January 01, 1970
A c-direct category is a small category equipped with an ordinal degree function such that every morphism is level or degree-raising. Every c-direct category is c-Reedy. The c-Reedy model structure on any functor category from a c-direct category to a model category coincides with the projective model structure. In this framework, a realization functor is a colimit-preserving functor satisfying some mild homotopical conditions from the category of presheaves on a c-direct category with cofibrant representables to a model category. We prove that any two such realization functors are weakly equivalent on cofibrant presheaves. For categories of cubes, we prove that thick categories have cofibrant representables. As an application, we introduce the \(\@undefined\)-branching space of an \(\mathcal{A}\)-set for any thick category of cubes \(\mathcal{A}\). It is obtained as a coend over a c-direct category with cofibrant representables constructed from \(\mathcal{A}\). We prove that, on free \(\mathcal{A}\)-sets generated by precubical sets, this new definition coincides with the earlier one. We prove that, for cofibrant \(\mathcal{A}\)-sets, the resulting space is independent of \(\@undefined\) up to homotopy.
It is well known that a direct category, i.e. a small category equipped with an ordinal degree function such that every nonidentity morphism is degree-raising, is a Reedy category and that the Reedy model structure on any functor category from a direct category to a model category coincides with the projective model structure. This observation does not apply directly to the framework of (symmetric) transverse sets introduced for the geometric study of concurrent processes in [1]–[3], because the indexing categories contain nonidentity level morphisms which are not isomorphisms, and their nonidentity endomorphisms need not be automorphisms. The generalized Reedy point of view of [4] therefore does not apply either.
To deal with these level noninvertible morphisms, we introduce c-direct categories (Definition 1). A c-direct category is a small category equipped with an ordinal degree function such that every morphism is either level or degree-raising. Every direct category is c-direct, and every c-direct category is c-Reedy in Shulman’s sense [5] (Theorem 5). Moreover, the c-Reedy model structure on any functor category indexed by a c-direct category coincides with the projective model structure (Theorem 12).
We then generalize the notion of a realization functor introduced in [6] and [2] to this setting (Definition 2) and prove invariance results (Theorems 13 and 15), extending the techniques used in [2], [6] for precubical sets and symmetric transverse sets. The only additional hypothesis is that the c-direct category has cofibrant representables.
Every category of cubes in the sense of Definition 7 is c-direct. By Theorem 19, if such a category is thick, then it has cofibrant representables. This yields a colimit-preserving definition of the \(\@undefined\)-branching space of an \(\mathcal{A}\)-set for every thick category of cubes \(\mathcal{A}\) (Definition 11). By Theorem 39, this definition agrees with the earlier one of [7], recalled in Definition 12, on free \(\mathcal{A}\)-sets generated by precubical sets. Moreover, Theorem 34 shows that the construction of [7] is colimit-preserving only for precubical sets. This explains why the new construction introduced here is necessary.
Section 2 introduces c-direct categories and records the basic calculations used later. After recalling the notion of a c-Reedy category, Section 3 establishes that every c-direct category is c-Reedy and proves that the c-Reedy model structure on functor categories coincides with the projective model structure. Section 4 explores the notion of a realization of a presheaf over a c-direct category with cofibrant representables and proves the resulting invariance theorems. Section 5 recalls the definition of a thick category of cubes, gives a characterization of thickness, and proves that thick categories of cubes have cofibrant representables. Section 6 introduces the \(\@undefined\)-branching space of an \(\mathcal{A}\)-set and proves its invariance up to homotopy with respect to \(\@undefined\). Section 7 explains why the geometric method for defining a branching space used in [7] gives rise to a colimit-preserving functor only in the precubical case. Finally, Section 8 proves that the \(\@undefined\)-branching space of an \(\mathcal{A}\)-set freely generated by a precubical set coincides with the earlier definition given in [7].
Let \(\mathcal{A}\) be a small category. An \(\mathcal{A}\)-set is a presheaf on \(\mathcal{A}\), that is, a functor \[K:\mathcal{A}^\mathrm{op}\longrightarrow\mathrm{Set}.\] The category of \(\mathcal{A}\)-sets is denoted by \[\mathcal{A}^\mathrm{op}\mathrm{Set}.\]
Definition 1. A small category \(\mathcal{A}\) is called c-direct if there exists a degree function \[d:\operatorname{Ob}(\mathcal{A})\longrightarrow \mathrm{Ord}\] with values in the class of ordinals such that, for every morphism \(f:x\to y\) of \(\mathcal{A}\), one has \[d(x)\@undefined d(y).\]
Every direct category is c-direct. This notion should not be confused with Shulman’s stratified categories: by [5], a small category is stratified precisely when it admits an ordinal degree function for which every morphism non-strictly decreases degree. Thus, with the same degree function, the opposite of a c-direct category is stratified in Shulman’s sense. The comparison relevant for the present paper is instead with Shulman’s c-Reedy categories; see Theorem 5 below.
Notation 1. For the rest of the paper, a c-direct category \(\mathcal{A}\) is fixed.
For \(n\@undefined 0\), let \(\mathcal{A}_{<n}\) denote the full subcategory of \(\mathcal{A}\) spanned by the objects \(x\) such that \(d(x)<n\). For objects \(y,z\) of \(\mathcal{A}\), set [5] \[\partial_n\mathcal{A}(y,z) = \int^{x\in \mathcal{A}_{<n}} \mathcal{A}(x,z)\times \mathcal{A}(y,x).\] There is a canonical map \[\gamma: \partial_n\mathcal{A}(y,z)\longrightarrow \mathcal{A}(y,z)\] induced by composition, namely \[[u: x\to z,\,v: y\to x] \longmapsto u\circ v.\] Here \([-]\) denotes the equivalence class in the coend.
Proposition 2. The following hold.
If \(d(y)>d(z)\) or \(n\@undefined d(y)\), then \[\partial_n\mathcal{A}(y,z)=\varnothing .\]
If \(d(y)\@undefined d(z)\) and \(d(y)<n\), then the canonical map \[\gamma: \partial_n\mathcal{A}(y,z)\longrightarrow \mathcal{A}(y,z)\] is a bijection. Thus, through this canonical identification, \[\partial_n\mathcal{A}(y,z)\cong \mathcal{A}(y,z).\]
Proof. Recall first that the coend is the quotient of the disjoint union \[\coprod_{x\in \operatorname{Ob}(\mathcal{A}_{<n})} \mathcal{A}(x,z)\times \mathcal{A}(y,x)\] by the relations \[(u\circ a,v)\sim (u,a\circ v)\] for every morphism \(a: x\to x'\) in \(\mathcal{A}_{<n}\), every \(u: x'\to z\), and every \(v: y\to x\). Assume first that \(d(y)>d(z)\). If a summand \[\mathcal{A}(x,z)\times \mathcal{A}(y,x)\] were nonempty, then there would be morphisms \[y\longrightarrow x\longrightarrow z.\] By the c-direct condition, these two morphisms imply \[d(y)\@undefined d(x)\@undefined d(z),\] which contradicts \(d(y)>d(z)\). Hence every summand is empty, and therefore \[\partial_n\mathcal{A}(y,z)=\varnothing .\] Assume now that \(n\@undefined d(y)\). If \(x\) is an object of \(\mathcal{A}_{<n}\), then \[d(x)<n\@undefined d(y),\] so \(d(y)>d(x)\). By the c-direct condition, \[\mathcal{A}(y,x)=\varnothing .\] Thus every summand \[\mathcal{A}(x,z)\times \mathcal{A}(y,x)\] is empty, and again \[\partial_n\mathcal{A}(y,z)=\varnothing .\] This proves the first assertion. It remains to prove the second assertion. Assume that \[d(y)\@undefined d(z) \qquad\text{and}\qquad d(y)<n.\] Then \(y\) is an object of \(\mathcal{A}_{<n}\). Define \[\sigma: \mathcal{A}(y,z)\longrightarrow \partial_n\mathcal{A}(y,z)\] by \[\sigma(f)=[f: y\to z,\operatorname{id}_y: y\to y].\] This is well-defined because \(y\in \mathcal{A}_{<n}\). By construction, \[\gamma\sigma(f) = \gamma([f,\operatorname{id}_y]) = f\circ \operatorname{id}_y = f.\] Hence \[\gamma\sigma=\operatorname{id}_{\mathcal{A}(y,z)}.\] Conversely, let \[[u: x\to z,\,v: y\to x]\] be an element of \(\partial_n\mathcal{A}(y,z)\), with \(x\in \mathcal{A}_{<n}\). Since also \(y\in \mathcal{A}_{<n}\) and \(\mathcal{A}_{<n}\) is a full subcategory of \(\mathcal{A}\), the morphism \(v: y\to x\) belongs to \(\mathcal{A}_{<n}\). Applying the defining coend relation to the morphism \(v: y\to x\) gives \[[u,v]=[u\circ v,\operatorname{id}_y].\] Therefore \[[u,v] = \sigma(u\circ v) = \sigma\gamma([u,v]).\] Thus \[\sigma\gamma=\operatorname{id}_{\partial_n\mathcal{A}(y,z)}.\] The maps \(\gamma\) and \(\sigma\) are inverse bijections. Hence \[\partial_n\mathcal{A}(y,z)\cong \mathcal{A}(y,z)\] canonically, through the composition map \(\gamma\). ◻
Proposition 3. Fix an object \(x\) of \(\mathcal{A}\). Consider the presheaf \[F_x := \int^{y\in \mathcal{A}_{<d(x)}} \mathcal{A}(y,x)\cdot \mathcal{A}(-,y),\] where \(\mathcal{A}(-,y)\) is the representable presheaf and where \(\cdot\) denotes the copower of a presheaf by a set. For every object \(z\) of \(\mathcal{A}\), there is a canonical bijection \[F_x(z) \cong \begin{cases} \mathcal{A}(z,x), & \text{if } d(z)<d(x),\\[4pt] \varnothing, & \text{if } d(z)\@undefined d(x). \end{cases}\] Thus \(F_x\) is the strict lower-degree part of the representable presheaf \(\mathcal{A}(-,x)\). It is denoted by \(\partial\mathcal{A}(-,x)\).
Proof. Since colimits in the presheaf category are computed pointwise, for every object \(z\) of \(\mathcal{A}\), we have \[F_x(z) = \int^{y\in \mathcal{A}_{<d(x)}} \mathcal{A}(y,x)\times \mathcal{A}(z,y).\] This coend is the quotient of the disjoint union \[\coprod_{y\in \mathcal{A}_{<d(x)}} \mathcal{A}(y,x)\times \mathcal{A}(z,y)\] by the usual coend relations. Explicitly, if \[a: y\longrightarrow y'\] is a morphism in \(\mathcal{A}_{<d(x)}\), and if \[u: y'\longrightarrow x, \qquad v: z\longrightarrow y,\] then \[(u\circ a,v)\sim (u,a\circ v).\] There is a canonical map \[\gamma_z: F_x(z)\longrightarrow \mathcal{A}(z,x)\] induced by composition: \[[u: y\to x,\;v: z\to y] \longmapsto u\circ v.\] Suppose first that \(d(z)\@undefined d(x)\). For every object \(y\in \mathcal{A}_{<d(x)}\), one has \[d(y)<d(x)\@undefined d(z),\] hence \(d(z)>d(y)\). By the c-direct condition, \[\mathcal{A}(z,y)=\varnothing.\] Therefore every summand \[\mathcal{A}(y,x)\times \mathcal{A}(z,y)\] is empty, and consequently \[F_x(z)=\varnothing.\] Suppose now that \(d(z)<d(x)\). Then \(z\) belongs to the full subcategory \(\mathcal{A}_{<d(x)}\). Hence every morphism \[f: z\longrightarrow x\] defines an element of \(F_x(z)\) by the factorization through \(z\): \[z \xrightarrow{\operatorname{id}_z} z \xrightarrow{f} x.\] This gives a map \[\sigma_z: \mathcal{A}(z,x)\longrightarrow F_x(z), \qquad f\longmapsto [f,\operatorname{id}_z].\] It is immediate that \[\gamma_z\sigma_z(f)=f\] for every \(f: z\to x\).
