)\to H^j(C(f)){\mathrm{\mid }}_S\to \ker (f^{j+1})\to 0$, where $f^j:H^j(A){\mathrm{\mid }}_S\to H^j(B){\mathrm{\mid }}_S$. To check its end terms near $s\in S$, trivialize the finitely ranked source and target. The images of the finitely many constant basis sections have locally constant coordinate functions. Shrinking around $s$ makes these finitely many functions constant, without requiring a connected neighbourhood. The morphism is then a fixed matrix, whose kernel and cokernel are the constant sheaves on the corresponding finite-dimensional vector spaces, as follows either from locally constant functions or from [F8] stalkwise. Thus both end terms are locally constant of finite rank.

(Cone constructibility.) Steps 2.3 and 1.2 show that each $H^j(C(f)){\mathrm{\mid }}_S$ is locally constant of finite rank. If $A$ and $B$ have cohomology only in finite intervals, their cone has cohomology only in the union of the interval for $B$ and the interval for $A$ shifted down one degree, by [F4]. A bounded roof also gives a bounded cone representative. Thus $C(f)\in D_{\mathfrak{S}}^b(X;k)$.

(Triangulated subcategory.) The zero complex belongs to the subcategory, and finite sums do too because their cohomology sheaves are the finite sums of the restricted local systems. By steps 2.2, 3.1 the full additive subcategory $D_{\mathfrak{S}}^b(X;k)$ contains all isomorphs of its objects, is closed under shifts in both directions, and whenever $f$ is a morphism of the subcategory, a cone of $f$ lies in the subcategory. In a triangulated category the third object of any distinguished triangle with two objects in such a subcategory is isomorphic to a cone of a morphism between objects of the subcategory, so the subcategory is closed under the two-out-of-three operation; by the equivalent characterisation of [F5] it is a triangulated subcategory. If $A$ and $B$ with their cone are only asserted to be constructible with respect to one common finite smooth algebraic frontier stratification, the same argument applies with that stratification in place of $\mathfrak{S}$, giving the final sentence: their cone is bounded and constructible with respect to that common stratification. Only step 2.1 uses AC, exactly as declared in [A1].

Kernels, images and cokernels of morphisms of diagrams of $k$-vector spaces are formed objectwise, so a morphism of diagrams is an epimorphism exactly when it is surjective at every object, and a complex of diagrams is exact exactly when it is exact at every object. In particular exactness questions about ${\mathrm{Diag}}_k(P)$ reduce to the vector spaces $N(p)$, and no diagram-level machinery beyond the objectwise one is used below.

For $p\in P$ define the diagram $Q_p$ by $Q_p(q)=k$ for $p\le q$ and $Q_p(q)=0$ otherwise, with the identity as structure map whenever both values are nonzero. A natural transformation $\phi :Q_p\to N$ is determined by ${\phi }_p(1)\in N(p)$, since for $q\ge p$ the component ${\phi }_q$ must equal $N(p\to q)\circ {\phi }_p$ and for $q\ngeqq p$ the source $Q_p(q)$ is zero; conversely every vector $v\in N(p)$ extends by these formulas. Hence $\mathrm{Hom}(Q_p,N)\cong N(p)$ by evaluation at $p$. Pulling back along the identification, a surjection $\pi :N\to N^{\mathrm{\prime }}$ induces the surjection $N(p)\to N^{\mathrm{\prime }}(p)$ by step 1.1, so every map $Q_p\to N^{\mathrm{\prime }}$ lifts to $N$ and $Q_p$ is projective. For a finite-dimensional $V$, a natural transformation $Q_p{\otimes }_{k}V\to N$ is likewise determined by its value at $p$, a $k$-linear map $V\to N(p)$, and conversely every such map extends uniquely by the structure maps of $N$, so $\mathrm{Hom}(Q_p{\otimes }_{k}V,N)\cong {\mathrm{Hom}}_k(V,N(p))$; since ${\mathrm{Hom}}_k(V,-)$ carries surjections to surjections — lift the images of a finite basis and extend linearly — the diagram $Q_p{\otimes }_{k}V$ is projective as well. A finite direct sum of projectives is projective, because $\mathrm{Hom}({⨁}_t{X}_t,N)\cong {\prod }_t\mathrm{Hom}(X_t,N)$ and a finite product of surjections of vector spaces is surjective by induction on the number of factors. Only finitely many basis vectors and summands occur at each step, so no choice principle is used.

