Proof. Simplexwise chain map. We first construct the cylinder in \(\omega\mathrm{Cat}\), before applying \(\tau_n\). If \(T\) is a nonempty finite totally ordered set, let \(M[T]\) be its augmented normalized simplex chain complex. Thus its positive basis in degree \(p\) consists of the strictly increasing tuples \([t_0,\ldots,t_p]\). A weakly increasing tuple with a repeated entry means zero. For \(p>0\) the differential is the alternating sum of deletions; the augmentation sends every vertex to \(1\). The alternating deletion formula remains valid for a tuple with repetitions: it is the differential in the normalized quotient of the unnormalized simplex complex. In particular all calculations below remain valid when a displayed tuple is zero.

Fix a simplex \(\sigma=(v_0\le\cdots\le v_m)\) of \(NE\) and put \[T_\sigma=\{f(v_i),g(v_i):0\le i\le m\}\subset E.\] This subset is totally ordered. Indeed, the indices whose vertices belong to \(A\) form a prefix. On that prefix the two image lists coincide. If there are remaining vertices, all their \(g\)-images are one element \(b\), and every \(f(v_i)\) is at most \(f(v_m)\le b\). Consequently \(T_\sigma\) consists of the ordered \(f\)-image list, possibly with this one upper element adjoined. If every vertex is old, \(T_\sigma\) is just the \(f\)-image list. The order embedding \(T_\sigma\hookrightarrow E\) gives the canonical map \[\nu M[T_\sigma]\longrightarrow U^{\infty}_{E}.\] Here \(\nu M[T_\sigma]\) is the oriental obtained by enumerating \(T_\sigma\) increasingly.

Let \(e_0,e_1\) denote the vertices and \(u\) the edge of \(M[1]\), so \(du=e_1-e_0\). We define a graded homomorphism \[ C_\sigma:M[m]\otimes M[1]\longrightarrow M[T_\sigma] \tag{10}\] on the tensor basis. For \(x=[i_0,\ldots,i_p]\), with \(i_0<\cdots<i_p\), set \(w_j=v_{i_j}\) and prescribe \[\begin{align*} C_\sigma(x\otimes e_0)&=F(w):=[f(w_0),\ldots,f(w_p)],\\ C_\sigma(x\otimes e_1)&=G(w):=[g(w_0),\ldots,g(w_p)],\tag{11}\\ C_\sigma(x\otimes u)&=h(w):=[f(w_0),\ldots,f(w_p),g(w_p)]. \end{align*}\] Each image is a positive basis element or zero: all displayed tuples are weakly increasing in \(T_\sigma\). Both endpoint prescriptions are the normalized chain maps of monotone maps on vertices. They preserve augmentation, and hence so does \(C_\sigma\) in degree zero.

It remains to check the differential on \(x\otimes u\), including its sign. If \(p=0\), the required identity is precisely \[d[f(w_0),g(w_0)]=[g(w_0)]-[f(w_0)].\] For \(p>0\), abbreviate \[\begin{split} B&=[f(w_0),\ldots,f(w_{p-1}),g(w_p)],\\ B'&=[f(w_0),\ldots,f(w_{p-1}),g(w_{p-1})]. \end{split}\] We claim that \(B-B'=G(w)\) in normalized chains. If \(w_{p-1}\in A\), all \(w_0,\ldots,w_{p-1}\) lie in \(A\), so that \(B=G(w)\), whereas \(B'=0\) because its last two entries coincide. If \(w_{p-1}\notin A\), then also \(w_p\notin A\). Constancy of \(g\) on the complement gives \(B=B'\) and makes the last two entries of \(G(w)\) coincide, so again \(B-B'=G(w)\). Expanding \(dh(w)\) and canceling the terms obtained by deleting one of \(f(w_0),\ldots,f(w_{p-1})\) therefore gives \[\begin{align*} dh(w) &=\sum_{j=0}^{p}(-1)^j h(w_0,\ldots,\widehat{w_j},\ldots,w_p) +(-1)^p\bigl(B-B'-F(w)\bigr)\\ &=\sum_{j=0}^{p}(-1)^j h(w_0,\ldots,\widehat{w_j},\ldots,w_p) +(-1)^p\bigl(G(w)-F(w)\bigr). \tag{12}\end{align*}\] This is exactly the image under \(C_\sigma\) of \[d(x\otimes u)=dx\otimes u+(-1)^p x\otimes(e_1-e_0).\] Thus (10) is a positive, augmentation-preserving chain map in every degree. The strong Steiner tensor comparison gives the strict functor \[ \mathcal O_{m}\otimes D_{1} \cong\nu(M[m]\otimes M[1]) \xrightarrow{\nu C_\sigma}\nu M[T_\sigma] \longrightarrow U^{\infty}_{E}. \tag{13}\]

Compatibility with simplex operators. We next verify compatibility with all simplex operators. Let \(\alpha:[q]\to[m]\) be monotone and set \(\rho=\sigma\alpha\). There is an order embedding \(j:T_\rho\hookrightarrow T_\sigma\). We have an equality of whole directed chain maps \[ M(j)C_\rho=C_\sigma\bigl(M(\alpha)\otimes\mathrm{id}_{M[1]}\bigr). \tag{14}\] To check it, take a basis tuple \([a_0,\ldots,a_p]\) in \(M[q]\). If \(\alpha\) is injective on this tuple, both sides of (14), on tensoring with \(e_0\), \(e_1\), or \(u\), are exactly the same tuple in (11), selected by the indices \(\alpha(a_0),\ldots,\alpha(a_p)\). If \(\alpha\) is not injective on it, the right side is zero by normalization. On the left, equal selected indices yield repeated \(f\)-images in the \(e_0\) and \(u\) prescriptions, and repeated \(g\)-images in the \(e_1\) prescription, so all three images are zero. These cases include every face, every degeneracy, and every composite of them, also when \(\sigma\) itself has repeated vertices. Naturality of the strong Steiner comparison and functoriality of \(\nu\) turn (14) into an equality of the strict functors (13) after mapping into \(U^{\infty}_{E}\).

The zero chains in this argument retain their lower boundary data after applying \(\nu\). In the cell table (3), the entries satisfy \(d x_r^\pm=x_{r-1}^+-x_{r-1}^-\). If the top entry of a \(q\)-cell is zero, with \(q>0\), its two \((q-1)\)-entries therefore coincide; the resulting cell is the identity on that common \((q-1)\)-cell. That lower cell need not be an identity. For example, take \(E=\{a<b\}\), \(A=\{a\}\), \(f\) constant at \(a\), and \(g=\mathrm{id}\). The degree-two tensor generator has image \(h(a,b)=[a,a,b]=0\), but its associated square maps to the identity \(2\)-cell on the nonidentity edge \([a,b]\). Equation (14) compares every degree of the chain maps, so it establishes agreement on these retained boundaries as well.

