Thomason Model Structures in Every Strict Higher Dimension

Proof. We prove the special case needed here directly; the general internal-hom truncation assertion is also [6, Proposition A.29]. An i-cell of H(Y) is a functor P:\Disk{i}\otimes\Disk{1}\to Y. Let s:\Disk{i-1}\to\Disk{i} be the source face inclusion, and let p:\Disk{i}\to\Disk{i-1} send the top i-cell to the identity of the top (i-1)-cell. Thus ps=\id and q=sp collapses \Disk{i} onto its source face. Under (2.5), the identity of the source of P is represented by P(q\otimes\id). We show that these functors agree when i>n.

The atoms of \Disk{i}\otimes\Disk{1} outside the source face \Disk{i-1}\otimes\Disk{1} come from exactly the following pairs: the top i-generator or the target (i-1)-generator of \Disk{i}, paired with e_0,e_1, or u. Their dimensions are respectively i,i,i+1 and i-1,i-1,i. If i>n+1, all have dimension greater than n. If i=n+1, only the target n-generator paired with e_0 or e_1 has dimension at most n. At either endpoint, the top (n+1)-cell maps under P to an identity, so its source and target n-cells have the same image. These are precisely the two equalities needed to compare P with P(q\otimes\id) on the remaining degree-n atoms. All atoms in the source face are fixed by q\otimes\id.

The two functors therefore agree on all atoms of dimension at most n. They agree on higher atoms as well: in an n-category the image of a cell of dimension greater than n must be the iterated identity on its n-source, whose image has already been determined. Generation by atoms proves P=P(q\otimes\id). Every cell of H(Y) above n is thus an identity.

Now transpose a cylinder to a functor X\to H(Y). Since H(Y) is an n-category, it factors uniquely through \trunc X. Transposing back proves (2.6). Naturality of the reflection and of the adjunction proves the restriction assertion. The endpoint assertions follow either by naturality of evaluation or by uniqueness of the factorization into Y.□

Proof. For each m define the classical Alexander–Whitney diagonal [10, IV.2, Remark 2.5]

It is a positive chain map M[m]\to M[m]\otimes M[m]. For clarity, in the differential of the right side, deleting a vertex strictly before or after the shared vertex gives the corresponding term in \operatorname{AW}_m d, with coefficient (-1)^j for deletion of i_j. The remaining terms cancel in pairs: deleting the final vertex of the first factor at cut a contributes

whereas deleting the initial vertex of the second factor at cut a-1 contributes the same tensor with coefficient (-1)^{a-1}. These cancellations use exactly the sign in (2.3). The degree-zero assertion is immediate. Every summand in (2.7) is positive. On vertices the augmentation is one; applying the augmentation to the first or second tensor factor selects, respectively, only the cut a=0 or a=p. Thus both counits give the identity.

The diagonal is natural for every monotone ordinal map. For a tuple whose image is strictly increasing, this follows term by term. If the image has equal consecutive entries in positions j,j+1, then at every cut a\le j the second factor is degenerate, and at every cut a\ge j+1 the first factor is degenerate. Both sides of naturality are therefore zero after normalization. This verifies degeneracies as well as faces. Only this cosimplicial naturality is asserted.

Apply \nu and (2.4) to obtain natural functors \delta_m:\Orient{m}\to\Orient{m}\otimes\Orient{m}. For an m-simplex F:\Orient{m}\to X and a monotone map \theta:[m]\to[1], put

where \Orient{\theta}:\Orient{m}\to\Orient{1}=\Disk{1} is the cosimplicial structure map. The naturality just checked makes J_X simplicial. If \theta is constant at a, this map factors through the right counit and the endpoint e_a, giving the stated endpoint formula. Applying \pi_X also gives the right counit, so N_\infty(\pi_X)J_X is the projection. Postcomposing with N_\infty P proves the homotopy assertion, and naturality in X together with the projection formula proves stationarity.□

Proof. Strict n-categories are models of a small finitary essentially algebraic theory. The sorts are the cell dimensions; the operations are source, target, identity, and the binary compositions. A composition is defined by finitely many matching-boundary equations, and every strict category axiom is an equation involving finitely many cells. In the \omega-case there are countably many sorts, but all operations and equations still involve finite data. Proposition A.3 proves local finite presentability, completeness, cocompleteness and a common regular presentability bound directly from this signature. Its presentation construction includes composites made possible by new boundary identifications. The general essentially algebraic approach is developed in [1, 2]; for its finitary partial-operation formulation, see also [12, pp. 52–53].

