Dense subsites and change of topology

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

A sheaf can be recovered from a sufficiently rich collection of test objects, provided their morphisms and coverings are retained. We prove that assertion using covering sieves and compatible sections. The construction requires no fiber products in the site. We then compare topologies and explain why gluing over finite covers can retain information that a usual basis misses.

1. Cover-density and restriction of covering sieves

Let $C$ be a small category with a Grothendieck topology $J$, given by covering sieves. Presheaves are contravariant and set-valued. Use the complete arbitrary-sieve sheafification and local-isomorphism proofs in Sieves, local isomorphisms and presheaf localization, §§1–3 and 6. In particular associated sheaf $a_J$ is left exact, a sheaf is separated on every covering sieve, and $\begin{gathered} S\in J(c)\\ \Longleftrightarrow a_JS\longrightarrow a_Jh_c\text{ is invertible}. \end{gathered} \tag{1.1}$ Here $h_c=C(-,c)$; it need not already be a sheaf. No subcanonicity or enough-points assumption is made.

Let $i\colon D\hookrightarrow C$ be a full subcategory. For an object $c\in C$, let $N_c\subset h_c$ be the sieve generated by all arrows $d\to c$ with $d\in D$. Assume $N_c\in J(c)\qquad\text{for every }c\in C. \tag{1.2}$ We call this cover-density. The term refers to this covering condition. It is different from the restricted-Yoneda definition of density recalled in Dense probes and reconstruction from colimits, §1.

If $S$ is a sieve on $c$, write $S|_D$ for its arrows with domains in $D$. For a collection $T$ of such arrows, write $\langle T\rangle_C$ for its generated $C$-sieve: its arrows are all composites $t v$, with $t\in T$. This notation also applies when the target $c$ is not in $D$.

Lemma 1.1. If $S\in J(c)$, then $\langle S|_D\rangle_C\in J(c)$.

Proof. Put $Q=\langle S|_D\rangle_C$. For $s\colon x\to c$ in $S$, every arrow $h\colon d\to x$ with $d\in D$ has $sh\in S|_D$. Hence $N_x\subset s^*Q$. By (1.2), $s^*Q$ covers $x$. Transitivity of the topology along the covering sieve $S$ now makes $Q$ cover $c$. This includes empty covering sieves. $\square$

For $d\in D$, define a covering sieve on $D$ by $\begin{gathered} R\in J_D(d)\\ \Longleftrightarrow\langle R\rangle_C\in J(d). \end{gathered} \tag{1.3}$ Fullness identifies a $D$-sieve $R$ with its arrows in $C$ whose domains lie in $D$.

Proposition 1.2. Formula (1.3) is a Grothendieck topology. For any $C$-sieve $S$ on $d\in D$, $S\in J(d)\quad\Longleftrightarrow\quad S|_D\in J_D(d). \tag{1.4}$

Proof. The maximal $D$-sieve generates the maximal $C$-sieve, since it contains $1_d$. For pullback stability, take $R\in J_D(d)$ and $f\colon e\to d$ in $D$. The $C$-sieve $f^*\langle R\rangle_C$ covers $e$. Let $g\colon x\to e$ belong to it, so $fg=r v$ for some $r\colon d'\to d$ in $R$ and $v\colon x\to d'$. For every $h\colon t\to x$, $t\in D$, fullness makes $gh$ and $vh$ arrows of $D$. Thus $fgh=rvh\in R$, so $gh\in f^*_D R$. Consequently the pullback by $g$ of $\langle f^*_DR\rangle_C$ contains $N_x$, and covers. Transitivity along $f^*\langle R\rangle_C$ proves that this generated sieve covers $e$, as required.

For transitivity in $D$, suppose $R\in J_D(d)$, and a $D$-sieve $T$ on $d$ has $r^*_DT\in J_D(d')$ for every $r\colon d'\to d$ in $R$. Its generated covering sieve is contained in $r^*_C\langle T\rangle_C$, so that latter sieve covers. Every arrow of $\langle R\rangle_C$ factors as $rv$, and pullback stability makes its pullback of $\langle T\rangle_C$ cover too. Transitivity in $C$ gives $\langle T\rangle_C\in J(d)$, proving the axiom. Closure under larger sieves follows directly from (1.3). Finally Lemma 1.1 proves the forward direction of (1.4), and the reverse follows because $\langle S|_D\rangle_C\subset S$. $\square$

2. Restriction really gives a sheaf

Write $rF=F|_D$. We prove that $r$ takes $J$-sheaves to $J_D$-sheaves, keeping the matching maps explicit.