Assembly and stationarity. The oriental presentation and cocontinuity of the Gray tensor in its first variable now give \[U^{\infty}_{E}\otimes D_{1} \cong \mathop{\mathrm{colim}}_{\sigma:\Delta[m]\to NE} (\mathcal O_{m}\otimes D_{1}).\] The compatible maps (13) consequently induce \(P^\infty_{f,g}:U^{\infty}_{E}\otimes D_{1}\to U^{\infty}_{E}\). This colimit is taken in strict \(\omega\)-categories; the argument uses tensor cocontinuity, not preservation of arbitrary colimits by \(\nu\). The two endpoint maps are \(U^\infty(f)\) and \(U^\infty(g)\) by their simplexwise formulas.

On a simplex entirely in \(A\), the maps \(F\) and \(G\) coincide and \(h(w)=0\) for every selected tuple, because \(f(w_p)=g(w_p)\). More precisely, the entire chain map is \[C_\sigma=F\circ(\mathrm{id}_{M[m]}\otimes\epsilon), \qquad \epsilon(e_0)=\epsilon(e_1)=1,\quad\epsilon(u)=0,\] where the chain augmentation of the interval is denoted by the same letter as its categorical realization. Applying \(\nu\) and the simplex colimit proves strict stationarity on \(U^{\infty}_{A}\) relative to \(U^\infty(v)\). It is this factorization, rather than the vanishing of individual top chains alone, that proves stationarity.

Finite-dimensional reflection. For finite \(n\), let \(q_E:U^{\infty}_{E}\to U_{E}\) be the reflection unit. By Lemma 2, the composite \(q_E P^\infty_{f,g}\) factors uniquely as \[U^{\infty}_{E}\otimes D_{1} \xrightarrow{q_E\otimes\mathrm{id}}U_{E}\otimes D_{1} \xrightarrow{P_{f,g}}U_{E}.\] Its endpoint identities follow by precomposing with \(q_E\) and using the universal property of reflection. To check (9), precompose both sides with \(q_A\otimes\mathrm{id}\). The resulting equality is the stationarity just proved before reflection. The injectivity in the natural bijection of Lemma 2, with target \(U_{E}\), gives the claimed equality after reflection. For \(n=\infty\) no reflection is needed. If \(E\) or \(A\) is empty, the corresponding colimit is empty and the same statements hold with the unique maps from it. Finally Lemma 4 gives the stated simplicial homotopy and its stationarity. ◻

Proof. Let \(H(Y)\) be the right internal hom representing strict functors \((-)\otimes D_{1}\to Y\). It is an \(n\)-category by Lemma 2; for \(n=\infty\) this is automatic. The cylinders \[bP:B\otimes D_{1}\to Y, \qquad j\pi_X:X\otimes D_{1}\to Y\] transpose to maps \(B\to H(Y)\) and \(X\to H(Y)\) in \(n\mathrm{Cat}\). Their restrictions to \(B_0\) agree, since on its cylinder both give \[bi\pi_{B_0}=ja\pi_{B_0}.\] The pushout universal property in \(n\mathrm{Cat}\) thus gives a unique map \(Y\to H(Y)\) extending both transposes. Its transpose is \(Q\). The endpoint formulas, the equality \(Q(j\otimes\mathrm{id})=j\pi_X\), and the restriction to \(B\) follow from naturality of the adjunction and the same pushout universal property. For a map \(X\to X'\) commuting with the attaching maps, the induced map \(Y\to Y'\) carries this cylinder to the corresponding cylinder: the two maps \(Y\to H(Y')\) agree on \(X\) and \(B\) and hence agree. This proves naturality. Lemma 4 proves the final homotopy assertion. No injectivity, freeness, or cofibrancy hypothesis on \(i\), \(a\), or \(X\) entered this construction. ◻

Proof. We construct a right adjoint \(Q_0\) to \(F_0\). Given \(B\), put all its objects over \(0\) and adjoin a new object \(*\) over \(1\). Keep the hom categories \(B(b,b')\), and set \[Q_0(B)(b,*)=\mathbf{1},\qquad Q_0(B)(*,b)=\varnothing,\qquad Q_0(B)(*,*)=\mathbf{1}.\] Here the homs are strict \((n-1)\)-categories for finite \(n\), sets when \(n=1\), and strict \(\omega\)-categories when \(n=\infty\); \(\mathbf{1}\) and \(\varnothing\) denote their terminal and initial objects. Compositions inside \(B\) are retained. Every remaining composition either has empty domain or has terminal codomain, so is uniquely determined. Associativity, units, and interchange hold by the laws in \(B\) or by uniqueness. This defines \(Q_0(B)\) and its map to \(D_{1}\) in all the stated dimensions.

For \(p:C\to D_{1}\), a functor \(C_0\to B\) extends uniquely over \(D_{1}\) to \(C\to Q_0(B)\). All objects of \(C_1\) must go to \(*\); their internal cells and the cells from \(C_0\) to \(C_1\) have unique images in the specified terminal homs. There are no cells from \(C_1\) to \(C_0\), since their \(1\)-boundaries would map to an arrow \(1\to0\). Restriction and extension are inverse and natural, giving \[\mathop{\mathrm{Hom}}_{n\mathrm{Cat}/D_{1}}(C,Q_0(B)) \cong \mathop{\mathrm{Hom}}_{n\mathrm{Cat}}(C_0,B).\] Thus \(F_0\) is a left adjoint. For \(F_1\), place \(B\) over \(1\) and adjoin a new initial object over \(0\), with terminal homs from that object into \(B\), empty reverse homs, and terminal endomorphism hom. The same restriction argument gives a right adjoint \(Q_1\). In particular these constructions include \(B=\varnothing\) and prove preservation of the empty colimit as well.

For a set \(S\), let \(K(S)\) be the ordinary chaotic category on \(S\), with one arrow for each ordered pair of objects, regarded as an \(n\)-category with identity cells above dimension one. Every object function \(\operatorname{Ob}C\to S\) has a unique strict extension \(C\to K(S)\). Consequently \(\operatorname{Ob}\) is left adjoint to \(K\) and preserves colimits. ◻

Proof. Recall the oriental presentation from Section 2: its atoms are indexed by strictly increasing tuples, with \(0\)-source the first vertex and \(0\)-target the last vertex. All cells are finite composites and identities of these atoms. In particular every \(1\)-cell has nondecreasing endpoints.

