Subsequent full independent model review: reviewer model not specified, Ultra; no corrections requested. This review covers the recorded version of this lesson and its specified prerequisite interfaces.

Dense probes and reconstruction from colimits

A detecting family tells us whether a map is invertible. A dense family does more: it recovers every map from its action on the probes. This distinction matters when the goal is to reconstruct a category inside a presheaf category. The missing information is compatibility among tests, and colimits provide a way to assemble it.

Fix a universe, take all Hom sets and the specified small diagrams in that universe, and let \(\mathsf C\) be locally small. Use the category-of-elements convention and canonical presheaf density from Ind-objects through their elements, the componentwise detecting and separating conventions from Generators and small quotient families, and the coimage and strict-epimorphism results from Coimages, images and composition of quotients. An essentially small full subcategory may be replaced by a small skeleton. Closure of probes under a construction means closure up to isomorphism.

1. The nerve and its realization

For a small full subcategory \(\mathsf A\subset\mathsf C\), write \(\widehat{\mathsf A}=\mathsf{Set}^{\mathsf A^{\rm op}}\) and define the restricted Yoneda functor, or nerve, by

Maps in \(\mathsf A\) act by precomposition; maps in \(\mathsf C\) act by postcomposition. We call \(\mathsf A\) dense if \(N\) is fully faithful. This is the restricted-Yoneda convention in Riehl, Definition 6.5.11. A dense family is detecting, because a fully faithful functor reflects isomorphisms: lift the inverse of the image map and use faithfulness for its two inverse identities.

If \(T\in\mathsf A\), fullness of the inclusion makes \(N(T)\) the representable presheaf \(h_T\). Yoneda therefore gives

Enlargement lemma. If \(\mathsf A\subset\mathsf B\subset\mathsf C\) are small full subcategories and \(\mathsf A\) is dense, then \(\mathsf B\) is dense.

Proof. Restriction from \(\mathsf B\) to \(\mathsf A\) shows that the larger nerve is faithful. For a natural transformation \(\alpha:N_{\mathsf B}(X)\to N_{\mathsf B}(Y)\), density on \(\mathsf A\) determines a unique map \(f:X\to Y\). If \(V\in\mathsf B\), \(v:V\to X\), \(T\in\mathsf A\), and \(s:T\to V\), naturality gives

The faithful smaller nerve distinguishes the maps \(\alpha_V(v)\) and \(fv\). Thus \(\alpha=N_{\mathsf B}(f)\). This proves fullness as well as uniqueness. \(\square\)

Suppose now that \(\mathsf C\) admits all small colimits. For \(P\in\widehat{\mathsf A}\), its element category \(\mathsf E_P\) is small: its objects are pairs \((T,p)\) with \(p\in P(T)\). Define

This is the inclusion specialization of the canonical nerve–realization adjunction in Riehl, Proposition 6.5.12. Its exact interface is

The last bijection uses the already retained presheaf density \(P=\operatorname{colim}_{\mathsf E_P}h_T\), whose trivial-site specialization comes from Stacks, Lemma 7.12.6. For an arrow \(P\to Q\), the induced element-category functor and colimit cocone give \(L(P)\to L(Q)\); their uniqueness proves identities and composition. The bijections in (1.4) are natural in both variables.

Write \(\eta:P\to NL(P)\) for the unit and \(\epsilon:LN(X)\to X\) for the counit. The counit is the colimit map whose component at \((T,a:T\to X)\) is \(a\). The canonical adjunction criterion Stacks, Lemma 4.24.4 says that \(N\) is fully faithful exactly when this actual counit is invertible. Thus density is equivalent to reconstruction by the specified probe diagram:

The assertion includes the structural map, rather than only an abstract isomorphism between its source and target.

2. Which presheaves are nerves?

Assume throughout this section that \(\mathsf C\) admits small colimits and \(\mathsf A\) is dense. For \(P\in\widehat{\mathsf A}\), set

This is a functor on \(\mathsf C^{\rm op}\). Say that it preserves limits in \(\mathsf C^{\rm op}\), meaning that it sends every small colimit in \(\mathsf C\) to the corresponding limit of sets.

Theorem 2.1. The essential image of \(N\) consists exactly of the \(P\) with this property. Every functor \(F:\mathsf C^{\rm op}\to\mathsf{Set}\) sending small colimits to limits is representable.