For a diagram $N$ put ${\mathcal{B}}_n(N)={⨁}_{p_0<\cdots <p_n}{Q}_{p_n}{\otimes }_{k}N(p_0)$ for $n\ge 0$, with the face maps ${\mathrm{\partial }}_i:{\mathcal{B}}_n(N)\to {\mathcal{B}}_{n-1}(N)$ ($n\ge 1$, $0\le i\le n$) given on the summand of the chain $p_0<\cdots <p_n$ by: for $i=0$ apply the structure map $N(p_0)\to N(p_1)$ and delete $p_0$; for $0<i<n$ delete the inner vertex $p_i$, keeping the coefficient in $N(p_0)$ and the representable index $p_n$; for $i=n$ delete $p_n$ and use the natural map $Q_{p_n}\to Q_{p_{n-1}}$ (the identity where the source is $k$, and zero otherwise) while keeping the coefficient. Let $\mathrm{\partial }={\sum }_i(-1{)}^i{\mathrm{\partial }}_i$ for $n\ge 1$ and set $\mathrm{\partial }=0$ in bar degree zero, before adjoining the augmentation. The simplicial identities ${\mathrm{\partial }}_i{\mathrm{\partial }}_j={\mathrm{\partial }}_{j-1}{\mathrm{\partial }}_i$ for $i<j$ hold because diagram transport $p\mapsto N(p)$ is functorial and the maps between representables compose, so ${\mathrm{\partial }}^2=0$; this is a sign-alternating sum of a simplicial identity. Since chains of $P$ have at most $h$ strict inequalities, ${\mathcal{B}}_n(N)=0$ for $n>h$. Augment by the structure maps $ϵ:{\mathcal{B}}_0(N)\to N$, $Q_p{\otimes }_{k}N(p)\to N$, which are natural.

Evaluate the augmented complex at $q\in P$. Its degree-$n$ part consists of the chains $p_0<\cdots <p_n\le q$ with a vector of $N(p_0)$, the faces delete a vertex as in step 3.1, the last face being free to delete $q$, and the augmentation sends a length-zero chain $(p_0)$ to $N(p_0\to q)v$. Define a degree-$(-1)$-to-$0$ map $s_{-1}:N(q)\to {\mathcal{B}}_0(N)(q)$ by $v\mapsto (q,v)$, and for $n\ge 0$ a degree-$n$-to-$n+1$ map $s_n$ on the summand of the chain $p_0<\cdots <p_n\le q$ by $s_n(v)=(-1{)}^{n+1}(p_0<\cdots <p_n<q,v)$ when $p_n<q$, and $s_n=0$ when $p_n=q$. One checks $\mathrm{\partial }{s}_n+s_{n-1}\mathrm{\partial }=\mathrm{i}\mathrm{d}$ on each evaluated component: the appended simplex contributes through its last face with sign $(-1{)}^{n+1}\cdot (-1{)}^{n+1}=1$, every other face is matched by the corresponding appended term in $s\mathrm{\partial }$ with opposite sign, and in degree zero the augmentation section cancels exactly the last-face term; when $p_n=q$ and $n\ge 1$, only the last face contributes to $s_{n-1}\mathrm{\partial }$, with coefficient $(-1{)}^n(-1{)}^n=1$, while for $n=0$ the term $s_{-1}ϵ$ is the identity. Moreover $ϵ$ is surjective on the evaluated complexes because $(q,v)\mapsto v$, so the augmented evaluated complex is exact in every degree. Exactness is objectwise by step 1.1, so $ϵ:\mathcal{B}(N)\to N$ is a quasi-isomorphism; and if every $N(p)$ is finite-dimensional then every ${\mathcal{B}}_n(N)$ is a finite direct sum of projectives by step 2.1, so $\mathcal{B}(N)$ is a bounded projective resolution of $N$.

Let $A$ be a bounded complex of diagrams with finite-dimensional values, vanishing outside internal degrees $[a,b]$, with internal differentials $d_A$. Apply the bar construction of step 4.1 to each internal degree $u$, so ${\mathcal{B}}_n(A^u)$ has bar degree $n\in [0,h]$ and carries the internal differential $d_A:{\mathcal{B}}_n(A^u)\to {\mathcal{B}}_n(A^{u+1})$. The face maps $\mathrm{\partial }$ commute with the maps induced by $d_A$, since they are built from the structure maps of the diagrams and from the fixed maps between representables. Take the total complex $\mathrm{Tot}$ with ${\mathrm{Tot}}^m={⨁}_{u-n=m}{\mathcal{B}}_n(A^u)$ and differential $D=d_A+(-1{)}^u\mathrm{\partial }$ on the summand ${\mathcal{B}}_n(A^u)$; the sign $(-1{)}^u$ makes the horizontal and vertical parts anticommute, so $D^2=0$. Each column $u$ maps quasi-isomorphically to $A^u$ by step 4.1, so the rows of the mapping cone of $\mathrm{Tot}\to A$ are the mapping cones of the augmentations $\mathcal{B}(A^u)\to A^u$ and are acyclic; a bounded double complex with acyclic rows has acyclic total complex, by filtering by the finitely many internal degrees and inducting on the number of rows with the long exact homology sequence of the quotient by the top row. Hence $\mathrm{Tot}\to A$ is a quasi-isomorphism, $\mathrm{Tot}$ is a complex of finite-dimensional projective diagrams by steps 2.1 and 4.1, and its total degrees $u-n$ lie in $[a-h,b]$.