Suppose a cell has \(0\)-source \(a\) and \(0\)-target \(b\). In any expression for it, every occurring atom has all its vertices in \([a,b]\). This follows by induction on the expression. For an atom it is the tuple description. In a \(0\)-composite the factors have endpoints \((a,c)\) and \((c,b)\), with \(a\leq c\leq b\). In a composition in positive dimension both factors have the same \(0\)-endpoints as the composite. Identities preserve these endpoints, and an identity on a vertex has support at that vertex. These observations prove the induction. If \(a,b\in I\), the whole interval \([a,b]\) lies in \(I\), so the expression uses only the atoms of that face. Thus the face generates every cell on those objects.

For finite \(n\geq1\), reflection leaves objects unchanged and every cell of dimension at most \(n\) is represented by a cell before reflection. The same argument applied to a representative proves surjectivity onto the full subcategory after reflection; higher cells are identities. For injectivity, the inclusion \(I\hookrightarrow[m]\) has a monotone retraction: clamp to the last vertex of \(I\) for an initial segment, or to its first vertex for a final segment. By cosimplicial functoriality it induces a retraction on orientals, and then on their reflections. The face map is therefore injective on every cell set. This proves the full-subcategory assertion. If \(I\) is empty there are no objects and hence no cells; no retraction is needed. ◻

Proof. Use the canonical simplex presentation \[ U_{P}=\mathop{\mathrm{colim}}_{\sigma:\Delta[m]\to NP}\tau_n\mathcal O_{m}. \tag{15}\] By Lemma 9, its object set is the colimit of the vertex sets \([m]\). Sending the \(j\)th vertex of a simplex \(\sigma\) to \(\sigma(j)\) identifies this colimit with \(P\): each element occurs as a zero-simplex, and the face inclusions from zero-simplices identify every vertex with that occurrence.

For a sieve \(S\), its characteristic map \(P\to[1]\) assigns \(0\) on \(S\) and \(1\) on its complement. It is monotone and induces the displayed label on \(U_{P}\), since \(U_{[1]}=D_{1}\). The presentation (15) is consequently a colimit in \(n\mathrm{Cat}/D_{1}\) as well: slice colimits have the underlying colimits in \(n\mathrm{Cat}\). In its \(\sigma\)-piece the vertices mapping to \(S\) form an initial segment \(I_\sigma\). By Lemma 10, the \(0\)-fiber of that piece is its truncated segment oriental. Taking fibers commutes with the colimit by Lemma 9.

We verify that this colimit of segment orientals is canonically \(U_{S}\). Restricting each nonempty simplex segment to \(S\) gives a map from its segment oriental to \(U_{S}\); the empty segment has the unique such map. These maps are compatible with every simplex operator. In the reverse direction, each simplex entirely in \(S\) supplies its whole oriental to the fiber colimit. The two composites are identities. On a simplex of \(S\) this is immediate. On a nonempty segment of an arbitrary simplex \(\sigma\), it follows from the face inclusion selecting \(I_\sigma\), which is a morphism in the simplex category of \(NP\); on an empty segment there is nothing to check. The isomorphism so obtained is the canonical map \(U_{S}\to U_{P}\), followed into its fiber. Fibers are full sieves, which proves the claim. The argument for a cosieve uses label \(1\) on \(S\), final vertex segments, and \(F_1\) throughout. All constructions commute with maps preserving the labels. ◻

Proof. The compatibility assumption makes the defining pushout a pushout in \(n\mathrm{Cat}/D_{1}\). Applying \(F_\epsilon\) and then Lemmas 9 and 11 gives (16). This identifies the canonical map from the displayed pushout to \(Y\) with the inclusion of a full fiber. The object formula follows from the object part of Lemma 9 and Lemma 11. Its last isomorphism uses only the injectivity of \(A\hookrightarrow P\): old vertices of \(P\) are replaced by their images in \(X\), while vertices in \(P\setminus A\) remain distinct new objects. There is no injectivity requirement on \(\operatorname{Ob}a\).

Two labels on \(Y\) give a functor \(Y\to D_{1}\times D_{1}\). The monotone functions \(\max,\min:[1]\times[1]\to[1]\) are strict functors, so their composites are labels on \(Y\). An object has maximum label \(0\) exactly when both labels are \(0\), and minimum label \(1\) exactly when both labels are \(1\). Full subcategories on the same objects are equal, which proves the assertions about intersections. The composite labels restrict to the corresponding maximum or minimum labels on both legs of the original pushout, so the same fiber formula applies. ◻

Proof. For (i), label membership in each \(P_i\) by \(0\) and label every object of \(X\) by \(0\). Apply Proposition 12 to the two labels and their maximum. For (ii), label membership in each \(T_i\) by \(1\); use constant label \(1\) on \(X\) for \(T_1\) and constant label \(0\) on \(X\) for \(T_2\). These agree on the attaching source by the stated conditions. The \(1\)-fiber of the second old-side label is empty, and so is that of the minimum of the two old-side labels. The same is true on their attaching sources. Formula (16) therefore gives \(U_{T_2}\) and \(U_{T_1\cap T_2}\) exactly. In both cases (17) proves the object cover. Finally, when \(A\) is a sieve, label \(A\) and \(X\) by \(0\) and \(P\setminus A\) by \(1\). The \(0\)-fiber formula is \(X\amalg_{U_{A}}U_{A}=X\). ◻

Proof. A full inclusion induces a monomorphism of Street nerves, and the intersection of the two nerves is \(N_nC_{12}\): a functor whose image belongs to both subcategories has a unique factorization through their full intersection. It remains to prove that every simplex \(z:\mathcal O_{m}\to C\) belongs to one of the two nerves. In the sieve case choose a piece containing \(z(m)\). For each vertex \(j\leq m\), the oriental has an arrow \(j\to m\), whose image shows that \(z(j)\) belongs to that piece. Fullness then puts every cell of the simplex in the same piece. In the cosieve case choose a piece containing \(z(0)\) and use the arrows \(0\to j\). This includes \(m=0\) and proves the union statement degree by degree. A union of simplicial subsets is their ordinary pushout over the intersection. Since its legs are monomorphisms, the homotopy-pushout assertion follows from Lemma 6. ◻