Proof of the image assertion. If \(P=N(Y)\), full faithfulness identifies \(H_P\) with \(\operatorname{Hom}_{\mathsf C}(-,Y)\), which has the required property.

Conversely, suppose \(H_P\) has that property. For any \(Q\in\widehat{\mathsf A}\), realization, probe Yoneda, and presheaf density identify

The map is precomposition with \(\eta_Q\), since its restrictions are precisely the element-indexed cocone maps. At \(Q=P\), lift \(\operatorname{id}_P\) to \(v:NL(P)\to P\). Then \(v\eta_P=\operatorname{id}_P\). The object \(NL(P)\) already lies in the essential image, so (2.1) also holds with target \(NL(P)\). The maps \(\eta_Pv\) and \(\operatorname{id}_{NL(P)}\) have the same precomposition with \(\eta_P\), hence are equal. Thus \(\eta_P\) is invertible and \(P\simeq NL(P)\). Both inverse identities have been checked.

Proof of representability. Restrict \(F\) to \(\mathsf A^{\rm op}\), obtaining \(P\). Reconstruction (1.5) and preservation give, naturally in \(X\),

The first map sends an element to its restrictions along all \(a:T\to X\); the second records those restrictions as a natural transformation. Thus \(H_P\simeq F\) has the required preservation property. The image assertion gives \(P\simeq N(L(P))\), and full faithfulness now gives \(F(X)\simeq\operatorname{Hom}_{\mathsf C}(X,L(P))\). All comparisons respect arrows in \(X\). \(\square\)

These arguments do not require ordinary quotient classes to be small. They use a small dense collection of probes instead.

3. Density from stability under base change

Assume that \(\mathsf C\) has small colimits and finite limits. A specified class of colimits is stable under base change if, for every \(T\to X\), pulling back a diagram in \(\mathsf C/X\) preserves its colimit. Colimits in a slice are computed in \(\mathsf C\): the compatible structure maps to \(X\) determine the colimit's map to \(X\).

Let \(\mathsf A\) be an essentially small full subcategory whose objects form a componentwise detecting family. The generator results already proved imply that \(N\) is conservative and faithful. For clarity, faithfulness uses the equalizer \(e:E\to Y\) of two maps \(Y\rightrightarrows Z\) with equal nerve. Every map from a probe factors uniquely through \(e\), so \(N(e)\) is invertible. Detection makes \(e\) invertible, and the original maps agree.

Theorem 3.1. The subcategory \(\mathsf A\) is dense under either of these two assumptions:

• all small colimits are stable under base change;
• filtered small colimits are stable under base change, and \(\mathsf A\) admits finite colimits which its inclusion in \(\mathsf C\) preserves.

Proof. We first prove a reconstruction test. Let \(i\mapsto T_i\) be a small diagram in \(\mathsf A/X\). Suppose that its specified comparison

is invertible. Put \(Y=\operatorname{colim}_iT_i\), with maps \(j_i:T_i\to Y\), and write \(u:Y\to X\) for the comparison. The factorization of (3.1) through \(N(Y)\) makes \(N(u)\) pointwise surjective.

For any two indices \(i,k\), the projections from \(T_i\times_XT_k\) give two maps into \(Y\). Their nerves agree: the two maps into the presheaf colimit in (3.1) have the same image in \(N(X)\), and (3.1) is invertible. Faithfulness makes the original maps into \(Y\) equal. Therefore the canonical inclusion of pullbacks is invertible:

Indeed, the equality just obtained supplies the inverse into the left pullback, with both projections unchanged; the two inverse identities follow from the pullback universal property.

Base-change stability applied twice identifies

The map induced by (3.2) is the diagonal \(Y\to Y\times_XY\), since both of its projections restrict to \(j_i\) on the first colimit cocone. It is invertible. A map with invertible diagonal is monic: for \(a,b:V\to Y\) with \(ua=ub\), their pullback map \(V\to Y\times_XY\) factors through that diagonal, forcing \(a=b\). Hence \(N(u)\) is pointwise injective as well as surjective. Detection makes \(u\) invertible.

Under the first assumption, this test works for arbitrary \(i\), including an empty index. Under the second, it works for filtered indices, because the two successive colimits in (3.3) are filtered.