Conversely, let \[[u,v]\in F_x(z)\] be represented by morphisms \[z\xrightarrow{v} y\xrightarrow{u} x\] with \(y\in \mathcal{A}_{<d(x)}\). Since \(d(z)<d(x)\), the object \(z\) also belongs to \(\mathcal{A}_{<d(x)}\). Because \(\mathcal{A}_{<d(x)}\) is full, the morphism \[v: z\longrightarrow y\] is a morphism of \(\mathcal{A}_{<d(x)}\). Therefore the coend relation applied to \(v\) gives \[[u,v]=[u\circ v,\operatorname{id}_z].\] Equivalently, \[[u,v]=\sigma_z\gamma_z([u,v]).\] Thus \(\sigma_z\gamma_z=\operatorname{id}_{F_x(z)}\), while we already have \(\gamma_z\sigma_z=\operatorname{id}_{\mathcal{A}(z,x)}\). Hence \(\gamma_z\) is a bijection \[F_x(z)\cong \mathcal{A}(z,x)\] when \(d(z)<d(x)\). Combining the two cases proves the assertion. ◻
Notation 4. For \(x\in \operatorname{Ob}(\mathcal{A})\), set \[\mathcal{A}[x] = \mathcal{A}(-,x), \qquad \partial\mathcal{A}[x] = \partial\mathcal{A}(-,x).\]
Let \(f\) be a morphism of \(\mathcal{A}\). The category of factorizations of \(f\) has for objects the pairs \[(h,g)\qquad such that \qquad hg=f\] and for morphisms \(k:(h,g)\to (h',g')\) the morphisms \(k\) of \(\mathcal{A}\) (which are called connecting morphisms) such that there is a commutative diagram \[\begin{tikzcd}[row sep=3em,column sep=7em] \bullet \arrow[r,"g'"] \arrow[equal,d] & \bullet \arrow[r,"h'"]& \bullet \arrow[equal,d] \\ \bullet \arrow[r,"g"] & \arrow[u,"k"]\bullet \arrow[r,"h"] & \bullet \end{tikzcd}\] We have ([5])
A morphism is level if its domain and codomain have the same degree.
The degree of a factorization \((h,g)\) of a morphism \(f\) is the degree of the intermediate object (i.e. the domain of \(h\) which is the codomain of \(g\)).
A factorization of a morphism \(f\) is fundamental if its degree is strictly less than the degrees of both the domain and codomain of \(f\).
A morphism is basic if it does not admit any fundamental factorization.
Denote by \(\mathcal{A}_{\mathrm{lev}}\) the wide subcategory of \(\mathcal{A}\) whose morphisms are the level morphisms, that is, \[\mathcal{A}_{\mathrm{lev}}(x,y) = \begin{cases} \mathcal{A}(x,y), & d(x)=d(y),\\ \varnothing, & d(x)\neq d(y). \end{cases}\] This is indeed a subcategory of \(\mathcal{A}\): identities are level, and the composite of two level morphisms is level. The \(\delta\)-th stratum of \(\mathcal{A}\), denoted by \[\mathcal{A}_{=\delta},\] is the subcategory of \(\mathcal{A}\) generated by the objects of degree \(\delta\) and by the basic morphisms between them [5]. Since \(\mathcal{A}\) is c-direct, all morphisms are therefore basic. Thus the \(\delta\)-th stratum is the full subcategory generated by all objects of \(\mathcal{A}\) of degree \(\delta\).
We use the following form of the c-Reedy axioms (see [5]). A small category \(\mathcal{C}\) equipped with wide subcategories \(\overrightarrow{\mathcal{C}}, \overleftarrow{\mathcal{C}}, \overleftrightarrow{\mathcal{C}}\) called respectively the direct, the inverse and the level subcategories and with a degree function is c-Reedy if:
\(\overleftrightarrow{\mathcal{C}}=\overrightarrow{\mathcal{C}}\cap \overleftarrow{\mathcal{C}}\);
every morphism of \(\overrightarrow{\mathcal{C}}\setminus \overleftrightarrow{\mathcal{C}}\) strictly raises degree;
every morphism of \(\overleftarrow{\mathcal{C}}\setminus \overleftrightarrow{\mathcal{C}}\) strictly lowers degree;
every morphism \(f\) of \(\mathcal{C}\) admits a factorization \[f=\overrightarrow{f}\circ \overleftarrow{f}\] with \(\overleftarrow{f}\in \overleftarrow{\mathcal{C}}\) and \(\overrightarrow{f}\in \overrightarrow{\mathcal{C}}\), and the category of such factorizations is non-empty and connected, with connecting morphisms in \(\overleftrightarrow{\mathcal{C}}\);
for every object \(x\), the restriction of \(\overleftarrow{\mathcal{C}}(x,-)\) to each lower degree is a coproduct of representables.
Theorem 5. Consider the wide subcategories \[\overrightarrow{\mathcal{A}}=\mathcal{A}, \qquad \overleftarrow{\mathcal{A}}=\overleftrightarrow{\mathcal{A}}=\mathcal{A}_{\mathrm{lev}}.\] Then \(\mathcal{A}\) is a c-Reedy category for this choice of the direct, inverse, and level subcategories.
Proof. We check the c-Reedy axioms. First, the intersection condition is immediate. Since \(\overrightarrow{\mathcal{A}}=\mathcal{A}\) and \(\overleftarrow{\mathcal{A}}=\mathcal{A}_{\mathrm{lev}}\), one has \[\overrightarrow{\mathcal{A}}\cap \overleftarrow{\mathcal{A}} = \mathcal{A}\cap \mathcal{A}_{\mathrm{lev}} = \mathcal{A}_{\mathrm{lev}} = \overleftrightarrow{\mathcal{A}}.\] Next, let \(f:x\to y\) be a morphism of \(\overrightarrow{\mathcal{A}}\setminus \overleftrightarrow{\mathcal{A}}\). Since \(\overrightarrow{\mathcal{A}}=\mathcal{A}\), this just means that \(f\) is a morphism of \(\mathcal{A}\) which is not level. The c-direct condition gives \[d(x)\@undefined d(y).\] Since \(f\) is not level, \(d(x)\neq d(y)\). Therefore \[d(x)<d(y).\] Thus every non-level morphism of \(\overrightarrow{\mathcal{A}}\) strictly raises degree. The corresponding condition for the inverse part is vacuous, because \[\overleftarrow{\mathcal{A}}=\overleftrightarrow{\mathcal{A}}.\] Hence there is no morphism in \(\overleftarrow{\mathcal{A}}\setminus\overleftrightarrow{\mathcal{A}}\). It remains to verify the factorization axiom. Let \[f:x\longrightarrow y\] be a morphism of \(\mathcal{A}\). There is a canonical factorization \[x \xlongrightarrow{\operatorname{id}_x} x \xlongrightarrow{f} y,\] where \(\operatorname{id}_x\) belongs to \(\overleftarrow{\mathcal{A}}=\mathcal{A}_{\mathrm{lev}}\) and \(f\) belongs to \(\overrightarrow{\mathcal{A}}=\mathcal{A}\). Thus the category of c-Reedy factorizations of \(f\) is nonempty. Let \[x \xlongrightarrow{\ell} z \xlongrightarrow{g} y\] be any other factorization of \(f\), with \[\ell\in \overleftarrow{\mathcal{A}}=\mathcal{A}_{\mathrm{lev}}, \qquad g\in \overrightarrow{\mathcal{A}}=\mathcal{A}, \qquad g\ell=f.\] Since \(\ell\) is level, it belongs to \(\overleftrightarrow{\mathcal{A}}\). Therefore \(\ell:x\to z\) defines a morphism in the factorization category from the canonical factorization \[x \xlongrightarrow{\operatorname{id}_x} x \xlongrightarrow{f} y\] to the factorization \[x \xlongrightarrow{\ell} z \xlongrightarrow{g} y.\] Indeed, the required equalities are \[\ell\operatorname{id}_x=\ell \qquad \text{and} \qquad g\ell=f.\] Thus \(\ell\) gives a morphism in the factorization category. Hence every factorization is connected to the canonical one. The factorization category is therefore connected. Finally, the remaining c-Reedy axiom concerning the inverse part in strictly smaller degree is automatic. For every object \(x\) and every degree \(\alpha<d(x)\), the restriction of \[\overleftarrow{\mathcal{A}}(x,-)\] to the full subcategory of objects of degree \(\alpha\) is the empty functor, because \(\overleftarrow{\mathcal{A}}=\mathcal{A}_{\mathrm{lev}}\) contains only morphisms preserving degree. The empty functor is the empty coproduct of representables. Thus the required representability condition is satisfied. All c-Reedy axioms are therefore satisfied. ◻
Notation 6. The category of functors from a small category \(\mathcal{C}\) to a locally small category \(\mathcal{M}\) together with the natural transformations is denoted by \(\mathcal{M}^\mathcal{C}\).
When \(\mathcal{M}\) is a cocomplete category, by [8], there is a bijective correspondence between the objects \(A\) of \(\mathcal{M}^\mathcal{C}\) and the colimit-preserving functors \[\widehat A:\mathcal{C}^\mathrm{op}\mathrm{Set}\longrightarrow \mathcal{M}\] from the presheaves over \(\mathcal{C}\) to \(\mathcal{M}\) with \[\widehat A(K)=\int^{x\in\mathcal{C}} K(x)\cdot A(x).\]
Notation 7. For the rest of the paper, \(\mathcal{M}\) denotes a model category such that the projective model structure on \[\mathcal{M}^{\mathcal{A}_{=\delta}}\] exists for all \(\delta\@undefined 0\) (e.g. any combinatorial or accessible model category).
Notation 8. Let \(n\@undefined 0\). The latching and matching object functors \(L_n,M_n:\mathcal{M}^{\mathcal{A}}\to \mathcal{M}^{\mathcal{A}_{=n}}\) are given by (see [5]) \[\begin{align} & (M_nA)_{y} = \int_{z\in \mathcal{A}} A(z)^{\partial_{n}\mathcal{A}(y,z)} \\ & (L_nA)_{y} = \int^{z\in \mathcal{A}}\partial_n\mathcal{A}(z,y) \cdot A(z) \end{align}\]
We obtain:
Theorem 9. Let \(\mathcal{M}\) be a model category. Suppose that the projective model structure on \(\mathcal{M}^{\mathcal{A}_{=n}}\) exists for all \(n\@undefined 0\). There exists a unique model structure on \(\mathcal{M}^{\mathcal{A}}\) such that
The weak equivalences are objectwise.
A map \(A\to B\) of \(\mathcal{M}^{\mathcal{A}}\) is a fibration (trivial fibration resp.) if for all \(n\@undefined 0\), the map \[A(x)\longrightarrow (M_nA)_{x}\times_{(M_nB)_{x}} B(x)\] is a fibration (trivial fibration resp.) of \(\mathcal{M}\) for all \(x\in\operatorname{Ob}(\mathcal{M}^{\mathcal{A}_{=n}})\).
A map \(A\to B\) of \(\mathcal{M}^{\mathcal{A}}\) is a cofibration (trivial cofibration resp.) if for all \(n\@undefined 0\), \[L_nB\sqcup_{L_nA} A\longrightarrow B\] is a projective cofibration (trivial cofibration resp.) of the projective model structure of \(\mathcal{M}^{\mathcal{A}_{=n}}\).
This model structure is called the c-Reedy model structure* of \(\mathcal{M}^{\mathcal{A}}\).*
Proof. By Theorem 5 and [5], the small category \(\mathcal{A}\) is almost c-Reedy in the sense of [5]. The proof is complete by [5]. ◻
Lemma 10. [2] Let \(\mathcal{C}\) be a small category. Consider a small diagram \(X:\mathcal{C}\to \mathcal{M}\) and the empty weight \(W:\mathcal{C}\to \mathrm{Set}\) with \(W(c)=\varnothing\) for all \(c\in \mathcal{C}\). Then there is the isomorphism \[\int_{c\in \mathcal{C}}X(c)^{W(c)} \cong \mathbf{1}.\]
Lemma 11. [2] Let \(\mathcal{C}\) be a small category. Consider a small diagram \(X:\mathcal{C}\to \mathcal{M}\) and a weight \(U:\mathcal{C}^{\mathrm{op}}\to \mathrm{Set}\). Let \(\mathcal{D}\) be the full subcategory of \(\mathcal{C}\) generated by the objects \(c\) such that \(U(c)\neq \varnothing\). Then there is the isomorphism \[\int^{c\in \mathcal{D}} U(c)\cdot X(c) \cong \int^{c\in \mathcal{C}} U(c)\cdot X(c).\]
Recall that the projective model structure is the unique model structure (if it exists) on a functor category \(\mathcal{M}^\mathcal{C}\) such that the weak equivalences and the fibrations are the objectwise ones.
Theorem 12. The projective model structure on \[\mathcal{M}^{\mathcal{A}}\] exists and coincides with the c-Reedy model structure. Let \(A\in \operatorname{Ob}(\mathcal{M}^{\mathcal{A}})\). Then \(A\) is projective cofibrant if and only if the map \[\widehat A(\partial\mathcal{A}[-]) \longrightarrow \widehat A(\mathcal{A}[-])\] is a projective cofibration of \(\mathcal{M}^{\mathcal{A}_{=n}}\) for every \(n\@undefined 0\).