Proof. Let \(x\) be greatest in \(E\setminus A\) and put \(t=r(x)\in A\). If \(b\in A\), \(y\in E\setminus A\), and \(b\le y\), then \(b\le y\le x\), so (18) gives \(b\le t\). Define maps \(k,g:E\to E\) by \[k(e)=\begin{cases}e,&e\in A,\\t,&e\notin A,\end{cases} \qquad g(e)=\begin{cases}e,&e\in A,\\x,&e\notin A.\end{cases}\] Both are monotone. Indeed, for a comparable pair \(e\le e'\), the old–old case is unchanged and the new–new case is constant. In the old–new case the preceding observation gives \(e\le t\le x\). The new–old case cannot occur because \(A\) is a sieve. We therefore have pointwise inequalities \[ \mathrm{id}_E\le g,\qquad k\le g, \tag{20}\] and all three maps restrict to \(\mathrm{id}_A\). The common upper map \(g\) is constant on \(E\setminus A\), exactly as required in Lemma 7. Figure 1 shows why this common upper map is useful: \(\mathrm{id}_E\) and \(k\) need not be pointwise comparable, although both lie below \(g\).

Apply Lemma 8 to the two comparisons in (20). Let \(G,K:Y\to Y\) be the pushout endomorphisms whose restrictions to \(X\) are \(j\) and whose restrictions to \(U_{E}\) are \(v\,c_nN(g)\) and \(v\,c_nN(k)\), respectively. The resulting right Gray cylinders, both stationary on \(X\), have endpoint pattern \[\mathrm{id}_Y\ \longrightarrow\ G\ \longleftarrow\ K.\] By Lemma 4, these give simplicial homotopies with the same endpoint pattern on Street nerves.

The image of \(k\) lies in \(A\), so \(k=i\bar k\) for a monotone retraction \(\bar k:E\to A\). The maps \(\mathrm{id}_X\) and \(a\,c_nN(\bar k):U_{E}\to X\) agree on \(U_{A}\) and hence define \(p:Y\to X\). They give \(pj=\mathrm{id}_X\). Comparing restrictions to the two pushout summands gives \(jp=K\): on \(X\) both restrict to \(j\), and on \(U_{E}\) one has \[jp\,v=j\,a\,c_nN(\bar k) =v\,c_nN(i)\,c_nN(\bar k) =v\,c_nN(k).\] After realization, concatenate the homotopy from \(\mathrm{id}_Y\) to \(G\) with the reverse of the homotopy from \(jp\) to \(G\). Thus \(|N_np|\) is a homotopy inverse to \(|N_nj|\), proving the assertion. This reversal is a reversal of an ordinary topological homotopy; no reversed Gray cylinder is asserted. ◻

Proof. First suppose that \(E\) is finite. We use induction on \(|E\setminus A|\), simultaneously for all attaching targets \(X\) and all attaching functors \(a\). When this cardinality is zero, \(j\) is an isomorphism. When the complement has a greatest element, Lemma 15 applies.

Otherwise the nonempty finite poset \(E\setminus A\) has at least two distinct maximal elements \(x_1,x_2\): a finite poset with a unique maximal element has that element as its greatest element. Each \(x_i\) is also maximal in \(E\), since an element above a new element cannot belong to the sieve \(A\). Put \[E_1=E\setminus\{x_1\},\qquad E_2=E\setminus\{x_2\},\qquad E_{12}=E\setminus\{x_1,x_2\}.\] These are sieves of \(E\) containing \(A\); moreover \(E_1\cup E_2=E\) and \(E_1\cap E_2=E_{12}\). The restriction of \(r\) to each \(E_T\), for \(T=1,2,12\), still takes values in \(A\) and satisfies (18). Each complement \(E_T\setminus A\) is smaller. Define \[Y_T=X\amalg_{U_{A}}U_{E_T},\qquad j_T:X\longrightarrow Y_T.\] By induction every \(N_nj_T\) is a weak homotopy equivalence.

Corollary 13(i), applied to the sieves \(E_1,E_2\) containing \(A\) and covering \(E\), identifies the canonical maps \(Y_i\to Y\) as full sieve inclusions. Their full intersection is canonically \(Y_{12}\), and they cover the objects of \(Y\). These identifications hold for the given arbitrary attaching functor \(a\).

Lemma 14 now identifies the bottom pushout in the following diagram with \(N_nY\): \[ \begin{tikzcd}[column sep=large] N_nX \arrow[d,"N_nj_1"'] & N_nX \arrow[l,equal] \arrow[r,equal] \arrow[d,"N_nj_{12}"] & N_nX \arrow[d,"N_nj_2"] \\ N_nY_1 & N_nY_{12} \arrow[l,hook] \arrow[r,hook] & N_nY_2 . \end{tikzcd} \tag{21}\] The vertical maps are the canonical maps induced by \(X\), so this is a strictly commuting map of spans. All three are weak equivalences by induction, and the indicated legs in both rows are monomorphisms. Lemma 6 proves that the induced map on pushouts is a weak equivalence. The top pushout is \(N_nX\), and its map to the bottom pushout is precisely \(N_nj\), by its restrictions to the two pieces. This completes the finite induction. In particular, no compatibility between the local cylinders on different pieces has been used.

Now let \(E\) be arbitrary, and write \(q=ir:E\to E\). Since \(q^2=q\), the collection \(\mathcal F\) of finite subsets \(F\subset E\) with \(q(F)\subset F\) is nonempty and filtered under inclusion: it contains \(\varnothing\), and finite unions remain stable. Every finite subset \(T\) is contained in the finite stable subset \(T\cup q(T)\). Give each \(F\) its full induced order and put \(A_F=A\cap F\). Then \(A_F\) is a sieve in \(F\), and stability gives \(r(F)\subset A_F\). The restriction \(r_F:F\to A_F\) is a right adjoint retraction, since (18) restricts to these full subposets. Notice that \(F\) itself need not be a sieve in \(E\).