Apply the test to \(\mathsf A/X\). Canonical presheaf density identifies \(\operatorname{colim}_{\mathsf A/X}N(T)\to N(X)\) with an isomorphism. In the second case, \(\mathsf A/X\) is filtered: its initial probe maps to \(X\); the coproduct of two probes supplies a common receiver; and the coequalizer in \(\mathsf A\) of parallel arrows over \(X\) is also their coequalizer in \(\mathsf C\), so its structure map to \(X\) exists. These are all three filteredness conditions. The reconstruction test makes \(\epsilon_X\) invertible, and the canonical counit criterion proves density. \(\square\)

Corollary 3.2. Suppose \(\mathsf C\) has small colimits, finite limits, filtered colimits stable under base change, and a detecting object \(G\). Then every colimit-to-limit functor \(\mathsf C^{\rm op}\to\mathsf{Set}\) is representable.

Proof. Start with an initial object and \(G\). Repeatedly adjoin one chosen pushout for every span among the objects already included, taking the full subcategory at each stage. Each stage is small by local smallness, and the union over the natural numbers is small.

This union contains finite coproducts, since a coproduct is a pushout over the initial object. It also contains coequalizers: for \(a,b\colon T\rightrightarrows V\), take the pushout of

A map from that pushout to \(W\) is exactly a map \(v:V\to W\) with \(va=vb\); the other component is then forced to equal \(va\). Thus its map from \(V\) is the coequalizer. Coproducts and coequalizers construct every finite colimit, so the inclusion preserves finite colimits. The family is detecting because it contains \(G\). Apply the second case of Theorem 3.1 and then Theorem 2.1. \(\square\)

If filtered colimits commute with finite limits, the required base-change stability follows. For a filtered slice diagram, express each pullback as a finite limit. The colimits of the constant \(X\) and \(T\) diagrams are \(X\) and \(T\), because a filtered category is nonempty and connected. Commutation therefore identifies the colimit of the pullbacks with the pullback of the colimit. Exactness is one sufficient condition, rather than an additional assumption in Corollary 3.2.

4. A different criterion using strict quotients

Finite-colimit closure of the probes can be replaced by finite-coproduct closure if ordinary epimorphisms are strict.

Theorem 4.1. Suppose \(\mathsf C\) has small colimits and finite limits, filtered colimits are stable under base change, and every epimorphism is strict. If an essentially small full detecting subcategory \(\mathsf A\) is closed under finite coproducts, then it is dense.

Proof. Again \(N\) is faithful and conservative. It also reflects pointwise surjective maps as epimorphisms. If \(N(f)\) is surjective and \(bf=cf\), every \(a:T\to\operatorname{codom}(f)\) from a probe lifts through \(f\); hence \(ba=ca\). Faithfulness of \(N\) gives \(b=c\).

First let \((Y_i\to X)_{i\in I}\) be an arbitrary small filtered slice diagram, and put \(Y_\infty=\operatorname{colim}_iY_i\). The diagonal \(I\to I\times I\) is cofinal. Its comma category at \((i,k)\) consists of common receivers of \(i,k\). It is nonempty by filteredness; two such receivers map to a common receiver, and filtered equalization of the two maps from \(i\) and the two from \(k\) makes those receiving maps compatible. Thus the comma category is connected, which is the cofinality convention retained from Stacks, Definition 4.17.1. Consequently, two applications of base-change stability give

Coequalizers commute with these colimits by their universal property: a map out of either iterated colimit is the same compatible family of maps equalizing the same pairs. Thus, with the specified projections and comparison maps,

Fix \(X\). Let \(J\) be the directed poset of finite subsets of the object set of \(\mathsf A/X\), including the empty subset. For \(B\in J\), let \(S_B\) be the coproduct of its probes, with its evaluation map to \(X\). Choose a representative in \(\mathsf A\) and the resulting coproduct injections; they determine coherent transition maps by the coproduct universal property.