For the cellwise colimit assertion one can argue directly. Form the filtered colimit of the sets of cells in each dimension. Any finite collection of representatives is represented at a common stage. If their boundaries agree in the colimit, those finitely many equalities hold at some later common stage, where the required composition is defined. Define its colimit value by that composition. A further common stage proves independence of choices, and each strict category law follows at a stage containing its finitely many entries. This constructs the colimit with its universal property. If all inputs are n-categories, every cell above n is already an identity, so the same is true of the colimit.

The oriental \Orient{m} has a finite polygraphic presentation: there is one generator for each nonempty strictly increasing tuple in [m], and its boundary is an expression in lower generators. A map from \Orient{m} into a filtered colimit is thus specified by finitely many cell representatives satisfying finitely many boundary equations. These representatives and equations occur at one stage. Equality of two such maps is likewise detected at one stage. Hence \Orient{m} is finitely presentable in \omCat. The inclusion \nCat\hookrightarrow\omCat preserves filtered colimits by the preceding cellwise calculation, so reflection and adjunction give finite presentability of \trunc\Orient{m}. This proves the assertion for \Street degreewise; compare also [4, Proposition 5.13(a)]. Since (\Ex K)_m=\Hom_{\sSet}(\Sd\Delta[m],K) and \Sd\Delta[m] is a finite simplicial set, \Ex preserves filtered colimits too.

Here are details of the weak-equivalence assertion. The functor Q=\Ex^\infty preserves filtered colimits, since \Ex does and colimits commute with colimits. A filtered colimit of Kan complexes is Kan: any horn has finitely many nondegenerate simplices, so it and its compatibility equations occur at one stage and can be filled there. For a filtered diagram of Kan complexes K_i, components commute with its colimit, since vertices and finite paths are finite data. If x is a vertex of \colim_iK_i, choose a representative x_i\in K_i. The coslice of arrows i\to j is filtered and its projection to the indexing category is final; it gives a diagram with compatible basepoints x_j. For every k\ge1,

Indeed, use the finite pointed sphere \Delta[k]/\partial\Delta[k] and its finite reduced simplicial cylinder to represent classes and pointed homotopies in a Kan complex. Their simplices and all required equalities, including the basepoint equality, occur at a sufficiently late stage. This proves both surjectivity and injectivity of the displayed map, including eventual equalities when transition maps are not injective. It follows that an objectwise weak equivalence between filtered diagrams of Kan complexes induces a weak equivalence of colimits. Apply this to Q of a diagram of arbitrary simplicial weak equivalences, and use the natural weak equivalences K\to QK and two-out-of-three. Finally apply preservation of filtered colimits by \Street.□

Proof. This is the simplicial gluing lemma [10, II.8.8 and II.8.12]; the following proof keeps track of the specified cofibrations. Put

By simplicial left properness, B\to E is a weak equivalence, since it is the pushout of c along the monomorphism i. The induced map E\to B' is consequently a weak equivalence by two-out-of-three. Both C'\to E and C'\to B' are cofibrations. They are therefore cofibrant objects of the undercategory C'/\sSet, with its inherited model structure. Cobase change along C'\to A' is a left Quillen functor

Indeed, its right adjoint is restriction of the structure map, which preserves fibrations and trivial fibrations. It preserves weak equivalences between cofibrant objects, so

is a weak equivalence. Also A\to A\amalg_CB is a monomorphism, being a pushout of i. Left properness applied to a gives a weak equivalence A\amalg_CB\to A'\amalg_CB. Composing the two proves the result.

To see the homotopy-pushout assertion, factor the possibly nonmonomorphic map C\to A as a cofibration C\hookrightarrow \widetilde A followed by a trivial fibration \widetilde A\to A. The span \widetilde A\leftarrow C\hookrightarrow B is a cofibrant replacement of the original span for calculating the derived pushout: its three objects are cofibrant and both legs are cofibrations. The already proved assertion identifies its pushout with A\amalg_CB up to weak equivalence. For the final assertion, one may either use left properness directly or apply the span assertion to X\leftarrow X\hookrightarrow Z and Y\leftarrow X\hookrightarrow Z, with vertical maps X\to Y,\id_X,\id_Z.□

Proof. Simplexwise chain map. We first construct the cylinder in \omCat, before applying \trunc. 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 \OrdN E and put

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

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

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

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

For p>0, abbreviate

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

This is exactly the image under C_\sigma of

Thus (3.2) is a positive, augmentation-preserving chain map in every degree. The strong Steiner tensor comparison gives the strict functor

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

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 (3.6), on tensoring with e_0, e_1, or u, are exactly the same tuple in (3.3), 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 (3.6) into an equality of the strict functors (3.5) after mapping into \Uinf{E}.