Take a $J_D$-covering sieve $R$ on $d$, and a matching family $(s_u\in F(d_u))_{u\colon d_u\to d\in R}$ on it. On an arrow $a\colon x\to d$ in $\langle R\rangle_C$, choose a factorization $a=uv$, and set $t_a=F(v)(s_u)\in F(x). \tag{2.1}$ If also $a=u'v'$, restrict the two proposed values along every $h\colon e\to x$ with $e\in D$. Fullness makes $vh$, $v'h$ arrows in $D$. The matching equations on $R$ identify both restrictions with $s_{ah}$. The sieve generated by these $h$ is $N_x$, which covers; separatedness of $F$ makes the two original values equal. Thus (2.1) is independent of factorization. Restriction along any further $k\colon y\to x$ gives $t_{ak}=F(k)t_a$, by the factorization $ak=u(vk)$. We have a matching family on the covering $C$-sieve $\langle R\rangle_C$. Its unique glue in $F(d)$ restricts to the prescribed $s_u$. Uniqueness on $R$ follows from uniqueness on that generated covering sieve. This is exactly the sheaf condition for $rF$.

3. The compatible-family extension and its adjunction

For a presheaf $G$ on $D$, define a presheaf $EG$ on $C$ by $\begin{gathered} (EG)(c)=\operatorname{Nat}(h_c|_D,G),\\ s=(s_a),\quad s_a\in G(d)\\ (a\colon d\to c,\ d\in D),\\ G(b)s_a=s_{ab}\\ (b\colon e\to d\text{ in }D). \end{gathered} \tag{3.1}$ All indices and sets here are small. For $f\colon c'\to c$, define $(EG)(f)(s)_a=s_{fa}$. Identity and composition follow componentwise; a map of $G$'s acts on every $s_a$ by its corresponding component.

There is an adjunction on presheaves $r\dashv E$. Explicitly, a transformation $q\colon rF\to G$ gives $\widetilde q_c(x)_a=q_d(F(a)x). \tag{3.2}$ The equations for $q$ and $F$ give the compatibility in (3.1) and naturality in $c$. Conversely a map $v\colon F\to EG$ gives $q_d(x)=v_d(x)_{1_d}$. Naturality of $v$ along $a\colon d\to c$ recovers (3.2); compatibility in (3.1) recovers $v_d(x)$ from its identity component when $d\in D$. Thus the two constructions are inverse and natural in both presheaves.

The unit and counit are $\begin{gathered} \eta_{F,c}(x)_a=F(a)x,\\ \epsilon_{G,d}(s)=s_{1_d}. \end{gathered} \tag{3.3}$ Fullness and Yoneda make $\epsilon_G\colon rEG\to G$ an isomorphism, with inverse $g\mapsto(G(a)g)_a$ at $d$. The first triangle evaluates the unit's family at the identity and returns $x$. In the second triangle, the $a$-component of $E\epsilon_G\eta_{EG}(s)$ is the identity component of the subfamily $(s_{ab})_b$, which is $s_a$. This proves both triangles with the stated maps.

4. Both sheaf conditions and the comparison equivalence

Theorem 4.1. Under (1.2), restriction induces an equivalence $\mathsf{Sh}(C,J)\xrightarrow{\ r\ }\mathsf{Sh}(D,J_D) \tag{4.1}$ with inverse $E$ as in (3.1), and with unit and counit (3.3).

Proof that $EG$ is a sheaf. Let $G$ be a $J_D$-sheaf. For a $J$-covering sieve $S$ on $c$, take a matching family $t_f\in(EG)(x)$, for $f\colon x\to c$ in $S$. For $a\colon d\to c$, $d\in D$, the sieve $R_a=(a^*S)|_D \tag{4.2}$ covers $d$ for $J_D$, by (1.4). On $b\colon e\to d$ in $R_a$, put $u_b=(t_{ab})_{1_e}\in G(e)$. The matching equations on the original family and compatibility of each $t_{ab}$ make the $u_b$ a matching family on $R_a$. Let $s_a\in G(d)$ be its unique glue.

For $v\colon d'\to d$ in $D$, one has $R_{av}=v^*_DR_a$. Restricting the glued $s_a$ along $v$ gives exactly the family used to glue $s_{av}$. Uniqueness implies $G(v)s_a=s_{av}$, so $s=(s_a)$ belongs to $(EG)(c)$.