Every \(a:T\to X\) is a summand of \(S_{\{(T,a)\}}\to X\). Hence \(N(\operatorname{colim}_B S_B)\to N(X)\) is pointwise surjective. Its source-to-\(X\) comparison \(e\) is epic, hence strict. Formula (4.2) and the coimage criterion already proved identify the specified cocone as

Set \(M_B=\operatorname{Im}(N(S_B)\to N(X))\), the pointwise image presheaf. For every \(Z\), there is a natural bijection

To check it, a map from \(Q_B\) is a map \(S_B\to Z\) equalizing the kernel pair. Probe Yoneda identifies maps from \(S_B\) with natural maps \(N(S_B)\to N(Z)\), since \(S_B\in\mathsf A\). The nerve preserves that kernel pair because each of its values is representable. Faithfulness of \(N\) ensures that the two original kernel-pair composites agree exactly when their nerves agree. Finally, the pointwise surjection \(N(S_B)\to M_B\) is the coequalizer of its kernel pair in presheaves, as checked in the earlier strict-morphism lesson. Its natural descent property gives (4.4), including uniqueness and both inverse maps.

The \(M_B\) are increasing subpresheaves of \(N(X)\), and every element belongs to the singleton-summand image. Thus \(\operatorname{colim}_B M_B=N(X)\). Combining this with (4.3) and (4.4) gives

The correspondence sends a map to its actual nerve, since every comparison was induced by the evaluation cocone. This proves full faithfulness. \(\square\)

Theorem 4.1 assumes filtered base-change stability, without asserting that strict epimorphisms are stable under pullback. Those are different properties. Its finite-coproduct condition is sufficient; the exercises below show that some much smaller probe collections are already dense.

5. Internal Homs, a classification, and a failure of density

Internal-Hom consequence. Suppose \(\mathsf C\) has small colimits, finite limits, all small colimits stable under base change, and a detecting object. Then for each \(X,Y\) there is an object \([X,Y]\) with natural bijections

Proof. Pullback along \(X\to 1\), followed by forgetting the slice structure, is \(Z\mapsto Z\times X\). The first functor preserves all small colimits by assumption, and the second creates them. Therefore \(\operatorname{Hom}_{\mathsf C}(-\times X,Y)\) sends small colimits to limits. Construct a small dense subcategory as in Corollary 3.2 and apply Theorem 2.1. Representability gives (5.1), with its evaluation map. Precomposition in \(X\) and postcomposition in \(Y\) determine the arrows between internal Homs by Yoneda; their identities and composites follow from the uniqueness of representing arrows. Thus the construction is contravariant in \(X\) and covariant in \(Y\). \(\square\)

Theorem 5.1. Under these same colimit and base-change assumptions, if a detecting object \(G\) has \(\operatorname{End}_{\mathsf C}(G)=\{\operatorname{id}_G\}\), then \(\mathsf C\) is equivalent to \(\mathsf{Set}\), to the ordered category \(0<1\), or to the terminal category.

Proof. The first case of Theorem 3.1 makes the single-object full subcategory on \(G\) dense. Its presheaf category is \(\mathsf{Set}\). Hence \(N=\operatorname{Hom}(G,-)\) embeds \(\mathsf C\) fully faithfully into sets, with left adjoint \(L\); its essential image \(\mathsf D\) is a full replete reflective subcategory. It contains a singleton because \(N(G)\) is a singleton.

If \(\mathsf D\) contains a set \(B\) with distinct elements \(b_0,b_1\), every reflection unit \(r:S\to N(L(S))\) is bijective. If \(r\) identified two distinct elements of \(S\), a function \(S\to B\) taking different values on those elements could not factor through it, contrary to reflection. If an element of \(N(L(S))\) were outside the image of \(r\), two functions from that set to \(B\), equal on the image and different at that element, would contradict the uniqueness of factorization. Thus \(r\) is both injective and surjective for every \(S\), including \(S=\varnothing\). The image is all sets.

Otherwise every set in \(\mathsf D\) has at most one element. Repleteness and the singleton already in \(\mathsf D\) leave exactly two possibilities: it also contains the empty set, giving the category \(0<1\), or it does not, giving the terminal category. \(\square\)

All three possibilities satisfy the hypotheses. For the ordered category \(0<1\), colimits are joins of the diagram's object values, with the empty join equal to \(0\); finite limits are meets. Pullback along \(t\le x\) is meet with \(t\), and meets in this two-element lattice distribute over arbitrary joins, including the empty one. Thus all colimits are stable under base change. The object \(1\) detects its only noninvertible map \(0\to1\), since the induced map \(\operatorname{Hom}(1,0)\to\operatorname{Hom}(1,1)\) is \(\varnothing\to\{*\}\). Its endomorphism is the identity. Sets satisfy the base-change condition because \(\mathsf{Set}/X\) is the category of \(X\)-indexed families of sets: colimits are computed in each fibre and pullback reindexes the fibres. The terminal category satisfies the conditions directly.