Use the restriction of \(a\) to define \[Y_F=X\amalg_{U_{A_F}}U_{F},\qquad j_F:X\longrightarrow Y_F.\] Every \(j_F\) is a Thomason equivalence by the finite case. The inclusions \(F\subset F'\) give compatible transition functors \(Y_F\to Y_{F'}\) under the same fixed \(X\). Every ordinary nerve simplex has finite vertex support, and therefore \[NE=\mathop{\mathrm{colim}}_{F\in\mathcal F}NF, \qquad NA=\mathop{\mathrm{colim}}_{F\in\mathcal F}NA_F.\] Applying the colimit-preserving left adjoint \(c_n\) and then commuting colimits gives \[ \mathop{\mathrm{colim}}_{F\in\mathcal F}Y_F \ \cong\ X\amalg_{U_{A}}U_{E}=Y. \tag{22}\] For clarity, the fixed copy of \(X\) in this formula is essential and legitimate: a compatible family of maps \(Y_F\to Z\) consists of one map \(X\to Z\), since \(\mathcal F\) is nonempty and connected, together with compatible maps \(U_{F}\to Z\) agreeing on each \(U_{A_F}\). These are exactly maps from the pushout on the right. Under this identification the colimit of the arrows \(j_F\) is \(j\).

By Lemma 5, \(N_n\) preserves these filtered colimits and simplicial weak homotopy equivalences are closed under filtered colimits. Hence \(N_nj\) is a weak homotopy equivalence. No injectivity of the transition functors \(Y_F\to Y_{F'}\) is required. The same proof applies directly when \(n=\infty\). ◻

Proof. First take \(S\) finite and induct on its cardinality. If \(S\) is empty, both simplicial sets are empty and the unit is an isomorphism. If \(S\) has a greatest element \(x\), the pointwise inequality \(\mathrm{id}_S\le\mathrm{const}_x\) contracts \(NS\). Lemma 7, with old sieve \(\varnothing\), gives a right cylinder from \(\mathrm{id}_{U_{S}}\) to \(c_nN(\mathrm{const}_x)\). The latter functor factors through \(U_{\{x\}}=\mathcal O_{0}\), the terminal strict \(n\)-category. Lemma 4 therefore contracts \(N_nU_{S}\). Both realizations are nonempty and contractible, so \(\eta_S\) is a weak homotopy equivalence.

In the remaining case choose two distinct maximal elements \(x_1,x_2\) of \(S\) and put \(S_i=S\setminus\{x_i\}\) and \(S_{12}=S\setminus\{x_1,x_2\}\). These are sieves, the two single deletions cover \(S\), and all three posets are smaller. Their ordinary nerves cover \(NS\): a simplex belongs to a sieve containing its last vertex. By Lemmas 11 and 14, their \(U_{\,\cdot\,}\) images give a full sieve cover of \(U_{S}\) and the analogous cover of its Street nerve. The joint zero label identifies the intersection of the two full sieves with \(U_{S_{12}}\). Naturality of the unit gives a commuting map of spans \[\begin{tikzcd}[column sep=large] NS_1 \arrow[d,"\eta_{S_1}"'] & NS_{12} \arrow[l,hook] \arrow[r,hook] \arrow[d,"\eta_{S_{12}}"] & NS_2 \arrow[d,"\eta_{S_2}"] \\ N_nU_{S_1} & N_nU_{S_{12}} \arrow[l,hook] \arrow[r,hook] & N_nU_{S_2} . \end{tikzcd}\] The vertical maps are weak equivalences by induction; all horizontal legs are monomorphisms. The induced map on pushouts is a weak equivalence by Lemma 6. Under the cover identifications it is exactly \(\eta_S\): its restriction to either \(NS_i\) agrees with that unit by naturality. This proves the finite case.

Finally every poset is the filtered union of its finite full subposets. Ordinary nerve preserves this union because each simplex has finite support, \(c_n\) preserves its colimit, and \(N_n\) preserves the resulting filtered colimit by Lemma 5. Naturality identifies \(\eta_S\) with the filtered colimit of the units on those finite subposets. Filtered-colimit closure of simplicial weak homotopy equivalences then proves the assertion. This proof uses only the prism, cover, and simplicial gluing lemmas, and in particular precedes and is independent of the model-structure transfer. ◻

Proof. First set \[V_Z(X)=X\amalg_{U_{A}}U_{Z},\qquad P_Z(X)=N_nX\amalg_{N_nU_{A}}N_nU_{Z}.\] Theorem 16, applied to \(A\hookrightarrow Z\), says that \(X\to V_Z(X)\) belongs to \(W_n\). Applying it with old target \(U_{A}\) also says that \(U_{A}\to U_{Z}\) belongs to \(W_n\). Since \(A\) is a sieve in \(Z\), Lemma 11 identifies \(U_{A}\to U_{Z}\) with an actual full inclusion. Its Street nerve is therefore a monomorphism and a weak equivalence, hence a trivial cofibration of simplicial sets. Its pushout \(N_nX\to P_Z(X)\) is a weak equivalence. The canonical triangle \[\begin{tikzcd}[column sep=large] N_nX \arrow[r] \arrow[dr] & P_Z(X) \arrow[d,"\gamma_{Z,X}"]\\ & N_nV_Z(X) \end{tikzcd}\] then shows by two-out-of-three that \(\gamma_{Z,X}\) is a weak equivalence.

To pass from \(Z\) to \(B\), write \[C=B\setminus A,\qquad T=Z\setminus A=Z\cap C.\] Both \(Z\) and \(C\) are cosieves in \(B\), and they cover \(B\): an element outside \(C\) belongs to \(A\subset Z\). Their intersection is \(T\). Lemma 11 therefore gives cosieves \(U_{Z},U_{C}\subset U_{B}\) with intersection \(U_{T}\). Apply Corollary 13(ii) with \(P=B\), \(T_1=Z\), and \(T_2=C\). Its remaining hypotheses hold because \(A\subset Z\) and \(A\cap C=\varnothing\). The canonical maps identify \[ V_Z(X),\quad U_{C}\ \subset\ V_B(X), \qquad V_Z(X)\cap U_{C}=U_{T}, \tag{24}\] as actual full cosieves and their full intersection. These two cosieves cover all objects of \(V_B(X)\). The corollary applies to the given arbitrary functor \(a\), including its identifications of old objects and cells.