Resolve the constant diagram $\underset{‾}{k}$ by its bar complex $\mathcal{B}(\underset{‾}{k})$, a bounded complex of finite-dimensional projective diagrams of the form of step 4.1 with $\underset{‾}{k}$ in internal degree $0$. These bounded projective models require no DC: Hom from a single projective into an acyclic complex is acyclic by exactness of its Hom functor; filtering a bounded projective complex by its finitely many degrees and applying the cohomology sequence to the resulting termwise split Hom sequences proves the same for that complex. Hence Hom from the model preserves quasi-isomorphisms and computes [F1] without any infinite sequence of lifts. Then $\mathbf{R}\lim M=\mathbf{R}\mathrm{Hom}(\underset{‾}{k},M)$ is computed by the Hom complex $Q\underset{}{\mathrm{Hom}}(\mathcal{B}(\underset{‾}{k}),M)$ of [F1], whose degree-$r$ term is ${\prod }_i\mathrm{Hom}(\mathcal{B}(\underset{‾}{k}{)}^i,M^{i+r})$. Here $\mathcal{B}(\underset{‾}{k}{)}^{-n}={⨁}_{p_0<\cdots <p_n}{Q}_{p_n}$ is nonzero only for $0\le n\le h$ and $M^j=0$ for $j\mathrm{\notin }[a,b]$, so nonzero factors require $-h\le i\le 0$ and $a\le i+r\le b$. Consequently $a\le i+r\le r$ and $r\le b-i\le b+h$, giving $r\in [a,b+h]$. Concretely the contribution of bar degree $n$ to total degree $r$ is the finite product ${\prod }_{p_0<\cdots <p_n}{M}^{r-n}(p_n)$, so every term is a finite product of finite-dimensional spaces, hence finite-dimensional. Thus $\mathbf{R}\lim M$ has a finite-dimensional cochain model concentrated in $[a,b+h]$.

For the pair $A,B$ concentrated in $[a,b]$ and $[c,d]$, resolve the source by the total bar model $P_A:=\mathrm{Tot}$ of step 5.1, concentrated in $[a-h,b]$, and apply [F1]: $\mathbf{R}\mathrm{Hom}(A,B)$ is computed by $Q\underset{}{\mathrm{Hom}}(P_A,B)$, whose degree-$r$ term is the finite product ${\prod }_i\mathrm{Hom}(P_A^i,B^{i+r})$. A nonzero factor requires $P_A^i\mathrm{\ne }0$ and $B^{i+r}\mathrm{\ne }0$, hence $a-h\le i\le b$ and $c\le i+r\le d$; so for $r<c-b$ every $i$ with $P_A^i\mathrm{\ne }0$ has $i+r\le b+r<c$ and for $r>d-a+h$ every such $i$ has $i+r>d$, and in both cases every factor of the term vanishes. Thus the model is concentrated in $[c-b,d-a+h]$. By the evaluation identity in step 2.1, each summand with $u-n=i$ and chain $p_0<\cdots <p_n$ contributes ${\mathrm{Hom}}_k(A^u(p_0),B^{i+r}(p_n))$. Thus every term is a finite product of Hom spaces between finite-dimensional vector spaces, hence finite-dimensional.

Let $K\mathrm{/}k$ be a field extension, and write the base-changed inputs as $K{\otimes }_{k}M$, $K{\otimes }_{k}A$, $K{\otimes }_{k}B$, diagrams over $K$. The models built in steps 5.2 and 6.1 have terms obtained from the values $M(p)$, ${\mathrm{Hom}}_k(A(p),B(q))$ by finite sums, finite products and the fixed face and structure maps; extending scalars commutes with finite direct sums, with finite products and, between finite-dimensional spaces, with Hom spaces, so $K{\otimes }_k$ of the model for $(M,A,B)$ is the model for the base-changed inputs, with the same signs and differentials. Tensoring with $K$ preserves kernels and images of linear maps between finite-dimensional $k$-vector spaces and hence commutes with the homology of a bounded complex of such spaces, so the cohomology of the base-changed model is the base change of the cohomology of the original model. This proves both base-change assertions. For the empty poset there are no chains, the diagram sections and the Hom complex vanish by the same product formulas, and the asserted model is zero.