A detecting probe need not be dense. Over any field \(k\), the single probe \(k\) detects linear isomorphisms, since \(\operatorname{Hom}_k(k,V)\simeq V\). Its endomorphisms test scalar multiplication but do not test addition. On \(V=k^2\), define

For every scalar \(a\), \(b(av)=ab(v)\): for \(a\ne0\), vanishing of the second coordinate is unchanged, and the case \(a=0\) holds separately. Thus \(b\) defines a natural transformation between the nerves of \(k^2\) and \(k\) on this single-object probe category. But \(b(1,0)=1\), \(b(0,1)=0\), and \(b(1,1)=0\), so it is not additive. The example works over every field, including the two-element field.

The category of all \(k\)-vector spaces has all small colimits and finite limits, every epimorphism is strict, and filtered colimits commute with finite limits: every finite tuple and every finite list of linear equations is represented and checked at a common stage. Hence filtered colimits are stable under base change. The single probe fails finite-coproduct closure, and the example shows that this condition cannot simply be deleted from Theorem 4.1. It also shows that the first alternative of Theorem 3.1 cannot be replaced by filtered stability without another condition on the probes.

6. Graded exercises with complete solutions

Exercise 1 (basic: one set as a dense probe). Fix a set \(A\) and take the full subcategory of \(\mathsf{Set}\) consisting of \(A\) alone. Determine exactly when it is dense. In the dense case, recover an arbitrary map \(X\to Y\) from an endomorphism-equivariant function \(X^A\to Y^A\).

Solution. If \(A=\varnothing\), every nerve is a singleton. The two maps \(1\to2\) have the same nerve, so the functor is not faithful.

Suppose \(A\ne\varnothing\). A natural transformation is a function \(T:X^A\to Y^A\) commuting with precomposition by every self-map of \(A\). Write \(c_x:A\to X\) for a constant function. The function \(T(c_x)\) is unchanged by every self-map of \(A\), hence is constant: precomposition with the constant map at any \(a\in A\) forces all its values to equal its value at \(a\). Denote that value by \(u(x)\). For \(f:A\to X\) and a constant self-map \(d_a\) at \(a\), equivariance gives

Evaluating at any point of \(A\) yields \(T(f)(a)=u(f(a))\). Thus \(T(f)=u\circ f\), and the constant functions recover \(u\) uniquely. Conversely every \(u:X\to Y\) has this equivariance. If \(X\) is empty, the nerve \(X^A\) is empty and the map out of it is unique, matching the unique map \(\varnothing\to Y\). If \(Y\) is empty and \(X\) is nonempty, both kinds of maps are absent. Therefore the nerve is fully faithful exactly when \(A\ne\varnothing\).

Exercise 2 (intermediate: one free module of rank at least two). Let \(R\) be an arbitrary unital ring, possibly noncommutative, and let \(r\ge2\) be finite. Prove that the single-object full subcategory on the left module \(R^r\) is dense in \(R\)-modules. Deduce that the two probes \(R,R^2\) are dense as well.

Solution. Evaluation on the standard basis identifies \(\operatorname{Hom}_R(R^r,X)\) with \(X^r\). If a left-linear endomorphism sends the \(j\)-th basis vector to \(\sum_i a_{ji}e_i\), precomposition sends a tuple to the tuple with entries \(\sum_i a_{ji}x_i\). Thus all coordinate selections, coordinate transfers, sums of two coordinates, and left scalar multiplications arise from probe endomorphisms. This basis description fixes the side of every scalar even when \(R\) is noncommutative.

Let \(T:X^r\to Y^r\) commute with these operations. The zero operation gives \(T(0)=0\). Let \(D_i\) keep only coordinate \(i\). Since \(T D_i=D_iT\), the output on a tuple supported in coordinate \(i\) is supported there. Call its coordinate \(u_i(x)\). Transfer from coordinate \(i\) to coordinate \(j\) gives \(u_i=u_j\); call the common function \(u\). Applying \(D_i\) to an arbitrary tuple now gives

The operation \((x_1,x_2,\ldots)\mapsto(x_1+x_2,0,\ldots)\) gives \(u(x+y)=u(x)+u(y)\). The operation \((x_1,\ldots)\mapsto(ax_1,0,\ldots)\) gives \(u(ax)=au(x)\) for every \(a\in R\). Together with \(u(0)=0\), this proves that \(u\) is a left-module homomorphism. Conversely every such homomorphism commutes with all the indicated linear combinations and hence with every probe endomorphism. This recovers all natural transformations and proves density, also for zero modules and the zero ring.