The zero chains in this argument retain their lower boundary data after applying \nu. In the cell table (2.2), 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=\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 (3.6) 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

The compatible maps (3.5) consequently induce P^\infty_{f,g}:\Uinf{E}\otimes\Disk{1}\to\Uinf{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

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 \Uinf{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:\Uinf{E}\to\U{E} be the reflection unit. By Lemma 2.1, the composite q_E P^\infty_{f,g} factors uniquely as

Its endpoint identities follow by precomposing with q_E and using the universal property of reflection. To check (3.1), precompose both sides with q_A\otimes\id. The resulting equality is the stationarity just proved before reflection. The injectivity in the natural bijection of Lemma 2.1, 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 2.3 gives the stated simplicial homotopy and its stationarity.□

Proof. Let H(Y) be the right internal hom representing strict functors (-)\otimes\Disk{1}\to Y. It is an n-category by Lemma 2.1; for n=\infty this is automatic. The cylinders

transpose to maps B\to H(Y) and X\to H(Y) in \nCat. Their restrictions to B_0 agree, since on its cylinder both give

The pushout universal property in \nCat thus gives a unique map Y\to H(Y) extending both transposes. Its transpose is Q. The endpoint formulas, the equality Q(j\otimes\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 2.3 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

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 \Disk{1} in all the stated dimensions.

For p:C\to\Disk{1}, a functor C_0\to B extends uniquely over \Disk{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

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

By Lemma 4.1, its object set is the colimit of the vertex sets [m]. Sending the jth 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]}=\Disk{1}. The presentation (4.1) is consequently a colimit in \nCat/\Disk{1} as well: slice colimits have the underlying colimits in \nCat. In its \sigma-piece the vertices mapping to S form an initial segment I_\sigma. By Lemma 4.2, the 0-fiber of that piece is its truncated segment oriental. Taking fibers commutes with the colimit by Lemma 4.1.

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 \OrdN P; 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 \nCat/\Disk{1}. Applying F_\epsilon and then Lemmas 4.1 and 4.3 gives (4.2). 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 4.1 and Lemma 4.3. 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\Disk{1}\times\Disk{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 4.4 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 (4.2) therefore gives \U{T_2} and \U{T_1\cap T_2} exactly. In both cases (4.3) 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 \Street C_{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:\Orient{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 2.5.□

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 (5.1) gives b\le t. Define maps k,g:E\to E by

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

and all three maps restrict to \id_A. The common upper map g is constant on E\setminus A, exactly as required in Lemma 3.1. Figure 1 shows why this common upper map is useful: \id_E and k need not be pointwise comparable, although both lie below g.

Apply Lemma 3.2 to the two comparisons in (5.3). 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_n\OrdN(g) and v\,c_n\OrdN(k), respectively. The resulting right Gray cylinders, both stationary on X, have endpoint pattern

By Lemma 2.3, 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 \id_X and a\,c_n\OrdN(\bar k):\U{E}\to X agree on \U{A} and hence define p:Y\to X. They give pj=\id_X. Comparing restrictions to the two pushout summands gives jp=K: on X both restrict to j, and on \U{E} one has

After realization, concatenate the homotopy from \id_Y to G with the reverse of the homotopy from jp to G. Thus |\Street p| is a homotopy inverse to |\Street j|, 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 5.1 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

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 (5.1). Each complement E_T\setminus A is smaller. Define

By induction every \Street j_T is a weak homotopy equivalence.

Corollary 4.5(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 4.6 now identifies the bottom pushout in the following diagram with \Street Y:

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 2.5 proves that the induced map on pushouts is a weak equivalence. The top pushout is \Street X, and its map to the bottom pushout is precisely \Street j, 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 (5.1) restricts to these full subposets. Notice that F itself need not be a sieve in E.