If $f\colon x\to c$ is in $S$, compare $(EG)(f)s$ with $t_f$ on an arbitrary component $a\colon d\to x$, $d\in D$. The composite $fa\in S$, so $R_{fa}$ is maximal and its glued value is $(t_{fa})_{1_d}$. Matching gives $(t_{fa})_{1_d}=(t_f)_a$. This proves that $s$ restricts to the whole given family. Conversely any possible glue must, for each $a$, restrict on $R_a$ to the specified $u_b$, hence equal $s_a$. This proves uniqueness and the sheaf condition.

Proof that the unit is invertible on sheaves. Let $F$ be a $J$-sheaf. Its unit is injective at $c$: equality on every $d\to c$ implies equality on $N_c$, hence equality by separatedness. For surjectivity take $s=(s_a)\in E(rF)(c)$. For $f\colon x\to c$ in $N_c$, choose $f=av$, with $a\colon d\to c$, $d\in D$, and set $t_f=F(v)s_a$. Independence of the factorization follows exactly as in §2: after any $h\colon e\to x$, $e\in D$, both restrictions equal $s_{fh}$, and $N_x$ covers. The $t_f$ are natural under restriction, so glue uniquely along $N_c$ to $x\in F(c)$. Its restrictions along every $a$ are $s_a$, proving $\eta_{F,c}(x)=s$.

Section 2 proves the other restriction sheaf condition. The presheaf adjunction therefore restricts to the sheaf categories, its unit and counit are invertible there, and its two triangles remain the componentwise identities already proved. This proves (4.1). $\square$

No chosen intersections, pullbacks, representable sheaves or stalks entered this construction. The right Kan extension on all presheaves need not preserve arbitrary colimits; the equivalence assertion is about the indicated sheaf categories.

5. Nested dense subcategories and coefficients

Suppose $D\subset E\subset C$ are full, $E$ is cover-dense in $(C,J)$, and $D$ is cover-dense in $(E,J_E)$. Then $D$ is cover-dense in $C$. Indeed for every $e\to c$ with $e\in E$, the pullback of the sieve generated by $D\to c$ contains the $C$-sieve generated by the covering $D$-arrows into $e$. That generated sieve covers by definition of $J_E$. Transitivity along the covering sieve generated by $E\to c$ proves density.

The directly induced topology on $D$ agrees with induction through $E$. A $D$-sieve $R$ covers through $E$ precisely when its $E$-generated sieve has a covering $C$-generated sieve. The latter is exactly $\langle R\rangle_C$: compositions through $E$, followed by arbitrary $C$-arrows, produce the same composites as direct generation, since $D\subset E$. This proves equality of the covering criteria.

The inverse equivalence also has a specified composition isomorphism. A family $s_a$ indexed by $a\colon d\to c$ corresponds to the nested family $(s_{ba})_{a\colon d\to e}\quad\text{at each }b\colon e\to c. \tag{5.1}$ Conversely evaluate the inner family at $1_d$, using $d\in E$, to recover $s_b$ for $b\colon d\to c$. The compatibility equations in (3.1) prove that these operations are inverse. For three nested dense inclusions either order of flattening gives $s_{cba}$; associativity of composition proves coherence. Identity inclusions give identity comparisons, and these comparisons carry all unit and counit components to those in (3.3).

Let $R$ now be a sheaf of associative unital rings on $(C,J)$. Restriction gives an exact equivalence $\mathsf{Mod}(R)\xrightarrow{\sim}\mathsf{Mod}(rR). \tag{5.2}$ For an $rR$-module sheaf $M$, the inverse has underlying compatible families $(EM)(c)$, with pointwise addition and $\begin{gathered} (t\cdot s)_a=R(a)(t)\,s_a\\ (t\in R(c)). \end{gathered} \tag{5.3}$ Every restriction map is linear with this action, and all module axioms hold on every component. The unit $R\to E(rR)$ is the ring isomorphism given by (3.3); thus this action agrees with transporting the ring and module structures through the equivalence. Module maps and both inverse comparisons are the same componentwise maps. The abelian sheaf-module constructions and their algebraic operations are supplied by Ringed sieve sites and extension of scalars, §§1–3. An equivalence of abelian categories preserves kernels and cokernels by their universal properties, hence this module equivalence is exact. It consequently induces inverse equivalences on bounded complexes, homotopy categories and $D^b$, with the same signed cone and cohomology maps. This is categorical exactness of sheaf-module functors; it is not sectionwise surjectivity of every sheaf epimorphism.