Apply Lemma 14 to the two cosieve covers just obtained. It gives ordinary simplicial pushouts \[\begin{align*} N_nU_{B} &=N_nU_{Z}\amalg_{N_nU_{T}}N_nU_{C}, \tag{25}\\ N_nV_B(X) &=N_nV_Z(X)\amalg_{N_nU_{T}}N_nU_{C}. \tag{26}\end{align*}\] All maps from the intersections to the pieces in these covers are monomorphisms. Reassociating the pushout in the definition of \(P_B(X)\) using Equation (25) gives \[P_B(X)=P_Z(X)\amalg_{N_nU_{T}}N_nU_{C}.\] Under these canonical identifications, Equation (23) is exactly the map induced by \(\gamma_{Z,X}\) and the identity of \(N_nU_{C}\). In particular the square \[\begin{tikzcd}[column sep=large] P_Z(X) \arrow[r,hook] \arrow[d,"\gamma_{Z,X}"'] & P_B(X) \arrow[d,"\gamma_{B,X}"]\\ N_nV_Z(X) \arrow[r,hook] & N_nV_B(X) \end{tikzcd}\] is a pushout: its lower right corner is also \(N_nV_Z(X)\amalg_{P_Z(X)}P_B(X)\). The top horizontal map is the cobase change of the specific monomorphism \[ N_nU_{T}\hookrightarrow N_nU_{C}. \tag{27}\] Left properness of simplicial sets, as recorded in Lemma 6, now makes \(\gamma_{B,X}\) a weak equivalence. Neither the map \(N_nU_{A}\to N_nX\) nor the map \(N_nU_{T}\to P_Z(X)\) needs to be a monomorphism.

The remaining assertion that \(N_nU_{A}\to N_nU_{B}\) is monic follows directly from the sieve identification in Lemma 11. Finally, \(\gamma_{B,X}\) is defined by the two canonical maps from \(N_nX\) and \(N_nU_{B}\). For \(f:X\to X'\) under \(U_{A}\) these maps commute with \(N_nf\) and with the induced pushout functor \(V_B(X)\to V_B(X')\). The pushout universal property therefore gives the strict naturality square \[ \begin{tikzcd}[column sep=large] P_B(X) \arrow[r,"\gamma_{B,X}"] \arrow[d] & N_nV_B(X) \arrow[d]\\ P_B(X') \arrow[r,"\gamma_{B,X'}"'] & N_nV_B(X'). \end{tikzcd} \tag{28}\] The comparison is thus canonical and natural, while the local contractions used in Theorem 16 establish that it is a weak equivalence. ◻

Proof. Put \(M=\{0,\ldots,m\}\) and let \(P\) be the poset of its nonempty subsets, ordered by inclusion. The subposet \(P_K\) of faces belonging to \(K\) is \[P_K= \begin{cases} \{F:\varnothing\ne F\subsetneq M\},&K=\partial\Delta[m],\\[2pt] \{F:\varnothing\ne F\subseteq M,\ M\setminus\{k\}\nsubseteq F\}, &K=\Lambda^k[m]. \end{cases}\] Indeed, a nonempty face belongs to the horn precisely when it is contained in \(M\setminus\{j\}\) for some \(j\ne k\). Thus in the horn case \(P_K\) is obtained from \(P\) by removing \(M\) and \(M\setminus\{k\}\). In both cases \(P_K\) is downward closed in \(P\).

The barycentric subdivision of an ordered simplicial complex has a vertex for every nonempty face and a nondegenerate simplex for every strict chain of such faces. Consequently \[\mathrm{Sd}\Delta[m]=NP,\qquad \mathrm{Sd}K=NP_K.\] One can see these identifications directly on a simplex: the vertices of its subdivision are its nonempty subsets, and its simplices are chains of those subsets. The identifications agree on every face, so they glue over the faces of \(K\). The nondegenerate simplices of \(NP\) are themselves the nonempty finite chains in \(P\), with a face obtained by deleting entries. Applying the same description a second time gives \[\begin{split} B&=\{\sigma:\sigma\text{ is a nonempty chain in }P\},\\ A&=\{\sigma\in B:\sigma\subseteq P_K\}, \end{split} \qquad \sigma\leq\tau\ \Longleftrightarrow\ \sigma\subseteq\tau.\] This proves the claimed identifications as simplicial sets, including degeneracies, and identifies their inclusion.

Every nonempty subchain of a chain in \(A\) again belongs to \(A\), so \(A\) is a sieve in \(B\). Its upward closure is exactly \[Z=\{\sigma\in B:\sigma\cap P_K\ne\varnothing\}.\] For the nontrivial implication in this equality, a face \(F\in\sigma\cap P_K\) gives \(\{F\}\in A\) below \(\sigma\). Define \[r:Z\longrightarrow A,\qquad r(\sigma)=\sigma\cap P_K.\] This is a nonempty chain, is monotone under inclusion, and fixes \(A\). For \(\alpha\in A\) and \(\sigma\in Z\) one has \[\alpha\subseteq\sigma \quad\Longleftrightarrow\quad \alpha\subseteq\sigma\cap P_K=r(\sigma).\] This is precisely the right adjunction to \(A\hookrightarrow Z\).

When \(m=0\) and \(K=\partial\Delta[0]\), the poset \(B\) is a singleton and \(A=Z=\varnothing\); the asserted adjunction is the identity of the empty poset. For a one-dimensional horn, \(P_K=\{\{k\}\}\), so \(A\) is a singleton and \(Z\) consists of the chains containing the face \(\{k\}\). These cases are included in the formulas above. There are no zero-dimensional horn generators. ◻

Proof. Use the pair \(A\subset B\) from Lemma 19, so that \(L_n d\) is \(U_{A}\to U_{B}\). The inclusion \(NA\to NB\) is a weak equivalence: the horn inclusion is a simplicial weak equivalence, and subdivision preserves weak equivalences. The latter follows by naturality and two-out-of-three from the objectwise weak equivalence \(\mathrm{Sd}\to\mathrm{id}\) of (Ara and Maltsiniotis 2014, Theorem 4.5). Naturality of the adjunction unit gives a square \[\begin{tikzcd}[column sep=large] NA \arrow[r] \arrow[d] & NB \arrow[d]\\ N_nU_{A} \arrow[r] & N_nU_{B}. \end{tikzcd}\] The vertical maps are weak equivalences by Proposition 17. Two-out-of-three makes the lower map a weak equivalence, and it is monic by Proposition 18. Thus it is a simplicial trivial cofibration. For any attaching functor \(U_{A}\to X\), its pushout \(N_nX\to P_B(X)\) is a weak equivalence. The composite \[N_nX\longrightarrow P_B(X) \xrightarrow{\gamma_{B,X}}N_nV_B(X)\] is the nerve of the attachment \(X\to V_B(X)\), and both factors are weak equivalences. This is exactly the assertion that the attachment belongs to \(W_n\). ◻

Proof. The map \(N_nX\to P_B(X)\) is a monomorphism, being the cobase change of \(N_nU_{A}\hookrightarrow N_nU_{B}\). There is a canonical pushout identification \[P_B(X')=N_nX'\amalg_{N_nX}P_B(X).\] Left properness of simplicial sets therefore makes \(P_B(X)\to P_B(X')\) a weak equivalence, since \(N_nf\) is one. In the naturality square of Equation (28), both horizontal maps are weak equivalences by Proposition 18. Two-out-of-three implies that its right vertical map is a weak equivalence. This is the nerve of the asserted functor. ◻

Proof. Corollary 21 proves the assertion for one attachment, with an arbitrary attaching target. A set of simultaneous attachments may be well ordered and attached one at a time: at each stage use the given map from its source into the original target, followed by the map to the current stage. The universal property of the resulting colimit is the universal property of the pushout of the coproduct. Thus it suffices to consider a continuous ordinal sequence \((X_\alpha)_{\alpha\leq\lambda}\) in which each successor map is a single such attachment.