Use the restriction of a to define

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

Applying the colimit-preserving left adjoint c_n and then commuting colimits gives

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 2.4, \Street preserves these filtered colimits and simplicial weak homotopy equivalences are closed under filtered colimits. Hence \Street j 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 \id_S\le\mathrm{const}_x contracts \OrdN S. Lemma 3.1, with old sieve \varnothing, gives a right cylinder from \id_{\U{S}} to c_n\OrdN(\mathrm{const}_x). The latter functor factors through \U{\{x\}}=\Orient{0}, the terminal strict n-category. Lemma 2.3 therefore contracts \Street\U{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 \OrdN S: a simplex belongs to a sieve containing its last vertex. By Lemmas 4.3 and 4.6, 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

The vertical maps are weak equivalences by induction; all horizontal legs are monomorphisms. The induced map on pushouts is a weak equivalence by Lemma 2.5. Under the cover identifications it is exactly \eta_S: its restriction to either \OrdN S_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 \Street preserves the resulting filtered colimit by Lemma 2.4. 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

Theorem 5.2, 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 4.3 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 \Street X\to P_Z(X) is a weak equivalence. The canonical triangle

then shows by two-out-of-three that \gamma_{Z,X} is a weak equivalence.

To pass from Z to B, write

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 4.3 therefore gives cosieves \U{Z},\U{C}\subset\U{B} with intersection \U{T}. Apply Corollary 4.5(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

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 4.6 to the two cosieve covers just obtained. It gives ordinary simplicial pushouts

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 (6.3) gives

Under these canonical identifications, Equation (6.1) is exactly the map induced by \gamma_{Z,X} and the identity of \Street\U{C}. In particular the square

is a pushout: its lower right corner is also \Street V_Z(X)\amalg_{P_Z(X)}P_B(X). The top horizontal map is the cobase change of the specific monomorphism

Left properness of simplicial sets, as recorded in Lemma 2.5, now makes \gamma_{B,X} a weak equivalence. Neither the map \Street\U{A}\to\Street X nor the map \Street\U{T}\to P_Z(X) needs to be a monomorphism.

The remaining assertion that \Street\U{A}\to\Street\U{B} is monic follows directly from the sieve identification in Lemma 4.3. Finally, \gamma_{B,X} is defined by the two canonical maps from \Street X and \Street\U{B}. For f:X\to X' under \U{A} these maps commute with \Street f and with the induced pushout functor V_B(X)\to V_B(X'). The pushout universal property therefore gives the strict naturality square

The comparison is thus canonical and natural, while the local contractions used in Theorem 5.2 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

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

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 \OrdN P are themselves the nonempty finite chains in P, with a face obtained by deleting entries. Applying the same description a second time gives

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

For the nontrivial implication in this equality, a face F\in\sigma\cap P_K gives \{F\}\in A below \sigma. Define

This is a nonempty chain, is monotone under inclusion, and fixes A. For \alpha\in A and \sigma\in Z one has

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 6.2, so that L_n d is \U{A}\to\U{B}. The inclusion \OrdN A\to\OrdN B 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 \Sd\to\id of [4, Theorem 4.5]. Naturality of the adjunction unit gives a square

The vertical maps are weak equivalences by Proposition 5.3. Two-out-of-three makes the lower map a weak equivalence, and it is monic by Proposition 6.1. Thus it is a simplicial trivial cofibration. For any attaching functor \U{A}\to X, its pushout \Street X\to P_B(X) is a weak equivalence. The composite

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 \Street X\to P_B(X) is a monomorphism, being the cobase change of \Street\U{A}\hookrightarrow\Street\U{B}. There is a canonical pushout identification

Left properness of simplicial sets therefore makes P_B(X)\to P_B(X') a weak equivalence, since \Street f is one. In the naturality square of Equation (6.6), both horizontal maps are weak equivalences by Proposition 6.1. Two-out-of-three implies that its right vertical map is a weak equivalence. This is the nerve of the asserted functor.□

Proof. Corollary 6.4 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 6.4. If \beta is a nonzero limit ordinal, the ordinal of stages \alpha<\beta is filtered. Apply Lemma 2.4 to the diagram of arrows X_0\to X_\alpha. Its colimit is X_0\to\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 2.4. 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

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 \id_B admits a diagonal s:B\to Z. The equations su=i and ps=\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

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

In the first equality of classes on the right of (7.2), 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 7.1 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=\id_B. The same fixed-source retract diagram puts u in \mathcal D. We have proved