Proof. We mimic the proof of [2]. The matching object functor \[M_n:\mathcal{M}^{\mathcal{A}}\to \mathcal{M}^{\mathcal{A}_{=n}}\] for all \(n\@undefined 0\) can be calculated as follows. There is the sequence of isomorphisms of \(\mathcal{M}\) \[(M_nA)_{y}\cong \int_{z\in \mathcal{A}} A(z)^{\partial_n\mathcal{A}(y,z)} \cong \int_{z\in \mathcal{A}} A(z)^{\varnothing} \cong \mathbf{1},\] the first isomorphism by definition of the matching object functor (Notation 8), the second isomorphism since \(\partial_n\mathcal{A}(y,z)=\varnothing\) by Proposition 2, and the third isomorphism by Lemma 10. Thus, the c-Reedy model structure of Theorem 9 on \(\mathcal{M}^{\mathcal{A}}\) coincides with the projective model structure which therefore exists. There is the sequence of isomorphisms of \(\mathcal{M}\) \[(L_nA)_{y} \cong \int^{z\in \mathcal{A}}\partial_n\mathcal{A}(z,y) \cdot A(z) \cong \int^{z\in \mathcal{A}_{<n}}\mathcal{A}(z,y) \cdot A(z) \cong \widehat{A}(\partial\mathcal{A}[y]),\] the first isomorphism by definition of the latching object functor (Notation 8), the second isomorphism by Lemma 11 and since \(\partial_n\mathcal{A}(z,y)=\varnothing\) for \(d(z)\@undefined n\) by Proposition 2, and finally the third isomorphism by Proposition 3 and since \(\widehat{A}\) is colimit-preserving. By Theorem 9, \(A\) is projective cofibrant if and only if for all \(n\@undefined 0\), the map \(L_nA\to A\) is a projective cofibration of the projective model structure of \(\mathcal{M}^{\mathcal{A}_{=n}}\). Since \(A(x) = \widehat{A}(\mathcal{A}[x])\) by definition of \(\widehat{A}\), the proof is complete. ◻
Definition 2. Let \(I\) be an object of \(\mathcal{M}^\mathcal{A}\). Let \(A\) be an object of \(\mathcal{M}^\mathcal{A}\). The functor \[\widehat A:\mathcal{A}^\mathrm{op}\mathrm{Set}\longrightarrow \mathcal{M}\] is a realization functor over \(I\) if the following conditions hold:
For all \(x\in\operatorname{Ob}\mathcal{A}\), the map \(\widehat A(\partial\mathcal{A}[x])\to \widehat A(\mathcal{A}[x])\) is a cofibration of \(\mathcal{M}\).
There is a natural transformation \[\begin{tikzcd}[cramped] A\arrow[twoheadrightarrow,r,"\simeq"] & I \end{tikzcd}\] which is an objectwise trivial fibration of \(\mathcal{M}\) (i.e. a projective trivial fibration).
Definition 3. An object \(K\) of \(\mathcal{A}^\mathrm{op}\mathrm{Set}\) is cellular if the canonical map \(\varnothing \to K\) is a cellular map, i.e. a transfinite composition of pushouts of the generating cofibrations \[\partial\mathcal{A}(-,x)\subseteq \mathcal{A}(-,x)\] for \(x\) running over \(\operatorname{Ob}(\mathcal{A})\). An object \(K\) of \(\mathcal{A}^\mathrm{op}\mathrm{Set}\) is cofibrant if it is a retract of a cellular object.
Let \(x\) be an object of \(\mathcal{A}\). We claim that \(\partial\mathcal{A}(-,x)\) is cofibrant if and only if \(\mathcal{A}(-,x)\) is cofibrant. The forward implication follows immediately from the definition, since \(\mathcal{A}(-,x)\) is obtained from \(\partial\mathcal{A}(-,x)\) by attaching one \(x\)-cell along the generating cofibration \[\partial\mathcal{A}(-,x)\subseteq \mathcal{A}(-,x).\] Conversely, for every ordinal \(\lambda\), let \(\operatorname{sk}_{<\lambda}\) denote the truncation functor on presheaves defined by \[\bigl(\operatorname{sk}_{<\lambda}K\bigr)(y)= \begin{cases} K(y) & \text{if } d(y)<\lambda,\\ \varnothing & \text{if } d(y)\@undefined\lambda . \end{cases}\] This is indeed a subpresheaf because \(\mathcal{A}\) is c-direct. Moreover \(\operatorname{sk}_{<\lambda}\) preserves cofibrant (and even cellular) presheaves: it preserves colimits and retracts, and it sends every generating cofibration \[\partial\mathcal{A}(-,z)\subseteq \mathcal{A}(-,z)\] either to itself, when \(d(z)<\lambda\), or to an identity map, when \(d(z)\@undefined\lambda\). Taking \(\lambda=d(x)\), one has \[\operatorname{sk}_{<d(x)}\mathcal{A}(-,x) = \partial\mathcal{A}(-,x).\] Therefore, if \(\mathcal{A}(-,x)\) is cofibrant, then \(\partial\mathcal{A}(-,x)\) is cofibrant as well.
Here is an example showing that the representables need not be cofibrant for an arbitrary c-direct category. Let \(G=C_{2}\), and let \(\mathcal{A}\) be the small category with two objects \(0\) and \(1\), with \(d(0)=0\) and \(d(1)=1\), defined by \[\mathcal{A}(0,0)=G,\qquad \mathcal{A}(1,1)=\{\operatorname{id}_{1}\},\qquad \mathcal{A}(0,1)=\{u\},\qquad \mathcal{A}(1,0)=\varnothing ,\] where the only nontrivial composition is given by \[u\circ g=u \qquad (g\in G).\] Then \(\mathcal{A}\) is c-direct. We claim that the representable \(\mathcal{A}(-,1)\) is not cofibrant. Indeed, if \(K\) is cofibrant, then \(K(0)\) is a free right \(G\)-set. This follows because attaching a \(0\)-cell adds a copy of the regular right \(G\)-set \(\mathcal{A}(0,0)=G\), whereas attaching a \(1\)-cell does not change the value at \(0\), since \[\partial\mathcal{A}(0,1)=\mathcal{A}(0,1)=\{u\}.\] Thus, by transfinite induction, the value at \(0\) of any cellular presheaf is a free right \(G\)-set, and the same remains true for retracts, since an equivariant retract of a free \(G\)-set has trivial stabilizers. However \[\mathcal{A}(-,1)(0)=\mathcal{A}(0,1)=\{u\}\] is the singleton right \(G\)-set with the trivial action \(u\cdot g=u\). Since \(G=C_{2}\) is nontrivial, this right \(G\)-set is not free. Hence \(\mathcal{A}(-,1)\) is not cofibrant.
Definition 4. The c-direct category \(\mathcal{A}\) has cofibrant representables if for all \(x\in \operatorname{Ob}(\mathcal{A})\), the representable \(\mathcal{A}(-,x)\) is cofibrant.
Theorem 13. Assume that \(\mathcal{A}\) has cofibrant representables. Let \[\widehat A \quadand\quad\widehat B\] be two realization functors over \(I\) such that there exists a commutative diagram of \(\mathcal{M}^\mathcal{A}\) \[\begin{tikzcd}[column sep=7em,row sep=3em] A \arrow[r,"\mu"] \arrow[d,twoheadrightarrow,"\simeq"]& B \arrow[d,twoheadrightarrow,"\simeq"]\\ I \arrow[r,equal] & I \end{tikzcd}\] Then, for every \(\mathcal{A}\)-set \(K\), the morphism \(A\to B\) gives rise to a natural map of \(\mathcal{M}\) \[\widehat\mu_K:\widehat{A}(K) \longrightarrow \widehat{B}(K)\] which is a weak equivalence of \(\mathcal{M}\) between cofibrant objects whenever \(K\) is cofibrant.
Proof. First of all, note that for all cofibrant \(\mathcal{A}\)-sets \(K\), \(\widehat A(K)\) and \(\widehat B(K)\) are cofibrant in \(\mathcal{M}\). Assume that \(K\) is cellular. Since \(\operatorname{sk}_{<\lambda}\) preserves cellularity, we have a transfinite sequence of cellular maps \[\operatorname{sk}_{<\lambda}(K)\longrightarrow \operatorname{sk}_{<\lambda+1}(K),\] and thus for each ordinal \(\lambda\) a pushout diagram of \(\mathcal{A}\)-sets \[\begin{tikzcd}[column sep=3em,row sep=3em] \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\partial\mathcal{A}(-,x) \arrow[r] \arrow[d] & \operatorname{sk}_{<\lambda}(K) \arrow[d] \\ \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\mathcal{A}(-,x) \arrow[r] & \arrow[lu, phantom, "\ulcorner"{font=\Large}, pos=0]\operatorname{sk}_{<\lambda+1}(K) \end{tikzcd}\] with \(\mathcal{C}_\lambda(K)\subseteq \operatorname{Ob}(\mathcal{A}_{=\lambda})\) and such that the comparison map \[\varinjlim_{\mu<\lambda} \operatorname{sk}_{<\mu}(K) \xlongrightarrow{\cong} \operatorname{sk}_{<\lambda}(K)\] is an isomorphism for all limit ordinal \(\lambda\), and finally such that the comparison map \[\varinjlim_{\lambda} \operatorname{sk}_{<\lambda}(K) \xlongrightarrow{\cong} K\] is an isomorphism. We obtain the commutative cube of \(\mathcal{M}\) \[\begin{tikzcd}[column sep=1.5em,row sep=2em] \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\widehat A(\partial\mathcal{A}[x]) \arrow[dd,rightarrowtail]\arrow[rd,"\simeq"]\arrow[rr] && \widehat A(\operatorname{sk}_{<\lambda}(K)) \arrow[dd,rightarrowtail] \arrow[rd,"\simeq"]\\ & \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\widehat B(\partial\mathcal{A}[x]) \arrow[rr,crossing over] & & \widehat B(\operatorname{sk}_{<\lambda}(K)) \arrow[dd,rightarrowtail]\\ \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\widehat A(\mathcal{A}[x]) \arrow[rd,"\simeq"]\arrow[rr] && \arrow[lu, phantom, "\ulcorner"{font=\Large}, pos=0]\widehat A(\operatorname{sk}_{<\lambda+1}(K)) \arrow[rd,dashed,shorten >=0.6em] \\ & \displaystyle\coprod_{x\in \mathcal{C}_\lambda(K)}\widehat B(\mathcal{A}[x]) \arrow[rr] \arrow[uu,leftarrowtail,crossing over] && \arrow[lu, phantom, "\ulcorner"{font=\Large}, pos=0]\widehat B(\operatorname{sk}_{<\lambda+1}(K)) \end{tikzcd}\] We consider the following statement: the map \[\widehat A(\operatorname{sk}_{<\lambda}(L)) \longrightarrow \widehat B(\operatorname{sk}_{<\lambda}(L))\] is a weak equivalence for \(\lambda\@undefined 0\) for all \(\mathcal{A}\)-sets \(L\). We proceed by transfinite induction on \(\lambda\@undefined 0\). For \(\lambda=0\), there is nothing to prove. We assume the statement for \(\lambda\@undefined 0\). For all \(x\in \mathcal{C}_\lambda(K)\), the commutative diagram \[\begin{tikzcd}[column sep=7em,row sep=3em] A(x) \arrow[r,"\mu"] \arrow[d,twoheadrightarrow,"\simeq"]& B(x) \arrow[d,twoheadrightarrow,"\simeq"]\\ I(x) \arrow[r,equal] & I(x) \end{tikzcd}\] implies the weak equivalence \[\widehat A(\mathcal{A}(-,x))=A(x) \simeq B(x) = \widehat B(\mathcal{A}(-,x))\] by the two-out-of-three property. By definition of \(\partial\mathcal{A}(-,x)\), one has \[\operatorname{sk}_{<\lambda}(\partial\mathcal{A}(-,x)) = \partial\mathcal{A}(-,x)\quadfor all\quad x\in \mathcal{C}_\lambda(K).\] Thus the induction hypothesis implies the weak equivalence \[\widehat A(\partial\mathcal{A}(-,x)) \simeq \widehat B(\partial\mathcal{A}(-,x))\] for \(x\in \mathcal{C}_\lambda(K)\). We deduce the same fact for the ordinal \(\lambda+1\) using the commutative cube above and using the cube lemma ([9] or [10]). It remains the case where \(\lambda\) is a limit ordinal. We have two towers of cofibrations between cofibrant objects of \(\mathcal{M}\) which are objectwise weakly equivalent. The colimit coincides with the homotopy colimit since the two towers are Reedy cofibrant for the Reedy model structure of \(\mathcal{M}^\lambda\). This proves the limit ordinal case and completes the proof. ◻
Recall that the injective model structure is the unique model structure (if it exists) on a functor category \(\mathcal{M}^\mathcal{C}\) such that the weak equivalences and the cofibrations are the objectwise ones.