6. A finer topology gives a canonical localization

Let $K$ be another topology on $C$, with $J\subset K$. Every $K$-sheaf is a $J$-sheaf. Associated sheaf gives an adjunction $\begin{gathered} L=a_K|_{\mathsf{Sh}(C,J)},\\ L\colon\mathsf{Sh}(C,J)\rightleftarrows\mathsf{Sh}(C,K)\colon v, \end{gathered} \tag{6.1}$ where $v$ is the full inclusion. This is just the sheafification Hom bijection restricted to the indicated objects, so its unit is the actual associated-sheaf map $F\to vLF$ and its counit is an isomorphism. The triangles are the sheafification triangles. Finite limits of sheaves are presheaf limits, and $a_K$ is left exact, so $L$ preserves finite limits. It preserves colimits by its adjunction, with the colimits formed in their respective sheaf categories.

Proposition 6.1. This canonical adjunction is an equivalence if and only if $J=K$.

Proof. Equality makes it the identity. Conversely for a $K$-covering sieve $S$ on $c$, the map $a_JS\to a_Jh_c$ becomes an isomorphism after $L$, because its image is $a_KS\to a_Kh_c$. An equivalence reflects isomorphisms, so the original map is invertible. Criterion (1.1) makes $S$ a $J$-cover. Thus $K\subset J$, proving equality. The identification $La_J=a_K$, including its unit, follows directly from the two sheafification adjunctions: maps into a $K$-sheaf have the same universal factor through either composite. $\square$

This criterion concerns the canonical inclusion and localization. It does not infer equality of sites from an unspecified abstract equivalence of two categories.

7. Topology change commutes with dense restriction

If $D$ satisfies (1.2) for $J$, it also does for $K$, since $J\subset K$. The induced topologies satisfy $J_D\subset K_D$. Use $r_J,r_K$ for the two restriction equivalences, and $L_D=a_{K_D}|_{\mathsf{Sh}(D,J_D)}$. There is a canonical isomorphism $r_K L\xrightarrow{\sim} L_D r_J \tag{7.1}$ whose composite with the restricted $K$-sheafification unit is the $K_D$-sheafification unit.

Proof. For a $K_D$-sheaf $G$, its extension $EG$, given by exactly (3.1), is a $K$-sheaf and hence a $J$-sheaf. The restriction equivalences and (6.1) give natural bijections $\begin{gathered} \operatorname{Hom}(r_KLF,G)\\ \simeq\operatorname{Hom}(LF,EG)\\ \simeq\operatorname{Hom}(F,EG)\\ \simeq\operatorname{Hom}(r_JF,G). \end{gathered} \tag{7.2}$ The last Hom is in $\mathsf{Sh}(D,J_D)$, viewing $G$ there by the full inclusion. Every bijection uses the unit or counit already specified. Thus $r_KLF$, with the restricted unit, is the reflection of $r_JF$ into the $K_D$-sheaves, exactly as is $L_Dr_JF$. The universal property gives the unique unit-preserving isomorphism (7.1). Its inverse is obtained by the opposite universal factorization; uniqueness makes both composites identities. For $J\subset K\subset H$, both resulting composite isomorphisms have the same composite unit, so the same uniqueness proves coherence. $\square$

In particular $J=K$ if and only if $J_D=K_D$. For the reverse implication, the bottom reflection is the identity; (7.1) and the restriction equivalences make the canonical top reflection an equivalence, and Proposition 6.1 applies. The forward implication follows from (1.3). Cover-density for the coarser topology is essential to this inference.

8. Four graded exercises with complete solutions

8.1. Rational intervals and a missing empty overlap — introductory

Exercise 1. Let $C$ be the open subsets of $\mathbb R$, with inclusions and ordinary open covers. Let $D$ consist of nonempty bounded open intervals with rational endpoints. Show that $D$ satisfies (1.2), although two disjoint objects have no pullback in $D$. For a set $A$, let $G(I)=A$ on every $I\in D$, with identity restrictions. Prove that it is a sheaf for the induced topology and compute $EG$ on every open subset, including the empty set.

Solution. Every $x\in U$, with $U$ open, lies in a rational-endpoint bounded interval contained in $U$. To obtain endpoints one may choose a sufficiently large positive integer denominator, then integers on either side of $x$ inside a small interval in $U$. Thus these intervals cover $U$. If two are disjoint, no nonempty interval includes into both. Since the empty object was excluded from $D$, their pullback over any common larger rational interval does not exist in $D$. The theorem requires no such pullback.