By transfinite induction, \(X_0\to X_\alpha\) belongs to \(W_n\) for every \(\alpha\). The assertion at zero is the identity, and the successor step is composition with the weak equivalence supplied by Corollary 21. If \(\beta\) is a nonzero limit ordinal, the ordinal of stages \(\alpha<\beta\) is filtered. Apply Lemma 5 to the diagram of arrows \(X_0\to X_\alpha\). Its colimit is \(X_0\to\mathop{\mathrm{colim}}_{\alpha<\beta}X_\alpha=X_\beta\), because the colimit of the constant source diagram is \(X_0\). This proves the limit step and the lemma. ◻

Proof. The underlying category is locally presentable, and hence complete and cocomplete, by Lemma 5. We describe the small object argument to make clear that the factorizations have no restriction on their sources. For either set \(\mathcal S=L_nI\) or \(\mathcal S=L_nJ\), choose an infinite regular cardinal \(\kappa\) such that every domain in \(\mathcal S\) is \(\kappa\)-presentable. Such a common bound exists because the category is locally presentable and \(\mathcal S\) is a set.

Given any map \(X\to Y\), start with \(X_0=X\). At stage \(\alpha+1\) attach to \(X_\alpha\) one copy of the codomain of each member of \(\mathcal S\) for every commutative square from that member to \(X_\alpha\to Y\). This is a set of squares, by local smallness. Use colimits at limit stages. The resulting factorization is \[X\xrightarrow{i}X_\kappa\xrightarrow{p}Y.\] Here \(i\) is a relative \(\mathcal S\)-cell map and \(p\in\mathcal S\text{-inj}\). Indeed a map from a generating domain to \(X_\kappa\) factors through some \(X_\alpha\), since the ordinal \(\kappa\) is \(\kappa\)-filtered. Its lifting square against \(p\) was therefore attached at stage \(\alpha+1\), which supplies the required lift. The construction may be made functorial by using all squares at each stage.

Pushouts, transfinite composites, and retracts of maps with a fixed left lifting property retain that property. Consequently every relative \(\mathcal S\)-cell map lies in \(\mathcal S\text{-cof}\). Conversely, if \(u:A\to B\) belongs to \(\mathcal S\text{-cof}\), factor it as \(A\xrightarrow{i}Z\xrightarrow{p}B\) by the construction above. The square with top map \(i\), left map \(u\), right map \(p\), and bottom map \(\mathrm{id}_B\) admits a diagonal \(s:B\to Z\). The equations \(su=i\) and \(ps=\mathrm{id}_B\) exhibit \(u\) as a retract of \(i\) in the arrow category, with the source fixed at \(A\). Thus the construction gives the weak factorization system \[(\mathcal S\text{-cof},\mathcal S\text{-inj}),\] and its left class consists exactly of retracts of relative \(\mathcal S\)-cell maps.

The adjunction identifies lifting squares and their fillers. The Kan–Quillen lifting characterizations therefore give exactly \[\begin{align*} (L_nI)\text{-inj} &=R_n^{-1}(I\text{-inj})=F_n\cap W_n, \tag{30}\\ (L_nJ)\text{-inj} &=R_n^{-1}(J\text{-inj})=F_n. \tag{31}\end{align*}\] In the first equality of classes on the right of (30), we used the equality of the two definitions of \(W_n\) proved above. The first weak factorization system is consequently \((C_n,F_n\cap W_n)\).

Let \(\mathcal D=(L_nJ)\text{-cof}\). As \(F_n\cap W_n\subseteq F_n\), one has \(\mathcal D\subseteq C_n\). Lemma 23 and retract closure give \(\mathcal D\subseteq W_n\) as well. To prove the reverse inclusion, take \(u:A\to B\) in \(C_n\cap W_n\) and factor it by the second small object argument as \(A\xrightarrow{i}Z\xrightarrow{p}B\), with \(i\in\mathcal D\) and \(p\in F_n\). Since \(i\) and \(u=pi\) belong to \(W_n\), two-out-of-three gives \(p\in F_n\cap W_n\). The lifting property of \(u\in C_n\) now supplies \(s:B\to Z\) with \(su=i\) and \(ps=\mathrm{id}_B\). The same fixed-source retract diagram puts \(u\) in \(\mathcal D\). We have proved \[\mathcal D=C_n\cap W_n.\] Thus the second weak factorization system is \((C_n\cap W_n,F_n)\). Together with two-out-of-three and retract closure for \(W_n\), these are all the model axioms. Local presentability and the displayed generating sets make the model structure combinatorial. ◻

Proof. For right properness, let \(p:Y\to Z\) belong to \(F_n\) and let \(w:Z'\to Z\) belong to \(W_n\). The right adjoint \(R_n\) carries the pullback \(Y\times_Z Z'\) to the corresponding simplicial pullback. Its map to \(R_nY\) is a weak equivalence by right properness of the Kan–Quillen model structure, since \(R_np\) is a Kan fibration and \(R_nw\) is a weak equivalence. The definition of \(W_n\) then gives \(Y\times_Z Z'\to Y\) in \(W_n\).

For left properness, first let \(i:X\to Y\) be a relative \(L_nI\)-cell map and let \(w:X\to X'\) belong to \(W_n\). As in Lemma 23, write \(i\) as a continuous sequence of individual boundary attachments \(X_\alpha\) starting at \(X\). Put \(X'_\alpha=X'\amalg_X X_\alpha\). We prove by induction that \(X_\alpha\to X'_\alpha\) belongs to \(W_n\). This is \(w\) at stage zero. At a successor stage, the second attachment has the first attaching map followed by \(X_\alpha\to X'_\alpha\), so Corollary 22 proves the induction step. At a nonzero limit ordinal, both sequences are their colimits, and Lemma 5 gives the assertion. Hence \(Y\to X'\amalg_XY\) belongs to \(W_n\).