Take \(r=2\) and apply the enlargement lemma to the full category on \(R,R^2\). This proves its density without requiring it to contain every finite free module.

Exercise 3 (advanced: equivariant function objects). For an arbitrary group \(H\), prove that the regular left \(H\)-set is a dense probe in all left \(H\)-sets. Check the base-change hypotheses for this category and construct its internal Hom, including its action and its empty-set cases.

Solution. Evaluation at the identity identifies \(\operatorname{Hom}_H(H,X)\) with the underlying set of \(X\). Every equivariant endomorphism of \(H\) is right multiplication by a unique \(h\). Precomposition by it acts on the evaluated element as left multiplication by \(h\). Therefore natural maps between the single-probe nerves are exactly equivariant functions \(X\to Y\), proving full faithfulness. An equivariant bijection has equivariant inverse, so this probe is detecting.

Small colimits are set colimits with the induced action; finite limits are set limits with the componentwise action. Both assertions follow because the action respects the defining equivalence relations or compatible tuples, and the corresponding set universal maps remain equivariant. Pullbacks and slice colimits have these same underlying sets. Set colimits are stable under base change by the fibre calculation above, so the canonical equivariant comparison is an underlying bijection and hence invertible. Thus all the required colimits are stable under base change.

Take the set of all functions \(f:X\to Y\), rather than only the equivariant ones, and give it the action

The identity acts trivially. Substitution shows that \(h\cdot(k\cdot f)=(hk)\cdot f\), so this is an \(H\)-set. For an equivariant \(a:Z\times X\to Y\), put \(\widetilde a(z)(x)=a(z,x)\). Then

Thus currying gives an equivariant map into this function object. Conversely, evaluation of any such map is equivariant, since \(f(hz)(hx)=h f(z)(x)\). Currying and evaluation are inverse and natural in all three variables, proving (5.1) explicitly. If \(X\) is empty, the function object is a singleton with trivial action for every \(Y\). If \(X\) is nonempty and \(Y\) is empty, it is empty. These include \(X=Y=\varnothing\), and the same inverse formulas verify all the cases.

Exercise 4 (expert: reflectors of sets and pullbacks). Classify full replete reflective subcategories of \(\mathsf{Set}\). Determine which of their reflectors, viewed as endofunctors of sets, preserve finite limits. Show that all three reflectors preserve finite products.

Solution. Let \(r:S\to R(S)\) be the reflection unit. If the reflective subcategory contains a set with at least two elements, the separation and uniqueness argument in Theorem 5.1 makes every \(r\) bijective, so it is all of \(\mathsf{Set}\). Otherwise all its objects have at most one element. The unit \(1\to R(1)\) makes \(R(1)\) nonempty, so repleteness puts \(1\) in the subcategory. It either also contains \(\varnothing\), or contains only singletons. This gives exactly the three categories in Theorem 5.1.

Up to natural isomorphism, their reflectors are the identity, the constant singleton functor, and the support functor

For the support case, maps to either a singleton or the empty set verify the reflection bijections: a map to the empty set exists exactly when the source is empty. The unit sends each element of a nonempty set to the sole point. The constant singleton similarly reflects into the singleton category.

The identity and constant singleton preserve finite limits. The support functor preserves the terminal object and binary products, because a product of two sets is nonempty exactly when both factors are nonempty. The canonical product comparison is therefore a bijection in every case. The other two reflectors plainly preserve finite products as well.

Support fails to preserve pullbacks. Map two singleton sets to a two-element set, choosing a different element for each map. Their pullback is empty, so its support is empty. After applying support, both arrows are \(1\to1\), whose pullback is \(1\). Thus support is not left exact. This does not contradict base-change stability inside the category \(0<1\): a reflection's preservation of limits of arbitrary set diagrams is a separate assertion.

Original lesson text: CC0 1.0. Canonical presheaf density and the nerve–realization adjunction are retained through checked interfaces; source prose is not reproduced. Self-checked; no independent review has occurred.