A sieve on an interval $I\in D$ is covering exactly when the union of its interval domains is $I$, by (1.3). A matching family for $G$ assigns an element of $A$ to each such interval, with equal values on inclusions. Overlapping members have equal values, by restriction to a rational interval inside their overlap. The overlap graph of an interval cover of $I$ is connected: otherwise the unions of two groups of graph components would separate $I$ into disjoint nonempty open subsets.

Here is the elementary connectedness fact used in that argument. If an interval had a partition into open subsets $V,W$, choose $v\in V$, $w\in W$, with $v<w$, exchanging the names if necessary. Put $t=\sup(V\cap[v,w])$. If $t\in V$, openness gives a larger point in that set, since $t\ne w$. If $t\in W$, openness excludes $V$ just to the left of $t$, contrary to its supremum property and $t\ne v$. Both cases are impossible. Thus the overlap graph is connected, all labels of a matching family agree, and that common value is its unique glue in $G(I)$. This also covers $A=\varnothing$: a cover of nonempty $I$ is nonempty and has no matching family with empty values.

For arbitrary open $U$, formula (3.1) assigns compatible labels to every rational interval in $U$. Two overlapping intervals have equal labels, as above. Each connected component of $U$ is an open interval: the class of a point under the relation that the segment between two points lies in $U$ is an open interval, and two such classes are separated by a missing point. The interval connectedness argument shows that all rational intervals in one component receive a single label. Conversely a choice of a label for each component gives the compatible family. Hence $(EG)(U)=A^{\pi_0(U)}, \tag{8.1}$ with restriction given by the map sending a component of a smaller open set to its containing component. Equivalently these are locally constant functions $U\to A$, where $A$ has the discrete topology. At $U=\varnothing$ there are no interval tests, and (3.1) is a singleton, agreeing with the empty power in (8.1). For nonempty disconnected $U$, the answer need not be just one copy of $A$.

8.2. Enough objects without enough morphisms — intermediate

Exercise 2. Let $C$ be the one-object category of the group $\{1,g\}$, with $g^2=1$, and let $D$ have the same object but only its identity arrow. Give both categories the topology in which only maximal sieves cover. Every $C$-object is covered by an identity coming from $D$. Explain why this does not give (4.1), and compute the right adjoint to restriction, its unit and its counit.

Solution. The inclusion is not full. Sheaves for the stated topology are all presheaves. A presheaf on $C$ is a set $X$ with an involution $\sigma$; a presheaf on $D$ is just a set. Restriction forgets the involution. It is not full: the identity map of the underlying two-element sets from a trivial involution to a swapping involution does not commute with the involutions.

The formula $ES=\operatorname{Nat}(h_*|_D,S)$ is still the presheaf right adjoint; now $h_*|_D$ has two elements, so $ES=S^{\{1,g\}}=S\times S, \tag{8.2}$ with involution exchanging the two coordinates. Indeed restriction along $g$ precomposes a family indexed by $a\in\{1,g\}$ with $a\mapsto ga$. The unit on $(X,\sigma)$ is $x\mapsto(x,\sigma x)$, and the counit on $S$ is $(s_1,s_g)\mapsto s_1$. Their two triangle identities follow by substitution, as for any restriction–right-Kan adjunction. Neither comparison is generally invertible: on a trivial two-element $X$ the unit has only the two diagonal values, while the target has four elements; the counit on a two-element $S$ is not injective. Fullness in the comparison theorem made $h_d|_D$ representable on $D$. It is exactly that step which fails here. Covering the object names does not retain the action of their missing arrows.

8.3. A finer topology on a fork — advanced

Exercise 3. Let $C$ have objects $p,a,b$, nonidentity arrows $a\to p$, $b\to p$, and no others. Let $J$ have only maximal covering sieves. Let $K$ additionally declare the sieve generated by the two leaf arrows to cover $p$. Describe (6.1), its unit and counit, and the dense-subsite square for $D=\{a,b\}$. Give a nonzero abelian $J$-sheaf killed by $K$-sheafification.