Lemma 14 (well known; e.g. [2]). Let \(\mathcal{C}\) be a small category. Let \(\mathcal{M}\) be a model category (not necessarily cofibrantly generated) such that both the projective model structure \((\mathcal{M}^\mathcal{C})_{proj}\) and the injective model structure \((\mathcal{M}^\mathcal{C})_{inj}\) exist. Then the identity of \(\mathcal{M}\) yields a left Quillen functor \[(\mathcal{M}^\mathcal{C})_{proj}\longrightarrow (\mathcal{M}^\mathcal{C})_{inj}.\] In particular, every projective cofibration is an injective cofibration.
Theorem 15. Assume that \(\mathcal{A}\) has cofibrant representables. Let \[\widehat A \quadand\quad\widehat B\] be two realization functors over \(I\). Then there exists a realization functor \(\widehat C\) over \(I\) and a commutative diagram of \(\mathcal{M}^\mathcal{A}\) \[\begin{tikzcd}[column sep=6em,row sep=3em] A \arrow[d,twoheadrightarrow,"\simeq"] &\arrow[l,"\mu"']C \arrow[r,"\nu"] \arrow[d,twoheadrightarrow,"\simeq"]& B \arrow[d,twoheadrightarrow,"\simeq"]\\ I& I \arrow[l,equal]\arrow[r,equal] & I \end{tikzcd}\] giving rise to two natural maps of \(\mathcal{M}\) \[\begin{tikzcd}[column sep=large] {\widehat{A}(K)} & {\widehat{C}(K)} \arrow[l, "{\widehat\mu_K}"'] \arrow[r, "{\widehat\nu_K}"] & {\widehat{B}(K)} \end{tikzcd}\] and, for every cofibrant \(\mathcal{A}\)-set \(K\), these two maps are weak equivalences of \(\mathcal{M}\) between cofibrant objects.
Proof. Consider the diagram of \(\mathcal{M}^\mathcal{A}\) \[\begin{tikzcd}[column sep=4em,row sep=4em] C \arrow[d,"\simeq"',twoheadrightarrow]\arrow[r,dashed,"\ell"] & B \arrow[d,twoheadrightarrow,"\simeq"]\\ A \arrow[r,twoheadrightarrow,"\simeq"] & I \end{tikzcd}\] where \(C\) is a cofibrant replacement of \(A\) in the projective model structure of \(\mathcal{M}^\mathcal{A}\). The lift \(\ell\) exists since \(C\) is cofibrant and since the map \(B\to I\) is a trivial projective fibration. By Theorem 12, the map \[\widehat C(\partial\mathcal{A}[-]) \longrightarrow \widehat C(\mathcal{A}[-])\] is a projective cofibration of \(\mathcal{M}^{\mathcal{A}_{=n}}\). By Lemma 14, the map \[\widehat C(\partial\mathcal{A}(-,x)) \longrightarrow \widehat C(\mathcal{A}(-,x))\] is therefore a cofibration of \(\mathcal{M}\) for all \(x\in \operatorname{Ob}(\mathcal{M}^{\mathcal{A}_{=n}})\). Thus \(\widehat C\) is a realization functor over \(I\). The proof is complete by Theorem 13. ◻
Let \[[0]=\{()\}, \qquad [n]=\{0<1\}^n \quad (n\@undefined 1),\] equipped with the product order. We write \[0_n=(0,\ldots,0), \qquad 1_n=(1,\ldots,1).\] Let \(\mathrm{PoSet}^+\) denote the category of posets and strictly increasing maps. For vertices \(x=(x_1,\ldots,x_n)\) and \(y=(y_1,\ldots,y_n)\) of \([n]\), put \[\overrightarrow d_1(x,y)= \begin{cases} \displaystyle\sum_{i=1}^n (y_i-x_i), & x\@undefined y,\\[5mm] +\infty, & \text{otherwise.} \end{cases}\] Thus \(\overrightarrow d_1(x,y)=1\) means that \(x<y\) and the two vertices differ in exactly one coordinate. The map \(\overrightarrow d_1:[0,1]^n\times [0,1]^n \longrightarrow [0,+\infty]\) yields a Lawvere metric on the topological \(n\)-cube \([0,1]^n\) by [2]. For \(n\@undefined 1\), \(1\@undefined i\@undefined n\), and \(\alpha\in\{0,1\}\), the coface map \[\delta_i^\alpha:[n-1]\longrightarrow[n]\] is defined by \[\delta_i^\alpha(x_1,\ldots,x_{n-1}) = (x_1,\ldots,x_{i-1},\alpha,x_i,\ldots,x_{n-1}).\]
Definition 5. The box category \(\square\) is the subcategory of \(\mathrm{PoSet}^+\) generated by the coface maps \(\delta_i^\alpha\). The presheaves on the box category are called precubical sets.
Definition 6. [1] A strictly increasing map \[f:[m]\longrightarrow[n]\] is cotransverse if it preserves adjacency in the following sense: \[\overrightarrow d_1(x,y)=1 \quad\Longrightarrow\quad \overrightarrow d_1(f(x),f(y))=1 \qquad (x,y\in[m]).\] The category \(\widehat{\square}_S\) has the cubes \([n]\), \(n\@undefined 0\), as objects and all cotransverse maps as morphisms: \[\widehat{\square}_S([m],[n]) = \{\text{cotransverse maps }[m]\to[n]\}.\] The presheaves on \(\widehat{\square}_S\) are called the (symmetric) transverse sets.
An immediate consequence of strict monotonicity is the following important fact:
Proposition 16. For \(n\@undefined 0\), every endomorphism of \([n]\) preserves \(0_n\) and \(1_n\): \[For all\quad n\@undefined 0,\quad \theta\in \widehat{\square}_S([n],[n]) \quad \Longrightarrow \quad \theta(0_n)=0_n \quad and \quad \theta(1_n)=1_n\,.\]
Proof. Indeed, let \[0_n=x_0<x_1<\cdots <x_n=1_n\] be a maximal chain in \([n]\). Since \(\theta\) is strictly increasing, the sequence \[\theta(x_0)<\theta(x_1)<\cdots <\theta(x_n)\] is a strict chain of length \(n\) in \([n]\). If \(r(y)=\overrightarrow d_1(0_n,y)\) denotes the rank of a vertex \(y\), then \[r(\theta(x_i))\@undefined r(\theta(x_0))+i\] for every \(i\). Taking \(i=n\), we get \[n\@undefined r(\theta(x_n))\@undefined r(\theta(x_0))+n.\] Thus \(r(\theta(x_0))=0\), and therefore \(\theta(0_n)=0_n\). The same inequalities also force \(r(\theta(x_n))=n\), and therefore \(\theta(1_n)=1_n\). ◻
Definition 7. [1] A category of cubes is a small category \(\mathcal{A}\) satisfying \[\square\subset \mathcal{A}\subset \widehat{\square}_S,\] where all three categories have the same objects \([n]\), \(n\@undefined 0\), and the inclusions are the identity on objects.
Proposition 17 (cotransverse–box factorization). [1] Every cotransverse map \[f:[m]\longrightarrow[n]\] factors uniquely as \[[m]\xrightarrow{\;\psi\;}[m]\xrightarrow{\;\delta\;}[n], \qquad f=\delta\psi,\] where \(\delta\in\square([m],[n])\) and \(\psi\in\widehat{\square}_S([m],[m])\) is a cotransverse endomorphism.
Definition 8. [3] A category of cubes \(\mathcal{A}\) is thick if the preceding cotransverse factorization is internal to \(\mathcal{A}\): whenever \[f:[m]\longrightarrow[n]\] is a morphism of \(\mathcal{A}\) and \[f=\delta\psi\] is its standard cotransverse factorization with \(\delta\in\square([m],[n])\) and \(\psi\in\widehat{\square}_S([m],[m])\), then \[\psi\in\mathcal{A}([m],[m]).\] Since \(\square\subset\mathcal{A}\), this implies that both factors \(\delta\) and \(\psi\) are morphisms of \(\mathcal{A}\).
Every category of cubes is c-direct with the degree function \(d([n])=n\) for \(n\@undefined 0\). Let \[i:\square\subset \mathcal{A}\] be the inclusion of the box category into a category of cubes. We denote by \[\omega_{\mathcal{A}}:\mathcal{A}^\mathrm{op}\mathrm{Set}\longrightarrow \square^\mathrm{op}\mathrm{Set}\] the restriction functor, and by \[\mathcal{L}_{\mathcal{A}}:\square^\mathrm{op}\mathrm{Set}\longrightarrow \mathcal{A}^\mathrm{op}\mathrm{Set}\] its left adjoint, that is, the left Kan extension along \(i^\mathrm{op}:\square^\mathrm{op}\to \mathcal{A}^\mathrm{op}\). The canonical map \[\mathcal{L}_{\mathcal{A}}(\square[n]) \longrightarrow \mathcal{A}[n]\] is an isomorphism of \(\mathcal{A}\)-sets for any category of cubes \(\mathcal{A}\) by [3].
Theorem 18. Let \(\mathcal{A}\) be a category of cubes. Then \(\mathcal{A}\) is thick if and only if, for every \(n\@undefined 1\), the canonical comparison map \[\mathcal{L}_{\mathcal{A}}(\partial\square[n])\longrightarrow \partial\mathcal{A}[n]\] is an isomorphism of \(\mathcal{A}\)-sets.
Proof. If \(\mathcal{A}\) is thick, the assertion follows from [3], which proves the isomorphism \[\mathcal{L}_{\mathcal{A}}(\partial\square[n])\cong \partial\mathcal{A}[n]\] for all \(n\@undefined 0\), hence in particular for all \(n\@undefined 1\). Conversely, suppose that \(\mathcal{A}\) is not thick. By the characterization of thickness recalled above, there exists a morphism \[f:[m]\longrightarrow [n]\] of \(\mathcal{A}\) whose unique factorization \[f=\delta\@undefined, \qquad \@undefined:[m]\to[m], \qquad \delta:[m]\to[n],\] with \(\delta\in\square([m],[n])\) and \(\@undefined\) cotransverse, satisfies \[\@undefined\notin \mathcal{A}([m],[m]).\] Choose such an \(f\) with \(n-m\) minimal. We first observe that \(m<n\). Indeed, if \(m=n\), then the cubical factor \(\delta\) is the identity of \([m]\), so that \(f=\@undefined\). Since \(f\in\mathcal{A}([m],[m])\), this would imply \(\@undefined\in\mathcal{A}([m],[m])\), a contradiction. Hence \(m<n\), and in particular \(n\@undefined 1\). Since \(m<n\), the morphism \(f\) defines an element \[f\in \partial\mathcal{A}[n]([m])=\mathcal{A}([m],[n]).\] We claim that this element is not in the image of \[\mathcal{L}_{\mathcal{A}}(\partial\square[n])([m])\longrightarrow \partial\mathcal{A}[n]([m]).\] Indeed, by the coend formula for the left Kan extension, \[\mathcal{L}_{\mathcal{A}}(\partial\square[n])([m]) \cong \int^{[q]\in\square} \partial\square[n]([q])\times \mathcal{A}([m],[q]).\] Since \(\partial\square[n]([q])\) is nonempty only for \(q<n\), every element of \(\mathcal{L}_{\mathcal{A}}(\partial\square[n])([m])\) is represented by a pair \[[m]\xrightarrow{g}[q]\xrightarrow{\alpha}[n]\] with \[g\in\mathcal{A}([m],[q]), \qquad \alpha\in\square([q],[n]), \qquad q<n.\] The comparison map sends the class of this pair to the composite \(\alpha g\in\mathcal{A}([m],[n])\). Suppose, for a contradiction, that \(f\) belongs to the image of the comparison map. Then there exist \(q<n\), a morphism \(g\in\mathcal{A}([m],[q])\), and a cubical map \(\alpha\in\square([q],[n])\), such that \[f=\alpha g.\] Factor \(g\) uniquely as \[g=\@undefined\psi, \qquad \psi:[m]\to[m], \qquad \@undefined:[m]\to[q],\] with \(\@undefined\in\square([m],[q])\) and \(\psi\) cotransverse. Then \[f=\alpha\@undefined\psi.\] Since \(\alpha\@undefined\in\square([m],[n])\), uniqueness of the cubical–cotransverse factorization of \(f\) gives \[\psi=\@undefined \qquad\text{and}\qquad \alpha\@undefined=\delta.\] Consequently \[g=\@undefined\@undefined.\] Thus \(g\in\mathcal{A}([m],[q])\) is another morphism of \(\mathcal{A}\) whose cotransverse endomorphism part is \(\@undefined\notin\mathcal{A}([m],[m])\). Since \(q<n\), we have \[q-m<n-m,\] contradicting the minimality of \(f\). Therefore \(f\) is not in the image of \[\mathcal{L}_{\mathcal{A}}(\partial\square[n])([m])\longrightarrow \partial\mathcal{A}[n]([m]).\] The comparison map is not surjective at \([m]\), and hence it is not an isomorphism of \(\mathcal{A}\)-sets. Thus, if the comparison map is an isomorphism for all \(n\@undefined 1\), then \(\mathcal{A}\) must be thick. ◻
Theorem 19. Let \(\mathcal{A}\) be a category of cubes. If \(\mathcal{A}\) is thick, then \(\mathcal{A}(-,[n])\) is cofibrant for every \(n\@undefined 0\).