(Flasque sheaves are acyclic.) For an injective $k$-sheaf $I$, the monomorphism $j_{U!}{k}_U\to j_{V!}{k}_V$ for opens $U\subseteq V$ is the identity on stalks in $U$ and has zero source elsewhere. The adjunction [F10], restricted to $k$-linear maps, identifies Hom from these sheaves with $I(U)$ and $I(V)$: a map from $k_U$ is determined by its value on $1$, and locally constant scalar multiples are defined by gluing. Thus injectivity makes $I(V)\to I(U)$ onto, so $I$ is flasque. Embed a flasque $F$ into an injective $I$ using [F7], with quotient $Q$. By [F8], $I(U)\to Q(U)$ is onto and $Q$ is flasque. The long exact sequence and injective acyclicity [F11] give $H^1(X,F)=0$ and $H^{n+1}(X,F)\cong H^n(X,Q)$ for $n\ge 1$. Induction, uniformly over flasque sheaves, proves their acyclicity. The same proof on any open subspace gives the assertion there. Repeating the argument for abelian sheaves, with ${\mathbb{Z}}_U$ in place of $k_U$ and [F14] for the injective data, proves abelian flasque acyclicity as well. No $k$-injective is claimed to be injective among abelian sheaves.

(Global representatives.) Let $C^n(U;k)$ be the functions on singular $n$-simplices with image in $U$, with the usual cochain differential, and let ${\mathcal{C}}^n$ be its sheafification. Locally represent $s\in {\mathcal{C}}^n(U)$ by cochains on an open cover $\{U_j\}$; local representability follows from the stalk bijection [F6]. The open subspace $U$ is itself metric, hence paracompact and Hausdorff by [F1], [F2]. Apply [F3] in $U$ to obtain a locally finite open cover $\{W_i\}$ and assigned indices $j(i)$ with ${\overline{W_i}}^U\subseteq U_{j(i)}$, and write $c_i$ for the representative on $U_{j(i)}$. For each singular simplex in $X$ whose image lies in some $W_i$, choose one such index and assign the value $c_i(\sigma )$; assign zero when no such index exists. These choices use [A1] and define a global cochain $c\in C^n(X;k)$. The germ verification below shows that its sheafification restricts to $s$.

(Prescribed germs and flasqueness.) Fix $x\in U$. Local finiteness implies that $J_x=\{i:x\in {\overline{W_i}}^U\}$ is finite and some neighbourhood meets no $W_i$ for $i\mathrm{\notin }{J}_x$: first take a neighbourhood meeting only finitely many $W_i$, then avoid the closures of those not containing $x$. Closure containment from step 1.2 puts $x$ in every $U_{j(i)}$ for $i\in J_x$. The $c_i$ therefore have germs at $x$, all representing $s_x$. The stalk bijection [F6] makes their presheaf germs equal, so on one common smaller neighbourhood $N$ the finitely many cochains agree. Further shrink $N$ into a $W_{i_0}$ containing $x$ and away from all $W_i$ with $i\mathrm{\notin }{J}_x$. Every simplex in $N$ is then assigned the value of $c_{i_0}$, independently of its chosen admissible index. Thus $c$ represents $s$ at every point of $U$. Every section over $U$ has a global representative; restricting that representative to any larger open set gives an extension. Hence every ${\mathcal{C}}^n$ is flasque. The canonical sheafification map $C^n(V;k)\to {\mathcal{C}}^n(V)$ is also onto for every open $V$.

(The flasque resolution.) The augmentation $k_X\to {\mathcal{C}}^0$ sends a locally constant function to the same function on points. By [F6], [F13], the cohomology of its stalk complex at $x$ is the colimit of the singular cohomology of neighbourhoods of $x$. Contractible open neighbourhoods are cofinal. A contraction and the cochain dual of [F5] identify their cohomology with that of a point. The ordinary singular cochain complex of a point has $k$ in every nonnegative degree, with differential $k\to k$ equal to zero in even degrees and the identity in odd degrees, because the boundary is the alternating sum of $n+1$ identical faces. Its cohomology is consequently $k$ in degree zero and zero otherwise. The augmentation induces the degree-zero stalk isomorphism, and all positive stalk cohomology vanishes. Therefore $k_X\to {\mathcal{C}}^0\to {\mathcal{C}}^1\to \cdots$ is exact. By steps 1.1 and 2.1 its terms are acyclic, and [F12] applies: the injective datum [F7] is defined on every successive kernel. Its section complex computes the $k$-linear derived cohomology. The same resolution consists of acyclic underlying abelian sheaves by step 1.1; applying [F12] with [F14] identifies its section cohomology also with the published abelian-sheaf convention.