Solution. A presheaf is a diagram $M\to A$, $M\to B$. All such diagrams are $J$-sheaves. The described $K$ is a topology: pullback of the leaf sieve along a leaf arrow is maximal; the other nonmaximal sieve on $p$ which could satisfy transitivity would have to contain both leaf arrows, hence is the same covering sieve. A covering sieve on a leaf is maximal. The matching families on the leaf cover are arbitrary pairs $(x,y)\in A\times B$. Thus a $K$-sheaf is exactly a diagram whose actual pair of restrictions gives a bijection $M\xrightarrow{\sim}A\times B. \tag{8.3}$ The reflection sends $(M\to A,M\to B)$ to the diagram with vertex $A\times B$ and its two projections. Its unit is the identity on leaves and sends $m$ to its two restrictions at the vertex. Its counit on a $K$-sheaf is the inverse of that restriction bijection at the vertex and the identities on leaves. To check the adjunction directly, a map from the original diagram to a $K$-sheaf is determined by its leaf maps: the required vertex map is their pair followed by the inverse restriction bijection. This gives both natural inverse Hom factors.

The full subcategory $D$ is $K$-dense, with the trivial topology on its two discrete objects; extension is exactly the product diagram just constructed. It is not $J$-dense: the leaf sieve is proper and hence not a $J$-cover. For instance $M=\varnothing$, $A=B=\{*\}$, has a unit which is not onto at $p$; with $M=\{0,1\}$ and the same leaves, its unit is not injective there. The localization is therefore not an equivalence, as Proposition 6.1 predicts.

For abelian sheaves take $M=\mathbb Z$, $A=B=0$, and the zero restrictions. This is a nonzero $J$-sheaf, while its $K$-associated sheaf is zero at every object. No original vertex section survives the new covering requirement. This computation uses the new sheaf condition; treating the two leaf arrows as already covering for $J$ would give a different category.

8.4. Finite gluing can hide from an ordinary basis — advanced

Exercise 4. On $C=\operatorname{Op}(\mathbb R)$, let $J$ be the topology generated by finite open covers and $K$ the topology of all open covers. Define $\begin{gathered} H(U)=\{f\colon U\to\mathbb Z:\\ f\text{ locally constant}\},\\ F(U)=\{f\in H(U):\\ f(U)\text{ finite}\}. \end{gathered} \tag{8.4}$ Show that $F,H$ are distinct $J$-sheaves, that $a_KF=H$ with its specified inclusion unit, and that their restrictions to the rational-interval category of Exercise 1 are identical. Locate the failed density hypothesis.

Solution. Finite covers are stable under pullback, since intersections of opens exist, and under composition, since a finite union of finite index sets is finite. They define the stated topology; a sieve covers for $J$ exactly when it contains a finite open cover. Compatible locally constant functions on any open cover glue uniquely as ordinary functions, and the glue is locally constant on the same neighborhoods. Therefore $H$ is a $K$-sheaf and hence a $J$-sheaf. On a finite cover, gluing functions of finite image has image contained in the finite union of those finite images. Thus $F$ also satisfies finite-cover gluing and is a $J$-sheaf. The empty open has its unique empty function, so empty covers there cause no exception.

They are different on $U=\bigcup_{n\ge1}(3n,3n+1). \tag{8.5}$ The constant function with value $n$ on the $n$-th component belongs to $H(U)$ and restricts to a member of $F$ on each displayed interval. Its infinite image keeps it out of $F(U)$. The inclusion $F\to H$ is locally injective for $K$, because it is already injective. It is locally surjective: the level sets of any locally constant section form an open cover, and its restriction to each nonempty level set is a constant finite-image section. The local-isomorphism criterion in the prerequisite makes $a_KF\to a_KH=H$ an isomorphism. It carries the associated-sheaf unit to the actual inclusion $F\hookrightarrow H$, as both maps factor that inclusion through the same adjunction.

Every interval in $D$ is connected by the argument in Exercise 1, so all its locally constant integer-valued functions are constant. Consequently $F|_D=H|_D$, with value $\mathbb Z$ and identity restrictions. Nevertheless their map is not invertible on $U$. Restriction cannot be an equivalence for the coarser topology $J$.

Indeed the $C$-sieve $N_{\mathbb R}$ generated by these bounded intervals has only bounded open domains. A finite family of such domains cannot cover $\mathbb R$. Hence it is not a $J$-cover, and (1.2) fails for $J$. It does hold for $K$, as Exercise 1 proved. This is why ordinary-basis reconstruction loses the finite-image condition: that condition survives finite gluing but not the larger gluing rule. The topology-change comparison in §7 required density for the coarser topology, precisely the hypothesis missing here.