Proof. We prove, by induction on \(n\), that if \(\mathcal{A}\) is thick, then \(\mathcal{A}(-,[n])\) is cofibrant for all \(n\@undefined 0\). For \(n=0\), the boundary \(\partial\mathcal{A}(-,[0])\) is empty, so \(\mathcal{A}(-,[0])\) is obtained from the initial presheaf by attaching one \(0\)-cell. Assume now that \(\mathcal{A}(-,[m])\) is cofibrant for all \(m<n\). Since \(\partial\square[n]\) is obtained in \(\square^\mathrm{op}\mathrm{Set}\) by attaching the ordinary lower-dimensional cubical faces \(\square[m]\) with \(m<n\), its image \(\mathcal{L}_{\mathcal{A}}(\partial\square[n])\) is obtained from the presheaves \(\mathcal{A}(-,[m])\), \(m<n\), by the corresponding finite colimit construction. By the induction hypothesis, these presheaves are cofibrant, and cofibrant presheaves are closed under such cellular colimits. Since \(\mathcal{A}\) is thick, we have \[\mathcal{L}_{\mathcal{A}}(\partial\square[n]) \cong \partial\mathcal{A}(-,[n])\] by Theorem 18. Hence \(\partial\mathcal{A}(-,[n])\) is cofibrant. Therefore \(\mathcal{A}(-,[n])\) is cofibrant, because it is obtained from \(\partial\mathcal{A}(-,[n])\) by attaching one \(n\)-cell along the generating cofibration \[\partial\mathcal{A}(-,[n]) \subseteq \mathcal{A}(-,[n]).\] ◻
Remark 20. The converse of the preceding statement is false in general. Let \[\gamma^-:[2]\longrightarrow [2], \qquad \gamma^-(x,y)=(\min(x,y),\max(x,y)),\] and let \[u=\delta^0_3\gamma^-:[2]\longrightarrow [3].\] Let \(\mathcal{A}\) be the subcategory of \(\widehat{\square}_S\) generated by the box category \(\square\) and by the single morphism \(u\). Then \(\mathcal{A}\) is a category of cubes. It is not thick: indeed, the standard cotransverse factorization of \(u\) is \[[2]\xrightarrow{\gamma^-}[2]\xrightarrow{\delta^0_3}[3],\] but \(\gamma^-\notin \mathcal{A}([2],[2])\). To see this, observe that every nonidentity generator of \(\mathcal{A}\) strictly raises degree. Hence every nonidentity morphism of \(\mathcal{A}\) strictly raises degree, and in particular \(\mathcal{A}([2],[2])=\{\operatorname{id}_{[2]}\}\). Nevertheless, all representable presheaves of \(\mathcal{A}\) are cofibrant. Indeed, the same observation shows that \(\mathcal{A}\) is a direct category for the degree \(d([n])=n\). For a direct category, each representable is cellular: the presheaf \(\mathcal{A}(-,[n])\) is obtained by attaching, in increasing degree \(p\@undefined n\), one \(p\)-cell for each morphism \([p]\to [n]\) of \(\mathcal{A}\). The boundary of such a cell only involves morphisms from objects of degree \(<p\), and therefore has already been attached at earlier stages. Thus \(\mathcal{A}(-,[n])\) is cellular, hence cofibrant, for every \(n\@undefined 0\), although \(\mathcal{A}\) is not thick.
Notation 21. A thick category of cubes \(\mathcal{A}\) is fixed for the rest of the paper.
Let \(\mathcal{A}^-\) be the subcategory of \(\mathcal{A}\) with the same objects and whose morphisms are the maps \[f:[m]\longrightarrow [n] \qquad such that\qquad f(0_m)=0_n.\] The small category \(\mathcal{A}^-\) is clearly c-direct. Let \(\square^{-}\) denote the wide subcategory of \(\square\) consisting of the cubical maps preserving the initial vertex. Equivalently, \(\square^{-}\) is generated by the coface maps \(\delta_i^0\). Thus a morphism \(\delta:[m]\to[n]\) of \(\square\) belongs to \(\square^{-}\) if and only if \(\delta(0_m)=0_n\). Let \[j:\square^{-}\longrightarrow \mathcal{A}^{-}\] be the inclusion. We denote by \[\mathcal{L}_{\mathcal{A}}^{-}:(\square^{-})^{\mathrm{op}}\mathrm{Set} \longrightarrow (\mathcal{A}^{-})^{\mathrm{op}}\mathrm{Set}\] the left Kan extension along \(j^{\mathrm{op}}\). Thus \[\mathcal{L}_{\mathcal{A}}^{-}(\square^{-}(-,[n])) \cong \mathcal{A}^{-}(-,[n])\] for every \(n\@undefined 0\).
Lemma 22. The cotransverse–box factorization of every morphism of \(\mathcal{A}^{-}\) is internal to \(\mathcal{A}^{-}\) after replacing \(\square\) by \(\square^{-}\). More explicitly, if \[f:[m]\longrightarrow[n]\] is a morphism of \(\mathcal{A}^{-}\), and if \[f=\delta\psi\] is its cotransverse–box factorization, with \[\psi:[m]\longrightarrow[m] \qquad\text{and}\qquad \delta:[m]\longrightarrow[n]\] where \(\delta\in\square([m],[n])\), then \[\delta\in\square^{-}([m],[n]) \qquad\text{and}\qquad \psi\in\mathcal{A}^{-}([m],[m]).\]
Proof. Since \(f\) belongs to \(\mathcal{A}^{-}\), one has \(f(0_m)=0_n\). Hence \[\delta(\psi(0_m))=0_n.\] Since \(0_m\@undefined\psi(0_m)\), we have \[\delta(0_m)\@undefined\delta(\psi(0_m))=0_n.\] As \(0_n\) is the least vertex of \([n]\), this implies \[\delta(0_m)=0_n.\] Thus \(\delta\in\square^{-}([m],[n])\). Moreover, if \(\psi(0_m)\neq 0_m\), then \(0_m<\psi(0_m)\), and the strict monotonicity of \(\delta\) gives \[\delta(0_m)<\delta(\psi(0_m))=0_n,\] which is impossible. Therefore \(\psi(0_m)=0_m\). Since \(\mathcal{A}\) is thick, the cotransverse endomorphism \(\psi\) belongs to \(\mathcal{A}([m],[m])\). Together with \(\psi(0_m)=0_m\), this gives \[\psi\in\mathcal{A}^{-}([m],[m]).\] ◻
For \(n\@undefined 0\), let \[\partial\square^{-}[n]\subseteq \square^{-}(-,[n])\] be the strict lower-degree part of the representable presheaf, namely \[\partial\square^{-}[n]([m]) = \begin{cases} \square^{-}([m],[n]), & m<n,\\ \varnothing, & m\@undefined n. \end{cases}\] Likewise, \(\partial\mathcal{A}^{-}[n]\) denotes the strict lower-degree part of \(\mathcal{A}^{-}(-,[n])\).
Lemma 23. For every \(n\@undefined 0\), the canonical comparison map \[\mathcal{L}_{\mathcal{A}}^{-}(\partial\square^{-}[n]) \longrightarrow \partial\mathcal{A}^{-}[n]\] is an isomorphism of \(\mathcal{A}^{-}\)-sets.
Proof. We evaluate the comparison map at \([r]\). By the coend formula for the left Kan extension, one has \[\mathcal{L}_{\mathcal{A}}^{-}(\partial\square^{-}[n])([r]) \cong \int^{[q]\in\square^{-}} \partial\square^{-}[n]([q])\times \mathcal{A}^{-}([r],[q]).\] Since \(\partial\square^{-}[n]([q])\) is nonempty only for \(q<n\), every element is represented by a pair \[[r]\xrightarrow{g}[q]\xrightarrow{\alpha}[n],\] with \[g\in\mathcal{A}^{-}([r],[q]), \qquad \alpha\in\square^{-}([q],[n]), \qquad q<n.\] The comparison map sends the class of such a pair to the composite \[\alpha g\in \mathcal{A}^{-}([r],[n]).\] Since \(g\) exists only if \(r\@undefined q\), this composite belongs to \(\partial\mathcal{A}^{-}[n]([r])\). We first prove surjectivity. Let \[f\in \partial\mathcal{A}^{-}[n]([r]).\] Thus \(r<n\), and \(f:[r]\to[n]\) is a morphism of \(\mathcal{A}^{-}\). Write the standard cotransverse–box factorization of \(f\) as \[f=\delta\psi,\] with \(\delta\in\square([r],[n])\) and \(\psi:[r]\to[r]\) cotransverse. By Lemma 22, \[\delta\in\square^{-}([r],[n]) \qquad\text{and}\qquad \psi\in\mathcal{A}^{-}([r],[r]).\] Since \(r<n\), the morphism \(\delta\) is an element of \(\partial\square^{-}[n]([r])\). Hence \(f\) is the image of the class represented by \[[r]\xrightarrow{\psi}[r]\xrightarrow{\delta}[n].\] We now prove injectivity. Suppose that two representatives \[[r]\xrightarrow{g}[q]\xrightarrow{\alpha}[n], \qquad [r]\xrightarrow{h}[q']\xrightarrow{\beta}[n],\] with \(\alpha,\beta\) in \(\square^{-}\), have the same image: \[\alpha g=\beta h.\] Factor \(g\) and \(h\) by their cotransverse–box factorizations: \[g=\@undefined\psi, \qquad h=\eta\chi,\] with \[\@undefined\in\square^{-}([r],[q]), \quad \eta\in\square^{-}([r],[q']), \quad \psi,\chi\in\mathcal{A}^{-}([r],[r]),\] again by the preceding lemma. In the coend, the defining relations give \[[\alpha,g]=[\alpha\@undefined,\psi], \qquad [\beta,h]=[\beta\eta,\chi].\] The equality \(\alpha g=\beta h\) becomes \[\alpha\@undefined\psi=\beta\eta\chi.\] Here \(\alpha\@undefined\) and \(\beta\eta\) are morphisms of \(\square^{-}\). By uniqueness of the standard cotransverse–box factorization, we obtain \[\alpha\@undefined=\beta\eta \qquad\text{and}\qquad \psi=\chi.\] Therefore \[[\alpha,g]=[\alpha\@undefined,\psi] = [\beta\eta,\chi]=[\beta,h].\] This proves injectivity, and hence the comparison map is an isomorphism. ◻
Theorem 24. For every \(n\@undefined 0\), the representable presheaf \[\mathcal{A}^{-}(-,[n]) : (\mathcal{A}^{-})^{\mathrm{op}}\longrightarrow\mathrm{Set}\] is cofibrant. Equivalently, \(\mathcal{A}^{-}\) has cofibrant representables.
Proof. We prove, by induction on \(n\), that \(\mathcal{A}^{-}(-,[n])\) is cellular, hence cofibrant. For \(n=0\), the boundary \(\partial\mathcal{A}^{-}[0]\) is empty. Therefore \(\mathcal{A}^{-}(-,[0])\) is obtained from the initial presheaf by attaching one \(0\)-cell. Assume now that \(\mathcal{A}^{-}(-,[m])\) is cellular for every \(m<n\). The presheaf \(\partial\square^-[n]\) is obtained in \((\square^-)^{\mathrm{op}}\mathrm{Set}\) by the finite cellular construction obtained by attaching the proper initial-vertex-preserving faces of \([n]\). Equivalently, it is built using only cells \(\square^-(-,[m])\) with \(m<n\), attached along their boundaries \(\partial\square^-[m]\). Applying the colimit-preserving functor \(\mathcal{L}_{\mathcal{A}}^{-}\), and using the preceding lemma for all \(m\@undefined n\), this gives a cellular construction of \[\mathcal{L}_{\mathcal{A}}^{-}(\partial\square^{-}[n]) \cong \partial\mathcal{A}^{-}[n]\] using only the generating cofibrations \[\partial\mathcal{A}^{-}[m]\subseteq \mathcal{A}^{-}(-,[m]), \qquad m<n.\] Hence \(\partial\mathcal{A}^{-}[n]\) is cellular. Finally, \(\mathcal{A}^{-}(-,[n])\) is obtained from \(\partial\mathcal{A}^{-}[n]\) by attaching one \(n\)-cell along the generating cofibration \[\partial\mathcal{A}^{-}[n]\subseteq \mathcal{A}^{-}(-,[n]).\] Therefore \(\mathcal{A}^{-}(-,[n])\) is cellular. This completes the induction. ◻
A directed path in \([0,1]^n\) starting from \(0_n\) is a continuous map \(\gamma=(\gamma_1,\dots,\gamma_n)\) from \([0,\ell]\) to \([0,1]^n\) such that each \(\gamma_i\) is nondecreasing. It is natural if (see [11] and [6]) \[t=\overrightarrow d_1(0_n,\gamma(t))\qquad for all\qquad t\in [0,\ell].\] Let \(n\@undefined 1\). A short natural directed path of \([0,1]^n\) is a natural directed path \(\@undefined:[0,\@undefined]\to [0,1]^n\) such that \(\@undefined(0)=0_n\) and \(0<\@undefined<1\). The real number \(\@undefined\) is called the natural length of the short natural directed path. The set of short natural directed paths of \([0,1]^n\) of natural length \(\@undefined\in ]0,1[\) is denoted by \(N_n(\@undefined)\). We set \(N_0(\@undefined)=\varnothing\). For \(n\@undefined 1\), by [7], the set \(N_n(\@undefined)\) equipped with the compact-open topology is \(\Delta\)-generated, \(\Delta\)-Hausdorff, metrizable, contractible, compact and sequentially compact.