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 7.1, 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 6.5 proves the induction step. At a nonzero limit ordinal, both sequences are their colimits, and Lemma 2.4 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 7.2, 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=\id_Y. Cobase change along w gives the corresponding maps s' and p' between X'\amalg_XY and X'\amalg_XZ, with p's'=\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 6.2 identifies \Sd^2K with the ordinary nerve of a poset. The assertion is therefore Proposition 5.3. 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

and assume the assertion for K_0. Write

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, \Sd^2K_+=V\amalg_A B. Naturality of \eta gives a map of spans and hence the map

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=\OrdN E and B=\OrdN F as in Lemma 6.2, the inclusion E\subseteq F is a sieve; Lemma 4.3 makes c_nA\to c_nB a sieve inclusion. Consequently \Street c_nA\to\Street c_nB is also a monomorphism. Lemma 2.5, 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 6.1 to the arbitrary attaching functor c_nA\to c_nV. It says that

is a weak equivalence. Because c_n preserves colimits, the target is \Street c_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, \Sd^2 and L_n preserve the colimit as left adjoints, while \Street preserves the filtered colimit by Lemma 2.4. 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 (7.4) for every K. The poset descriptions used in this proof concern the simplex and boundary pieces only.□

Proof. Equations (7.2) and (7.3) show that R_n preserves fibrations and trivial fibrations, so this is a Quillen adjunction.

Set S=\Sd and T=\Ex, and let a_K:K\to TSK be the unit of S\dashv T. The mate identity is

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

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

Proposition 7.4, 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

Using the definition of W_n and two-out-of-three now gives

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 7.2 and 7.3, this proves all the asserted model-categorical conclusions for each finite n\geq1 and for n=\infty.□

Proof. Generate simultaneously judgments \mathsf D(t) and t=u, meaning that t is defined and that t,u are defined with the same value. Use the following rules.

1. (i)

   Every generator is defined, and each member of R is an equation.

2. (ii)

   An equation t=u implies \mathsf D(t) and \mathsf D(u). Definedness of an operation term implies definedness of all its arguments. Definedness of x*_k y also implies its matching equation (A.1).

3. (iii)

   Total operations applied to defined arguments give defined terms. Defined arguments satisfying (A.1) give a defined composite.

4. (iv)

   Equality is symmetric and transitive. Reflexivity is the rule \mathsf D(t)\Rightarrow t=t; it is not imposed on undefined raw terms.

5. (v)

   Equal arguments transport definedness and equality through an operation. Precisely, from t_i=u_i for all arguments and \mathsf D(f(t_1,\ldots,t_r)), infer \mathsf D(f(u_1,\ldots,u_r)) and equality of these two operation terms.

6. (vi)

   Impose every ground instance of the strict category equations above when its arguments are defined and its stated matching conditions hold. The conclusion is equality of the two sides and therefore definedness of both. In particular, interchange retains all four conditions in (A.3).

Globularity, identity boundary and unit equations require just definedness of their variables. Composition boundary equations and preservation of composition by identities require (A.1); associativity requires its two matching equations. Thus every rule has finitely many premises. All rules are sound in a strict category, including the definedness conclusions in the last rule.

Start with the assumptions and repeatedly add the conclusions of all applicable rules. The union of the finite stages is the least closed set of judgments: any rule’s finitely many premises occur together at some stage. In particular, every resulting judgment has a finite derivation. This construction involves a set of judgments, since the signature and the set of raw finite terms are sets.

Let P(G,R)_d be the defined terms of sort d modulo derivable equality. The equality rules give an equivalence relation on this set. Total operations descend by rules (iii) and (v). For the partial operations, suppose the boundary classes of [x] and [y] match. Total boundary terms are defined, so equality of their classes means that s_k(x)=t_k(y) is derivable. Rule (iii) then declares x*_k y defined.

This construction does not depend on the representatives. Indeed, if x=x' and y=y' are derivable, total-operation congruence gives

The matching equality for x,y therefore gives the matching equality for x',y'. Rule (iii) defines both composites, and rule (v) equates them. Conversely, rule (ii) ensures that every admitted composite has matching boundaries. The operation domain in the quotient is consequently exactly the set of matching pairs. Rule (vi) gives all the strict category laws there.

Now let an assignment G\to Y satisfy R. Induction on derivations shows that each term declared defined evaluates in Y and each derived equation has equal values. For rule (v), equal arguments have identical boundary values, so existence of one partial operation value entails existence of the other. Evaluation therefore descends to a strict functor P(G,R)\to Y. It is unique by induction on terms, since definedness is hereditary to subterms and the images of the generators are fixed. Conversely, every strict functor gives such an assignment. This proves the universal property.