(Identification with singular cohomology.) The canonical sheafification map $C^{\ast }(X;k)\to \mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })$ is surjective by step 2.1. Let $N^{\ast }$ be its kernel. A cochain lies in $N^{\ast }$ exactly when it vanishes on all simplices contained in some member of some open cover of $X$: a cochain has zero germ at $x$ exactly when it vanishes on all simplices of some neighbourhood of $x$ by [F6], and a cochain vanishing on the small simplices of a cover has zero germ everywhere, while conversely the neighbourhoods of all points form such a cover. For an open cover $\mathcal{U}$ let $K_{\mathcal{U}}^{\ast }$ be the subcomplex of cochains vanishing on the $\mathcal{U}$-small singular chains. By [F4] applied with coefficients in the field $k$, the inclusion of the small chain complex into all chains is a homology isomorphism, so the quotient chain complex is acyclic; over a field an acyclic chain complex of possibly infinite-dimensional spaces splits degreewise into its boundaries and a complementary subspace, and dualizing such a splitting shows that the Hom of the quotient into $k$ is acyclic. Since restriction of functions to the subcomplex of small chains is surjective, this dual complex is exactly $K_{\mathcal{U}}^{\ast }$; hence $K_{\mathcal{U}}^{\ast }$ is acyclic. The complexes $K_{\mathcal{U}}^{\ast }$ are directed by refinement, and their union is $N^{\ast }$. A cocycle in the union lies in one $K_{\mathcal{U}}^{\ast }$ and is a coboundary there, so $N^{\ast }$ is acyclic. The long exact sequence of the short exact sequence of cochain complexes $0\to N^{\ast }\to C^{\ast }(X;k)\to \mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })\to 0$ therefore identifies $H_{\mathrm{sing}}^{}(X;k)$ with the cohomology of $\mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })$.

(Comparison and naturality.) Steps 3.1 and 3.2 identify $H^q(X,k_X)$ with $H_{\mathrm{sing}}^{}(X;k)$ in all nonnegative degrees; both are zero in negative degrees. The comparison uses the canonical augmentation and sheafification map, not the choices used to prove surjectivity. For a continuous map $f:X\to Y$ between spaces satisfying the hypotheses, pullback of singular cochains on each open $V\subseteq Y$ gives cochains on $f^{-1}V$, commuting with differentials, restriction and the constant-sheaf augmentation. Sheafifying gives the same pullback morphism of resolutions. Thus the induced cohomology comparisons commute with pullback, proving naturality, in both the $k$-linear and abelian conventions compared in step 3.1.

(Open subspaces and finite-polyhedral consequences.) Every open subspace is metric and inherits the contractible-open basis, so the argument applies there. A nonempty contractible such space has the cohomology of a point by [F5]. For the remaining geometric claims, realize a finite simplicial complex in barycentric coordinates $(t_v)$ on its finite vertex set. If $x$ has support simplex $\sigma$, choose $0<\delta <{\min }_{v\in \sigma }{t}_v(x)$ and put $N_{\delta }(x)=\{y:\mathrm{\mid }{t}_v(y)-t_v(x)\mathrm{\mid }<\delta (v\in \sigma ),{\sum }_{v\mathrm{\notin }\sigma }{t}_v(y)<\delta \}$. These are open neighbourhoods and form a basis as $\delta \downarrow 0$. Every carrier simplex of $y\in N_{\delta }(x)$ contains $\sigma$, so the segment from $y$ to $x$ stays in that carrier and satisfies the defining inequalities; these segments contract $N_{\delta }(x)$ to $x$. The open star of a simplex $\rho$ is $\{y:t_v(y)>0(v\in \rho )\}$. It is star-shaped toward the barycentre of $\rho$, since each carrier contains $\rho$ and the segment preserves these positive coordinates. Its nonempty intersection with a closed simplex $\tau$ requires $\rho \subseteq \tau$ and has the same contraction inside $\tau$. The intersections of the neighbourhoods $N_{\delta }(x)$ with these subspaces form contractible open neighbourhood bases: their contractions toward $x$ preserve the star's positive coordinates and membership in $\tau$. All these spaces inherit a metric from the finite coordinate space. Therefore their constant-sheaf cohomology is $k$ in degree zero and zero otherwise. An open star need not be convex in the whole realization; only these explicit contractions are used.