Definition 9. [2] Let \(f=(f_1,\dots,f_n):[n]\to [n]\) be a cotransverse map. Let \[\operatorname{T}(f):[0,1]^n\to [0,1]^n\] be the continuous map defined by \[\operatorname{T}(f)(x_1,\dots,x_n) = (\operatorname{T}(f)_1(x_1,\dots,x_n),\dots,\operatorname{T}(f)_n(x_1,\dots,x_n))\] with \[\operatorname{T}(f)_i(x_1,\dots,x_n) = \max_{(\@undefined_1,\dots,\@undefined_n)\in f_i^{-1}(1)} \min \{x_k\mid \@undefined_k=1\}\] for all \(1\@undefined i\@undefined n\).
Notation 25. [2] For \(\delta_i^\alpha:[n-1]\to [n] \in \square\), let \[\operatorname{T}(\delta_i^\alpha)= \begin{cases} [0,1]^{n-1} \to [0,1]^n\\ (\@undefined_1, \dots, \@undefined_{n-1})\mapsto (\@undefined_1,\dots, \@undefined_{i-1}, \alpha, \@undefined_i, \dots, \@undefined_{n-1}) \end{cases}\] for all \(n\@undefined 1\) and \(\alpha\in\{0,1\}\).
If \[f:[m]\longrightarrow [n]\] is a morphism of \(\mathcal{A}\), we denote by \[\operatorname{T}(f):[0,1]^m\longrightarrow [0,1]^n\] the associated continuous map of topological cubes, as defined above. Together with the identities \(\operatorname{T}(\operatorname{id}_{[n]})=\operatorname{id}_{[0,1]^n}\) and \(\operatorname{T}(gf)=\operatorname{T}(g)\operatorname{T}(f)\) [2], this defines a functor from \(\mathcal{A}\) to \(\mathbf{Top}\), sending \([n]\) to \([0,1]^n\), where, in this paper, \(\mathbf{Top}\) denotes either the category of \(\Delta\)-generated spaces or the category of \(\Delta\)-Hausdorff \(\Delta\)-generated spaces.
Definition 10. Let \(K\) be an \(\mathcal{A}\)-set. Let \[|K|_{\mathrm{geom}} = \int^{[n]\in \mathcal{A}} K_n\cdot |\mathcal{A}[n]|_{\mathrm{geom}}\,.\] This gives rise to a colimit-preserving functor \(|-|_{\mathrm{geom}}:\mathcal{A}^{\mathrm{op}}\mathrm{Set}\to \mathbf{Top}\) called the geometric realization.
Proposition 26. Let \(\@undefined\in ]0,1[\). For any \(u\in N_m(\@undefined)\) and any morphism \(f:[m]\longrightarrow [n]\) of \(\mathcal{A}^-\), one has \(T(f)u \in N_n(\@undefined)\). Equivalently, \[\operatorname{T}(f)(N_m(\@undefined))\subseteq N_n(\@undefined).\]
Proof. By [2], for any cotransverse map \(f:[m]\longrightarrow [n]\), the continuous map \[\operatorname{T}(f):[0,1]^m\longrightarrow [0,1]^n\] induces a map of Lawvere metric spaces which is also quasi-isometric for the Lawvere distance \(\overrightarrow d_1\). Thus for any \(u\in N_m(\@undefined)\), one has for all \(t\in[0,\@undefined]\) \[t=\overrightarrow d_1(0_m,u(t)) = \overrightarrow d_1(0_n,\operatorname{T}(f)(u(t))).\] Using [2], we deduce that \(\operatorname{T}(f)u\) is natural. ◻
Hence we obtain a functor \[N(\@undefined):\mathcal{A}^-\longrightarrow \mathbf{Top}, \qquad [n]\longmapsto N_n(\@undefined).\] Let \[\rho_\mathcal{A}:\mathcal{A}^{\mathrm{op}}\mathrm{Set}\longrightarrow(\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\] denote the restriction functor along the inclusion \(\mathcal{A}^-\subseteq\mathcal{A}\). Since colimits of presheaves are computed pointwise, \(\rho_\mathcal{A}\) preserves all colimits.
Lemma 27. The restriction functor \[\rho_\mathcal{A}:\mathcal{A}^{\mathrm{op}}\mathrm{Set}\longrightarrow(\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\] preserves cofibrant presheaves.
Proof. Since \(\rho_\mathcal{A}\) preserves colimits and retracts, it suffices to prove that it sends each generating cofibration \[\partial\mathcal{A}[n]\subseteq \mathcal{A}(-,[n])\] to a cofibration of \((\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\). Consider the commutative square \[\begin{tikzcd}[column sep=3em,row sep=3em] \partial\mathcal{A}^-[n] \arrow[r] \arrow[d] & \rho_\mathcal{A}(\partial\mathcal{A}[n]) \arrow[d] \\ \mathcal{A}^-(-,[n]) \arrow[r] & \rho_\mathcal{A}(\mathcal{A}(-,[n])) . \end{tikzcd}\] We claim that this square is a pushout. Evaluating at \([q]\), there are three cases.
If \(q<n\), then both \(\rho_\mathcal{A}(\partial\mathcal{A}[n])([q])\) and \(\rho_\mathcal{A}(\mathcal{A}(-,[n]))([q])\) are equal to \(\mathcal{A}([q],[n])\), so the assertion is immediate.
If \(q>n\), then all terms are empty, since \(\mathcal{A}\) is c-direct.
If \(q=n\), then \[\partial\mathcal{A}[n]([n])=\varnothing \qquad\text{and}\qquad \partial\mathcal{A}^-[n]([n])=\varnothing.\] Moreover \[\rho_\mathcal{A}(\mathcal{A}(-,[n]))([n])=\mathcal{A}([n],[n])\] and \[\mathcal{A}^-(-,[n])([n])=\mathcal{A}^-([n],[n]).\] These two sets are equal since every endomorphism of \([n]\) preserves \(0_n\) by Proposition 16. Hence the square is a pushout objectwise, and therefore a pushout of presheaves.
Thus \(\rho_\mathcal{A}(\partial\mathcal{A}[n]\subseteq\mathcal{A}(-,[n]))\) is a pushout of the generating cofibration \[\partial\mathcal{A}^-[n]\subseteq\mathcal{A}^-(-,[n]).\] It is therefore a cofibration. Hence \(\rho_\mathcal{A}\) sends cellular \(\mathcal{A}\)-sets to cellular \(\mathcal{A}^-\)-sets, and since it also preserves retracts, it sends cofibrant \(\mathcal{A}\)-sets to cofibrant \(\mathcal{A}^-\)-sets. ◻
Define \[\widehat N_\@undefined:(\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\longrightarrow\mathbf{Top}\] by \[\widehat N_\@undefined(H) = \int^{[n]\in\mathcal{A}^-} H([n])\cdot N_n(\@undefined).\]
Definition 11. For an \(\mathcal{A}\)-set \(K\), set \[B^-_\@undefined(K) = \widehat N_\@undefined(\rho_\mathcal{A}K).\] Equivalently, \[B^-_\@undefined(K) = \int^{[n]\in\mathcal{A}^-} K_n\cdot N_n(\@undefined).\] The topological space \(B^-_\@undefined(K)\) is called the \(\@undefined\)-branching space of \(K\).
Theorem 28. Let \(\mathbf{1}_\mathcal{A}\in \mathbf{Top}^{\mathcal{A}^-}\) be the unique functor such that \[\mathbf{1}_\mathcal{A}([0])=\varnothing\quad and\quad \mathbf{1}_\mathcal{A}([n])=\{n\} \quad for \quad n\@undefined 1.\] Let \(\@undefined\in]0,1[\). The functor \[\widehat N_\@undefined:(\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\longrightarrow\mathbf{Top}\] is a realization functor over \(\mathbf{1}_\mathcal{A}\) for \(\mathbf{Top}\) equipped with the m-model structure or with the h-model structure.
Proof. For \(n=0\), the Yoneda lemma gives \[\widehat N_\@undefined(\mathcal{A}^- [0])\cong N_0(\@undefined)=\varnothing =1_{\mathcal{A}}([0]),\] so the canonical map \[\widehat N_\@undefined(\mathcal{A}^- [0])\longrightarrow 1_{\mathcal{A}}([0])\] is an isomorphism. For \(n\@undefined 1\), the Yoneda lemma gives \[\widehat N_\@undefined(\mathcal{A}^-[n])\cong N_n(\@undefined).\] The space \(N_n(\@undefined)\) is contractible. Hence the canonical map \[\widehat N_\@undefined(\mathcal{A}^-[n])\longrightarrow \mathbf{1}_\mathcal{A}([n])\] is a trivial h-fibration and a trivial m-fibration for \(n\@undefined 1\). Together with the case \(n=0\), this proves condition (2) of Definition 2. By Theorem 24, the category \(\mathcal{A}^-\) has cofibrant representables. It remains to verify the cofibration condition. By Lemma 23, \[\mathcal{L}_{\mathcal{A}}^-(\partial\square^-[n])\cong \partial\mathcal{A}^-[n].\] For \(n=0\), both sides are empty and the corresponding map is the identity of \(\varnothing\). Assume now \(n\@undefined 1\). Therefore \(\widehat N_\@undefined(\partial\mathcal{A}^-[n])\) is identified with the corresponding boundary subspace of \(N_n(\@undefined)\). The inclusion \[\widehat N_\@undefined(\partial\mathcal{A}^-[n]) \subseteq \widehat N_\@undefined(\mathcal{A}^-[n])\] is an m-cofibration by [7], and therefore also an h-cofibration. Thus \(\widehat N_\@undefined\) is a realization functor over \(\mathbf{1}_\mathcal{A}\). ◻
Corollary 29. Let \(\@undefined,\@undefined'\in]0,1[\). There exists a zigzag of natural transformations \[B^-_\@undefined\;\longleftarrow\; \bullet \;\longrightarrow\; B^-_{\@undefined'}\] such that, for every cofibrant \(\mathcal{A}\)-set \(K\), the two maps are homotopy equivalences between m-cofibrant spaces.
Proof. By Theorem 28 and Theorem 15 applied to the c-direct category \(\mathcal{A}^-\), the corresponding statement holds for every cofibrant \(\mathcal{A}^-\)-set. If \(K\) is cofibrant as an \(\mathcal{A}\)-set, then \(\rho_\mathcal{A}K\) is cofibrant as an \(\mathcal{A}^-\)-set by Lemma 27. Since \[B^-_\@undefined(K)=\widehat N_\@undefined(\rho_\mathcal{A}K),\] the result follows. Finally, weak homotopy equivalences between m-cofibrant spaces are homotopy equivalences by [12]. ◻
Corollary 29 applies in particular to every free \(\mathcal{A}\)-set of the form \(\mathcal{L}_{\mathcal{A}}(K)\), where \(K\) is a precubical set. Indeed, every precubical set is cofibrant as a \(\square\)-set, and, since \(\mathcal{A}\) is thick, the functor \(\mathcal{L}_{\mathcal{A}}\) sends the generating cofibrations \[\partial\square[n]\subseteq \square[n]\] to \[\partial\mathcal{A}[n]\subseteq \mathcal{A}[n].\] Thus \(\mathcal{L}_{\mathcal{A}}(K)\) is cofibrant as an \(\mathcal{A}\)-set.