For a general \(i:X\to Y\) in \(C_n\), the first small object factorization and its lifting retract, used in Proposition 24, express \(i\) as a retract of a relative \(L_nI\)-cell map \(j:X\to Z\) with the same source. Explicitly there are \(s:Y\to Z\) and \(p:Z\to Y\) with \(si=j\), \(pj=i\), and \(ps=\mathrm{id}_Y\). Cobase change along \(w\) gives the corresponding maps \(s'\) and \(p'\) between \(X'\amalg_XY\) and \(X'\amalg_XZ\), with \(p's'=\mathrm{id}\). Thus the arrow \(Y\to X'\amalg_XY\) is a retract of \(Z\to X'\amalg_XZ\). The latter is in \(W_n\) by the preceding paragraph, and so is the former. Both properness arguments apply to arbitrary objects; no cofibrancy or fibrancy condition on the base objects was used. ◻

Proof. For \(K=\Delta[m]\) and \(K=\partial\Delta[m]\), Lemma 19 identifies \(\mathrm{Sd}^2K\) with the ordinary nerve of a poset. The assertion is therefore Proposition 17. It also holds for the empty simplicial set: \(c_n\varnothing\) is the empty strict category, whose Street nerve is empty.

Every simplicial set is obtained from the empty set by a continuous transfinite sequence of boundary attachments. One construction orders its nondegenerate simplices by dimension and well orders those in each dimension. The boundary of a simplex then maps into the stage already constructed. That attaching map may identify vertices or faces, or send faces to degenerate simplices. The construction still recovers \(K\), by the unique expression of each simplex as a degeneracy of a nondegenerate simplex.

Consider one successor step \[K_+=K_0\amalg_{\partial\Delta[m]}\Delta[m]\] and assume the assertion for \(K_0\). Write \[V=\mathrm{Sd}^2K_0,\qquad A=\mathrm{Sd}^2\partial\Delta[m], \qquad B=\mathrm{Sd}^2\Delta[m].\] Here \(A,B,V\) are simplicial sets; the representing posets for \(A,B\) will be denoted by \(E,F\) below. Since subdivision is a left adjoint, \(\mathrm{Sd}^2K_+=V\amalg_A B\). Naturality of \(\eta\) gives a map of spans and hence the map \[ q:V\amalg_A B\longrightarrow N_nc_nV\amalg_{N_nc_nA}N_nc_nB. \tag{33}\] The three maps on the objects of these spans are weak equivalences: on \(V\) by induction and on \(A,B\) by the simplex and boundary cases. The leg \(A\to B\) is a simplicial monomorphism. Writing \(A=NE\) and \(B=NF\) as in Lemma 19, the inclusion \(E\subseteq F\) is a sieve; Lemma 11 makes \(c_nA\to c_nB\) a sieve inclusion. Consequently \(N_nc_nA\to N_nc_nB\) is also a monomorphism. Lemma 6, applied with these two specified monomorphic legs, proves that \(q\) is a weak equivalence. It requires no monomorphism condition on \(A\to V\).

Apply Proposition 18 to the arbitrary attaching functor \(c_nA\to c_nV\). It says that \[ t:N_nc_nV\amalg_{N_nc_nA}N_nc_nB \longrightarrow N_n(c_nV\amalg_{c_nA}c_nB) \tag{34}\] is a weak equivalence. Because \(c_n\) preserves colimits, the target is \(N_nc_n(V\amalg_A B)\). The composite \(tq\) is exactly \(\eta_{V\amalg_A B}\): on each of the two pushout legs \(V\) and \(B\) this follows from naturality of \(\eta\), and equality on those legs determines the map from the pushout. Thus the actual canonical map \(\theta_{K_+}\) is a weak equivalence. This also explains why noninjective boundary maps cause no change to the argument.

At a nonzero limit ordinal, \(\mathrm{Sd}^2\) and \(L_n\) preserve the colimit as left adjoints, while \(N_n\) preserves the filtered colimit by Lemma 5. Therefore the unit at that stage is the filtered colimit of the previous units; it is a weak equivalence by the same lemma. Transfinite induction proves (32) for every \(K\). The poset descriptions used in this proof concern the simplex and boundary pieces only. ◻

Proof. Equations (30) and (31) show that \(R_n\) preserves fibrations and trivial fibrations, so this is a Quillen adjunction.

Set \(S=\mathrm{Sd}\) and \(T=\mathrm{Ex}\), and let \(a_K:K\to TSK\) be the unit of \(S\dashv T\). The mate identity is \[j_K=T(\ell_K)a_K:K\longrightarrow TK.\] The independently cited theorem above makes \(j_K\) a weak equivalence. Since \(\ell_K\) is a weak equivalence and \(T\) preserves weak equivalences, \(T(\ell_K):TSK\to TK\) is also one. Hence \(a_K\) is a weak equivalence for every simplicial set \(K\), by two-out-of-three. In particular, \(a_{SK}:SK\to TS^2K\) is a weak equivalence. The unit of \(S^2\dashv T^2\) is \[b_K=T(a_{SK})a_K:K\longrightarrow T^2S^2K, \qquad T(a_{SK}):TSK\longrightarrow T^2S^2K.\] Thus \(b_K\) is a weak equivalence because \(T\) preserves weak equivalences. Composition is written with the rightmost map applied first.

The unit of the composite adjunction \(L_n\dashv R_n\) is \[u_K=T^2(\theta_K)b_K: K\longrightarrow T^2N_nc_nS^2K=R_nL_nK.\] Proposition 26, preservation of weak equivalences by \(T\), and the preceding calculation show that \(u_K\) is a weak equivalence for every \(K\). For any strict functor \(f:L_nK\to Y\), its simplicial adjunct is \[f^\sharp=R_n(f)u_K:K\longrightarrow R_nY.\] Using the definition of \(W_n\) and two-out-of-three now gives \[f\in W_n \quad\Longleftrightarrow\quad R_n(f)\in W_{\mathrm{KQ}} \quad\Longleftrightarrow\quad f^\sharp\in W_{\mathrm{KQ}}.\] This equivalence holds for every \(K\) and \(Y\). In particular it holds for cofibrant \(K\) and fibrant \(Y\), which is the Quillen-equivalence criterion. Together with Propositions 24 and 25, this proves all the asserted model-categorical conclusions for each finite \(n\geq1\) and for \(n=\infty\). ◻