The constructed object is small. There is no inconsistency obstruction: the strict category with one cell in each sort satisfies every strong equation under the unique assignment. If G is empty then there are no raw ground terms, since all operations have positive arity. The construction gives the empty strict category in that case.□

Proof. Colimits. For a small diagram X:I\to\nCat, use a generator [i,a] for each cell a of each X_i. Impose all its operation-table equations:

where the last equation is included for every defined composite. For each arrow f:i\to j also impose [i,a]=[j,X(f)(a)]. All these generators and equations form sets. By Lemma A.1, maps from this presentation to Y are exactly compatible families of strict functors X_i\to Y. It is therefore the colimit.

Limits and local smallness. In each dimension, take the set of compatible families in the diagram. Source, target and identity are coordinatewise. Two families have matching boundaries exactly when their coordinates do, so coordinatewise composition is defined on exactly the required domain and again gives a compatible family. All laws and the limit universal property hold coordinatewise. This includes the empty limit, with a singleton in each sort. Strict functors form a subset of the product of the sets of cell maps, so the category is locally small.

Filtered colimits and finite witnesses. For a filtered diagram, form the colimit of each set of cells. Finitely many representatives occur at a common stage. If a matching equation holds in the colimit, it holds at a later common stage, where the composite can be formed. A further common stage proves independence of choices. Each category law holds at a stage containing its finite tuple and its matching equations. This constructs the colimit with its universal property, without any injectivity assumption on transition maps.

We will also use the finite-term version of this argument. If finitely many assigned generator values make a finite collection of terms defined and satisfy specified equations in the colimit, these facts hold at a common stage. Induct on the finitely many subterms: first lift the arguments, then the boundary equality needed for a partial composite, and then the composite. Finally lift the required equalities. Each step uses the equality criterion for filtered colimits of sets.

Finite presentations. If G and R are finite, a map P(G,R)\to\colim_iX_i is an assignment of finitely many generator values satisfying finitely many strong equations. The preceding finite-term argument lifts that assignment and all the relations to a single stage. Lemma A.1 then gives a functor to that stage. If two stage functors become equal in the colimit, their values on the finite set G agree at a later common stage; their entire functors agree there by the presentation universal property. Thus

Hence every finite presentation is finitely presentable. This does not assert that it has finitely many cells.

A generating set of finite presentations. Every strict category X has its full cell presentation: one generator g_a for each cell and all operation-table equations. Evaluation identifies its presented category with X. Indeed it is onto, and induction on a defined term, using the operation-table equations, equates that term to the generator of its value. This also proves injectivity.

Write this presentation as (G_X,R_X). Consider all finite pairs (G_0,R_0) with G_0\subseteq G_X, R_0\subseteq R_X, and with every generator occurring in R_0 contained in G_0. Ordered by inclusion, these pairs form a nonempty filtered poset: the empty pair is allowed, and finite unions give upper bounds. Their presented categories have colimit P(G_X,R_X). To verify this, a compatible family of maps from the finite presentations assigns a value to every generator and enforces every relation, since each appears in some finite pair. The presentation universal property gives exactly a map from P(G_X,R_X).

The finite pairs need not be closed under all operations. Finite presentations themselves supply the required closures. There is a set of finite presentation types: use finite numbered generator sets with specified sorts and finite sets of raw term equations. Thus every object is a filtered colimit of objects from a set of finitely presentable objects. Together with cocompleteness, this is local finite presentability. The same reasoning applies to the countable signature in the \omega case.

A common regular bound. More generally, if G and R have cardinality less than an infinite regular cardinal \kappa, the same proof makes P(G,R) \kappa-presentable. In a \kappa-filtered diagram one lifts fewer than \kappa generator values and the finitely many subterms of each of fewer than \kappa relations. All stage, morphism and equality witnesses involved form a diagram of size less than \kappa, so a cocone places them at one stage. Equality of maps is detected by the fewer than \kappa generator equalities in the same way.

The full cell presentation of X has cardinality at most

because the signature is countable and operations have arity at most two. Choose a larger infinite regular cardinal. The preceding argument makes X presentable at that bound. For a set of objects, bound the cardinalities of all their full presentations first and choose one larger infinite regular cardinal. This gives the claimed common bound.□