In this section, \(\mathcal{A}\) is a thick category of cubes. For an \(\mathcal{A}\)-set \(K\), write \[K_n=K([n])\] for the set of \(n\)-cubes of \(K\). The initial vertex of an \(n\)-cube \(c\) of an \(\mathcal{A}\)-set \(K\) is the vertex \[c^-=c(0_n).\] When \(c\in K_0\), one has \(c^-=c\). For \(\alpha\in K_0\), let \[\mathcal{C}^-_\alpha(K)=\{c\in K\mid \dim(c)\@undefined 1 \,and\, c^-=\alpha\}.\]
Definition 12. [7] Let \(\@undefined\in ]0,1[\). Let \(K\) be an \(\mathcal{A}\)-set. The naive \(\@undefined\)-branching space \(\mathcal{P}^-_\mathcal{A}(K,\@undefined)\) of \(K\) is the space \[\mathcal{P}^-_\mathcal{A}(K,\@undefined) = \coprod_{\alpha\in K^0} (\mathcal{P}^-_\mathcal{A})_\alpha(K,\@undefined)\] where \[(\mathcal{P}^-_\mathcal{A})_\alpha(K,\@undefined) = \big\{|c|_{\mathrm{geom}}\circ\@undefined\mid c\in \mathcal{C}^-_\alpha(K) and \@undefined\in N_{\dim(c)}(\@undefined)\big\}.\] It is equipped with the \(\Delta\)-Kelleyfication of the compact-open topology.
Proposition 30. Let \(\@undefined\in ]0,1[\). The assignment \(K\mapsto \mathcal{P}^-_\mathcal{A}(K,\@undefined)\) induces a functor from \(\mathcal{A}\)-sets to topological spaces.
Proof. Let \(f:K\to L\) be a map of \(\mathcal{A}\)-sets. For all \(c\in\mathcal{C}^-_\alpha(K)\), \(f(c)\in \mathcal{C}^-_{f(\alpha)}(L)\) and \(\dim(c)=\dim(f(c))\). This proves the claim, since \(0<\@undefined<1\). ◻
Let \[\begin{align} & c:[2]\longrightarrow [2],\qquad c(x_1,x_2)=(x_2,x_1),\\[1mm] & \gamma^+:[2]\longrightarrow [2], \qquad \gamma^+(x_1,x_2)=(\max(x_1,x_2),\min(x_1,x_2)),\\[1mm] & \gamma^-:[2]\longrightarrow [2], \qquad \gamma^-(x_1,x_2)=(\min(x_1,x_2),\max(x_1,x_2)).\\ \end{align}\] Recall that the rank of a vertex \(u=(u_1,\dots,u_n) \in [n]\) is \[r(u)=u_1+\cdots+u_n.\] For \(1\@undefined i\@undefined n\), let \[e_i=(0,\dots,0,1,0,\dots,0)\] be the vertex with the unique nonzero coordinate in position \(i\). A \(\square\)-map \(h:[2]\to[n]\) satisfying \(h(0,0)=0_n\) is necessarily of the form \[h(x,y)= (0,\dots,0,x,0,\dots,0,y,0, \dots,0)\] for a unique pair \(1\@undefined p<q\@undefined n\). Hence \[h(1,0)=e_p, \qquad h(0,1)=e_q, \qquad p<q.\]
Lemma 31. Let \(f:[n]\longrightarrow [n]\) be a cotransverse map which is not the identity. Then there exists a \(\square\)-map \[g:[2]\longrightarrow [n]\] such that the composite \[fg:[2]\longrightarrow [n]\] does not belong to the box category.
Proof. Since \(f\) is cotransverse, it is order-preserving and preserves the rank. In particular, it sends vertices of rank \(1\) to vertices of rank \(1\). Therefore there is a unique map \[\alpha:\{1,\dots,n\}\longrightarrow \{1,\dots,n\}\] such that \[f(e_i)=e_{\alpha(i)} \qquad (1\@undefined i\@undefined n).\] We first prove that \(\alpha\neq\operatorname{id}_{\{1,\dots,n\}}\). Indeed, suppose that \(\alpha=\operatorname{id}_{\{1,\dots,n\}}\). Let \(u\in[n]\), and write \[\operatorname{supp}(u)=\{i\mid u_i=1\}.\] For every \(i\in\operatorname{supp}(u)\), one has \(e_i\@undefined u\). Since \(f\) is order-preserving, \[e_i=f(e_i)\@undefined f(u).\] Thus \[\operatorname{supp}(u)\subseteq \operatorname{supp}(f(u)).\] Since \(f\) preserves the rank, we also have \[|f(u)|=|u|.\] Equivalently, \[\#\operatorname{supp}(f(u))=\#\operatorname{supp}(u).\] The inclusion of supports is therefore an equality. Hence \(\operatorname{supp}(f(u))=\operatorname{supp}(u)\), and so \(f(u)=u\). Since this holds for every vertex \(u\), we obtain \(f=\operatorname{id}_{[n]}\), contradicting the hypothesis. Consequently, \[\alpha\neq\operatorname{id}_{\{1,\dots,n\}}.\] Since \(\alpha\) is a non-identity endomap of the finite ordered set \(\{1,\dots,n\}\), there exist \(i<j\) such that \[\alpha(i)\@undefined\alpha(j).\] Indeed, if \(\alpha(i)<\alpha(j)\) for every \(i<j\), then \(\alpha\) would be strictly increasing. A strictly increasing endomap of \(\{1,\dots,n\}\) is necessarily the identity. Choose such a pair \(i<j\), and let \[g=g_{ij}:[2]\longrightarrow[n]\] be the \(\square\)-map defined by putting the first variable in the \(i\)-th coordinate and the second variable in the \(j\)-th coordinate: \[g(x,y)=(0,\dots,0,x,0,\dots,0,y,0,\dots,0).\] Then \[fg(0,0)=f(0_n)=0_n,\] and \[fg(1,0)=f(e_i)=e_{\alpha(i)}, \qquad fg(0,1)=f(e_j)=e_{\alpha(j)}.\] Suppose, for a contradiction, that \[fg\in\square([2],[n]).\] Since \(fg(0,0)=0_n\), the preceding description of \(\square\)-maps implies that there exist indices \(p<q\) such that \[fg(1,0)=e_p, \qquad fg(0,1)=e_q.\] Therefore \[p=\alpha(i), \qquad q=\alpha(j),\] and hence \[\alpha(i)<\alpha(j).\] This contradicts the choice of \(i<j\) with \(\alpha(i)\@undefined\alpha(j)\). Thus \[fg\notin\square([2],[n]),\] as required. ◻
Lemma 32. Assume that \(\square\subsetneq \mathcal{A}\). Then \[\{c,\gamma^+,\gamma^-\} \cap \mathcal{A}([2],[2]) \neq \varnothing.\]
Proof. Let \(f:[m]\to [n]\in \mathcal{A}\backslash \square\). One always has \[\mathcal{A}([0],[n]) = \square([0],[n]),\quad \mathcal{A}([1],[n]) = \square([1],[n])\quadfor all\quad n\@undefined 0\,.\] By the preceding observation, one must have \(m\@undefined 2\). If \(n=2\), then necessarily \(m=2\), and the conclusion follows directly. Assume that \(2\@undefined m<n\). Since \(\mathcal{A}\) is thick, \(f\) factors as a composite \[f:[m]\xlongrightarrow{\@undefined\in\mathcal{A}} [m] \xlongrightarrow{\delta\in\square} [n]\,.\] Since \(f\notin \square\), one deduces \(\@undefined\notin \square\). Thus we can suppose without loss of generality that \(m=n\). By Lemma 31, there exists a \(\square\)-map \(g:[2]\to [m]\) such that \[fg\notin \square([2],[n])\,.\] Using thickness again, write \(fg\) as a composite \[fg:[2]\xlongrightarrow{\@undefined\in\mathcal{A}} [2] \xlongrightarrow{\delta\in\square} [m]\,.\] Then \(\@undefined\notin\square\). Thus \[\@undefined\in \{c,\gamma^+,\gamma^-\}\,.\] This completes the proof. ◻
Let \(\theta\in \{c,\gamma^+,\gamma^-\}\). Consider the following coequalizer in \(\mathcal{A}^{\mathrm{op}}\mathrm{Set}\): \[\begin{tikzcd}[column sep=3em] {\mathcal{A}[2]} \arrow[r, shift left=1ex, "\operatorname{id}"] \arrow[r, shift right=1ex, "{\mathcal{A}(-,\theta)}"'] & {\mathcal{A}[2]} \arrow[r, "q"] & {Q_\theta} \end{tikzcd}\] In the geometric realization, the quotient \(Q_\theta\) identifies each point \(z\in[0,1]^2\) with \(\theta(z)\).
Lemma 33. Let \(0<\@undefined<1\), and put \(a=\@undefined/2\). For each \(\theta\in\{c,\gamma^+,\gamma^-\}\), there exist two natural directed paths \[u,v:[0,\@undefined]\longrightarrow [0,1]^2\] of natural length \(\@undefined\) such that:
\(q u=q v\) in \(\mathcal{P}^-_\mathcal{A}(Q_\theta,\@undefined)\);
\(u\) and \(v\) are not equivalent for the equivalence relation on \(\mathcal{P}^-_\mathcal{A}(\mathcal{A}[2],\@undefined)\) generated by \[w\sim \theta w .\]
Proof. First suppose that \(\theta=c\) or \(\theta=\gamma^+\). Define \[u(t)= \begin{cases} (t,a), & 0\@undefined t\@undefined a,\\ (a,t), & a\@undefined t\@undefined\@undefined, \end{cases}\] and \[v(t)= \begin{cases} u(t), & 0\@undefined t\@undefined a,\\ \theta u(t), & a\@undefined t\@undefined\@undefined. \end{cases}\] Since \(u(a)=(a,a)\), and since \((a,a)\) is fixed by both \(c\) and \(\gamma^+\), the path \(v\) is continuous. Moreover, for \(a\@undefined t\@undefined\@undefined\), we have \[c(a,t)=(t,a), \qquad \gamma^+(a,t)=(t,a).\] Thus, in both cases, \[v(t)=(t,a)\qquad (0\@undefined t\@undefined\@undefined).\] Therefore \(v\) is again a natural directed path of natural length \(\@undefined\). The equality \(qu=qv\) follows because, for \(0\@undefined t\@undefined a\), one has \(u(t)=v(t)\), while for \(a\@undefined t\@undefined\@undefined\), one has \(v(t)=\theta u(t)\), and the quotient \(q\) identifies \(z\) with \(\theta(z)\). It remains to check that \(u\) and \(v\) are not equivalent in the coequalizer of path spaces. If \(\theta=c\), then \(\theta^2=\operatorname{id}\), so the equivalence class of \(u\) generated by \(w\sim cw\) is exactly \[\{u,cu\}.\] The path \(v\) is neither \(u\) nor \(cu\): on the second half it is different from \(u\), and on the first half it is different from \(cu\). Hence \(u\) and \(v\) are not equivalent. If \(\theta=\gamma^+\), then \((\gamma^+)^2=\gamma^+\), so the equivalence class of \(u\) generated by \(w\sim\gamma^+w\) is exactly \[\{u,\gamma^+u\}.\] Again \(v\) is neither \(u\) nor \(\gamma^+u\). Hence \(u\) and \(v\) are not equivalent. It remains to treat \(\theta=\gamma^-\). Define \[u(t)= \begin{cases} (a,t), & 0\@undefined t\@undefined a,\\ (t,a), & a\@undefined t\@undefined\@undefined, \end{cases}\] and \[v(t)= \begin{cases} u(t), & 0\@undefined t\@undefined a,\\ \gamma^- u(t), & a\@undefined t\@undefined\@undefined. \end{cases}\] Since \(u(a)=(a,a)\), the path \(v\) is continuous. For \(a\@undefined t\@undefined\@undefined\), one has \[\gamma^-(t,a)=(a,t),\] so \[v(t)=(a,t)\qquad (0\@undefined t\@undefined\@undefined).\] Thus \(v\) is again a natural directed path of natural length \(\@undefined\), and the same quotient argument gives \(q u=q v\). Finally, since \((\gamma^-)^2=\gamma^-\), the equivalence class of \(u\) generated by \(w\sim\gamma^-w\) is exactly \[\{u,\gamma^-u\}.\] The path \(v\) is neither \(u\) nor \(\gamma^-u\). Therefore \(u\) and \(v\) are not equivalent. ◻
Theorem 34. Let \(\mathcal{A}\) be a thick category of cubes. The following two conditions are equivalent:
\(\mathcal{A}=\square\),
For every \(0<\@undefined<1\), the naive \(\@undefined\)-branching-space functor \[\mathcal{P}^-_\mathcal{A}(-,\@undefined):\mathcal{A}^{\mathrm{op}}\mathrm{Set}\longrightarrow\mathbf{Top}\] is colimit-preserving.
Proof. The direction \((1)\Rightarrow (2)\) is [7]. Assume now \(\mathcal{A}\neq\square\). Using Lemma 32, choose \(\theta\in\{c,\gamma^+,\gamma^-\}\cap\mathcal{A}([2],[2])\). Then the diagram of spaces \[\begin{tikzcd}[column sep=6em,row sep=3em] {\mathcal{P}^-_\mathcal{A}(\mathcal{A}[2],\@undefined)} \arrow[r, shift left=1ex, "\operatorname{id}"] \arrow[r, shift right=1ex, "{\mathcal{P}^-_\mathcal{A}(\mathcal{A}(-,\theta),\@undefined)}"'] & {\mathcal{P}^-_\mathcal{A}(\mathcal{A}[2],\@undefined)} \arrow[r, "q"] & {\mathcal{P}^-_\mathcal{A}(Q_\theta,\@undefined)} \end{tikzcd}\] is not a coequalizer by Lemma 33. This proves the implication \((2)\Rightarrow(1)\). ◻
We explore the case of \(\mathcal{A}\)-sets freely generated by precubical sets, i.e. of the form \(\mathcal{L}_{\mathcal{A}}(K)\). Let \[\rho_\square:\square^{\mathrm{op}}\mathrm{Set}\longrightarrow(\square^-)^{\mathrm{op}}\mathrm{Set}\] denote the restriction functor along the inclusion \(\square^-\subseteq\square\).
Lemma 35. Let \(\mathcal{A}\) be a thick category of cubes. For every \(n\@undefined 0\), the canonical map of \((\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\) \[\mathcal{L}_{\mathcal{A}}^-\rho_{\square}(\square[n]) \longrightarrow \rho_\mathcal{A}\mathcal{L}_{\mathcal{A}}(\square[n])\] is an isomorphism.
Proof. Since \(\mathcal{L}_{\mathcal{A}}(\square[n])\cong \mathcal{A}(-,[n])\), it suffices to prove that, for every \(r\@undefined 0\), the canonical map \[\Phi_{r,n}: \int^{[q]\in\square^-} \square([q],[n])\times \mathcal{A}^-([r],[q]) \longrightarrow \mathcal{A}([r],[n])\] induced by composition is a bijection. Explicitly, \[\Phi_{r,n}([\alpha,g])=\alpha g\] for \(g:[r]\to[q]\) in \(\mathcal{A}^-\) and \(\alpha:[q]\to[n]\) in \(\square\). We first prove surjectivity. Let \(f:[r]\to[n]\) be a morphism of \(\mathcal{A}\). Write its cotransverse–box factorization as \[f=\delta\psi, \qquad \psi:[r]\to[r], \qquad \delta:[r]\to[n],\] with \(\delta\in\square([r],[n])\). Since \(\mathcal{A}\) is thick, the cotransverse endomorphism \(\psi\) belongs to \(\mathcal{A}([r],[r])\). By Proposition 16, it preserves the initial vertex, and therefore \[\psi\in\mathcal{A}^-([r],[r]).\] Thus \(f\) is the image of the class represented by \[[r]\xrightarrow{\psi}[r]\xrightarrow{\delta}[n].\] We now prove injectivity. Suppose that two representatives \[[r]\xrightarrow{g}[q]\xrightarrow{\alpha}[n], \qquad [r]\xrightarrow{h}[q']\xrightarrow{\beta}[n]\] have the same image in \(\mathcal{A}([r],[n])\), so that \[\alpha g=\beta h.\] Since \(g\) and \(h\) are morphisms of \(\mathcal{A}^-\), Lemma 22 gives their cotransverse–box factorizations inside \(\mathcal{A}^-\): \[g=\@undefined\psi, \qquad h=\eta\chi,\] with \[\@undefined\in\square^-([r],[q]), \qquad \eta\in\square^-([r],[q']), \qquad \psi,\chi\in\mathcal{A}^-([r],[r]).\] The defining relations of the coend give \[[\alpha,g]=[\alpha\@undefined,\psi], \qquad [\beta,h]=[\beta\eta,\chi].\] The equality \(\alpha g=\beta h\) becomes \[\alpha\@undefined\psi=\beta\eta\chi.\] Here \(\alpha\@undefined\) and \(\beta\eta\) are morphisms of \(\square([r],[n])\). By the uniqueness of the cotransverse–box factorization, we obtain \[\alpha\@undefined=\beta\eta \qquad\text{and}\qquad \psi=\chi.\] Therefore \[[\alpha,g] =[\alpha\@undefined,\psi] =[\beta\eta,\chi] =[\beta,h].\] This proves injectivity. Hence \(\Phi_{r,n}\) is a bijection for all \(r,n\), and the canonical map is an isomorphism of \((\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}\). ◻
Lemma 36. Let \(\mathcal{A}\) be a thick category of cubes. There is a natural isomorphism of functors \[\mathcal{L}_{\mathcal{A}}^-\rho_{\square} \cong \rho_\mathcal{A}\mathcal{L}_{\mathcal{A}} : \square^{\mathrm{op}}\mathrm{Set}\longrightarrow (\mathcal{A}^-)^{\mathrm{op}}\mathrm{Set}.\]
Proof. Both functors preserve colimits. Indeed, \(\mathcal{L}_{\mathcal{A}}\) and \(\mathcal{L}_{\mathcal{A}}^-\) are left adjoints, and \(\rho_\mathcal{A}\) and \(\rho_{\square}\) preserve colimits because colimits of presheaves are computed pointwise. Since every precubical set is a colimit of representables, it is enough to compare the two functors on the representables \(\square[n]\). This is precisely Lemma 35. ◻
Lemma 37. For every \(\square^-\)-set \(H\), there is a natural homeomorphism \[\widehat{N_\@undefined}(\mathcal{L}_{\mathcal{A}}^- H) \cong \int^{[q]\in\square^-} H([q])\cdot N_q(\@undefined).\]
Proof. Using the coend formula for the left Kan extension, Fubini’s theorem for coends, and the co-Yoneda lemma, we obtain the following natural sequence of homeomorphisms: \[\begin{align} \widehat{N_\@undefined}(\mathcal{L}_{\mathcal{A}}^- H) &= \int^{[p]\in\mathcal{A}^-} (\mathcal{L}_{\mathcal{A}}^- H)([p])\cdot N_p(\@undefined) \nobreak\\ &\cong \int^{[p]\in\mathcal{A}^-} \left( \int^{[q]\in\square^-} H([q])\times \mathcal{A}^-([p],[q]) \right) \cdot N_p(\@undefined) \nobreak\\ &\cong \int^{[q]\in\square^-} H([q])\cdot \left( \int^{[p]\in\mathcal{A}^-} \mathcal{A}^-([p],[q])\cdot N_p(\@undefined) \right) \nobreak\\ &\cong \int^{[q]\in\square^-} H([q])\cdot N_q(\@undefined). \end{align}\] ◻
For a vertex \(a=(a_1,\ldots,a_n)\in [n]\), put \[r(a)=n-(a_1+\cdots+a_n)=\overrightarrow d_1(a,1_n).\] Let \[\kappa_a:[r(a)]\longrightarrow [n]\] be the unique box map whose image is the face spanned by the interval \([a,1_n]\), that is, the unique box map satisfying \[\kappa_a(0_{r(a)})=a \qquad\text{and}\qquad \kappa_a(1_{r(a)})=1_n.\] Explicitly, \(\kappa_a\) fixes to \(1\) the coordinates \(i\) such that \(a_i=1\), and inserts the \(r(a)\) variables, in their natural order, in the coordinates \(i\) such that \(a_i=0\). We call the representable \(\square^-(-,[r])\) the standard \(r\)-corner. With this terminology, the restriction of the representable precubical \(n\)-cube to \(\square^-\) is the coproduct of its corners: \[\rho_\square(\square[n]) \cong \coprod_{a\in [n]} \square^-(-,[r(a)]).\] Indeed, evaluating at \([q]\), this isomorphism sends a morphism \[\beta:[q]\longrightarrow [r(a)]\] of \(\square^-\) to the composite \[[q]\xrightarrow{\beta}[r(a)]\xrightarrow{\kappa_a}[n].\] This map is bijective. If \(\alpha:[q]\to[n]\) is a box map, then its initial vertex \(a=\alpha(0_q)\) determines the component. For every coordinate \(i\) with \(a_i=1\), the \(i\)-th coordinate of \(\alpha\) is constantly \(1\). On the remaining coordinates, \(\alpha\) is uniquely encoded by a morphism \[\beta:[q]\longrightarrow [r(a)]\] of \(\square^-\), and \(\alpha=\kappa_a\beta\). This decomposition is clearly natural with respect to precomposition by morphisms of \(\square^-\).
Lemma 38. For every precubical set \(K\), there is a natural homeomorphism \[\int^{[q]\in\square^-} K_q\cdot N_q(\@undefined) \xlongrightarrow{\cong} \mathcal{P}^-_\square(K,\@undefined).\] For \(K=\square[n]\), this homeomorphism sends the class of a pair \[\alpha:[q]\to[n], \qquad u\in N_q(\@undefined),\] with \(\alpha\in\square([q],[n])\), to the short natural directed path \[\operatorname{T}(\alpha)u\in \mathcal{P}^-_\square(\square[n],\@undefined).\]
Proof. The displayed map is well-defined because, for every morphism \(v:[p]\to[q]\) of \(\square^-\), one has \[\operatorname{T}(\alpha v)=\operatorname{T}(\alpha)\operatorname{T}(v),\] and the coend relation identifies \((\alpha v,u)\) with \((\alpha,\operatorname{T}(v)u)\). Both sides are colimit-preserving with respect to the precubical set \(K\): the left-hand side is colimit-preserving by the coend formula, and the right-hand side is colimit-preserving by [7]. Since every precubical set is a colimit of representables, it remains to prove the result for \(K=\square[n]\). Using the corner decomposition above, we obtain \[\begin{align} \int^{[q]\in\square^-}\square([q],[n])\cdot N_q(\@undefined) &\cong \int^{[q]\in\square^-} \left( \coprod_{a\in[n]}\square^-([q],[r(a)]) \right)\cdot N_q(\@undefined) \\ &\cong \coprod_{a\in[n]} \int^{[q]\in\square^-} \square^-([q],[r(a)])\cdot N_q(\@undefined) \\ &\cong \coprod_{a\in[n]} N_{r(a)}(\@undefined), \end{align}\] by the co-Yoneda lemma. Under this identification, the summand indexed by \(a\) is sent to \(P^-_\square(\square[n],\@undefined)\) by \[u\longmapsto \operatorname{T}(\kappa_a)u.\] This is exactly the space of short natural directed paths in the precubical cube \(\square[n]\) starting from the vertex \(a\): the map \(\kappa_a\) identifies the standard \(r(a)\)-corner with the face interval \([a,1_n]\). Taking the coproduct over all vertices \(a\in[n]\) gives precisely \(P^-_\square(\square[n],\@undefined)\). This proves the assertion for representables, and hence for all precubical sets by colimit preservation. ◻
Theorem 39. Let \(\mathcal{A}\) be a thick category of cubes, and let \(\@undefined\in]0,1[\). For every precubical set \(K\), there is a natural homeomorphism \[\widehat{N_\@undefined}\bigl(\rho_\mathcal{A}\mathcal{L}_{\mathcal{A}}(K)\bigr) \cong \mathcal{P}^-_\square(K,\@undefined).\] Equivalently, \[B^-_\@undefined(\mathcal{L}_{\mathcal{A}}(K)) \cong \mathcal{P}^-_\square(K,\@undefined),\] where \(B^-_\@undefined=\widehat{N_\@undefined}\rho_\mathcal{A}\).
Proof. By Lemma 36, there is a natural isomorphism \[\rho_\mathcal{A}\mathcal{L}_{\mathcal{A}}(K) \cong \mathcal{L}_{\mathcal{A}}^-\rho_{\square}(K).\] Applying \(\widehat{N_\@undefined}\) and then Lemma 37, we obtain \[\begin{align} \widehat{N_\@undefined}\bigl(\rho_\mathcal{A}\mathcal{L}_{\mathcal{A}}(K)\bigr) &\cong \widehat{N_\@undefined}\bigl(\mathcal{L}_{\mathcal{A}}^-\rho_{\square}(K)\bigr) \\ &\cong \int^{[q]\in\square^-} \rho_{\square}(K)([q])\cdot N_q(\@undefined) \\ &= \int^{[q]\in\square^-} K_q\cdot N_q(\@undefined). \end{align}\] The last coend is naturally homeomorphic to \(\mathcal{P}^-_\square(K,\@undefined)\) by Lemma 38. This proves the assertion. ◻
For the representable precubical set \(K=\square[n]\), the resulting homeomorphism is the explicit map \[\widehat{N_\@undefined}\bigl(\rho_\mathcal{A}\mathcal{L}_{\mathcal{A}}(\square[n])\bigr) \cong \int^{[q]\in\square^-}\square([q],[n])\cdot N_q(\@undefined) \longrightarrow \mathcal{P}^-_\square(\square[n],\@undefined)\] sending \([\alpha,u]\) to \(T(\alpha)u\). Notice that \(\alpha\) is an arbitrary morphism of \(\square\), not necessarily a morphism of \(\square^-\). This is why the construction recovers short natural paths starting from every vertex of the precubical cube \(\square[n]\), not only those starting from \(0_n\).