Cubical models of (∞,1) -categories
Brandon Doherty, Chris Kapulkin, Zachery Lindsey, Christian Sattler
aa r X i v : . [ m a t h . A T ] M a y CUBICAL MODELS OF ( ∞ , -CATEGORIES BRANDON DOHERTY, KRZYSZTOF KAPULKIN, ZACHERY LINDSEY, AND CHRISTIAN SATTLER
Abstract.
We construct a model structure on the category of cubical sets with connectionswhose cofibrations are the monomorphisms and whose fibrant objects are defined by the rightlifting property with respect to inner open boxes, the cubical analogue of inner horns. We showthat this model structure is Quillen equivalent to the Joyal model structure on simplicial setsvia the triangulation functor. As an application, we show that cubical quasicategories admit anelegant and canonical notion of a mapping space between two objects.
Introduction
The category sSet of simplicial sets carries two canonical model structures: the Kan-Quillen modelstructure [Qui67], presenting the homotopy theory of ∞ -groupoids, and the Joyal model structure[Joy09], presenting the homotopy theory of ( ∞ , -categories. Both of these model structures havemonomorphisms as their cofibrations and their fibrant objects are defined by a more or less restric-tive lifting condition, depending on whether or not the 1-simplices of a fibrant object are supposedto be invertible.The category cSet of cubical sets is also known to carry a model structure, called the Grothendieckmodel structure, constructed by Cisinski [Cis06, Cis14], presenting the theory of ∞ -groupoids. Thismodel structure is completely analogous to the Kan-Quillen model structure, but with open boxesreplacing horns in the definition of fibrant objects. The goal of the present work is provide a cubicalanalogue of the Joyal model structure, thus filling the bottom right corner in the table:category \ theory ∞ -groupoids ( ∞ , -categories sSet [Qui67] [Joy09] cSet [Cis14] present workOur main theorem (cf. Theorem 4.1 and Proposition 4.15, and Theorem 5.1) states Theorem.
The category cSet of cubical sets carries a model structure in which: • the cofibrations are the monomorphisms; • the fibrant objects are defined by having fillers for all inner open boxes.Moreover, this model structure is Quillen equivalent to the Joyal model structure on the category sSet of simplicial sets via the triangulation functor T : cSet → sSet . A few comments are in order.First, there are many different notions of a cubical set, depending on the choice of maps in theindexing category (cid:3) , called the box category . Here, we are working with the cubical sets withconnections (specifically the max-connection), as studied in [Cis14, Mal09, KLW19]. The categoryof combinatorial cubes with connections is both an EZ-Reedy category and a strict test category,which makes it convenient to work with. Either more restrictive or more lenient choices of maps inthe box category (such as the ones studied previously by Cisinski/Jardine [Cis06, Jar06] or in therecent work of Coquand and his group [CCHM18] on cubical type theory) result in a loss of someof these convenient properties.It is also exactly the category of cubical sets with connections that was recently shown [KLW19]to admit a co-reflective embedding of the category of simplicial sets via the straightening-over-the-point functor Q : sSet → cSet , an instance of a more general construction of straightening, studiedin [KV18]. And despite its perhaps less clear definition, Q ends up being much easier to work withthan the triangulation functor. Indeed, in order to show that T is a Quillen equivalence, we firstprove it about Q and establish that the derived functors of T and Q are each other’s inverses.Lastly, the concept of an inner open box appearing in the statement of our main theorem is thecubical analogue of the notion of an inner horn in simplicial sets. Its definition is somewhat subtle,which is the reason behind our taking a slight detour in the construction of the model structureon cubical sets. At this point however, we shall simply note this subtlety here and give a precisedefinition in Section 4.In order to establish a model structure on cSet , we consider first a model structure on markedcubical sets. A marked cubical set is a cubical set with a distinguished subset of edges (to bethought of as “equivalences”), containing all degenerate ones. We then use the minimal markingfunctor, taking a cubical set to a marked cubical set in which the marked edges are precisely thedegeneracies, to left-induce a model structure on cubical sets.In order to establish that the triangulation functor is a Quillen equivalence between our modelstructure on cubical sets and the Joyal model structure on simplicial sets, we introduce a cubicaltheory of cones, which generalizes the straightening-over-the-point construction. Our cubical conesserve as a convenient way of relating simplicial and cubical shapes, and we believe that these toolswill find applications beyond present work.This paper is organized as follows. In Section 1, we collect the necessary results on model categories,cubical sets, and marked cubical sets. Trying to keep the exposition as self-contained as possible, weincluded statements of frequently used results and those that may be harder to find in the existingliterature.In Section 2, we construct the model structure on the category of marked cubical sets, using JeffSmith’s theorem. Then, in Section 3 we show that it is right-induced by a model structure on thecategory of structurally marked cubical sets constructed using the Cisinski theory.In Section 4, we use the minimal marking functor to construct the desired model structure onthe category of cubical sets. We then analyze the resulting classes of maps, characterizing weakequivalences and fibrations between fibrant objects, and construct the mapping space between two -cubes in a fibrant object. UBICAL MODELS OF ( ∞ , -CATEGORIES 3 Finally, in Section 5, we develop the theory of cones and use it to show that our model structure isQuillen equivalent with the Joyal model structure. This last argument is fairly combinatorial andincludes a number of routine computations involving cubical identities. For the clarity of exposition,most of these computations are therefore relegated to appendix A, to be verified only by the mostmasochistic of the readers.Introduction 11. Cubical sets and marked cubical sets 32. Model structure on marked cubical sets 143. Model structure on structurally marked cubical sets 314. Joyal model structure on cubical sets 345. Comparison with the Joyal model structure 43Appendix A. Verification of identities on θ Cubical sets and marked cubical sets
Model categories.
Here we will review various general results from the theory of modelcategories which we will use throughout subsequent sections. We begin with a result which allowsus to construct model structures having specified classes of cofibrations and weak equivalences.
Theorem 1.1 (Jeff Smith’s Theorem, [Bar10, Prop. 2.2]) . Let C be a locally presentable category.Let W be a class of morphisms forming an accessibly embedded, accessible subcategory of C → , and I a set of morphisms in C . Suppose that the following conditions are satisfied. • W satisfies the two-out-of-three axiom. • W contains all maps having the right lifting property with respect to the maps in I . • The intersection of W with the saturation of I is closed under pushouts and transfinitecomposition.Then C admits a cofibrantly generated model structure with weak equivalences W and generatingcofibrations I . Next we review some of the machinery of Cisinski theory [Cis06], which allows for the easy con-struction of model structures on presheaf categories having monomorphisms as cofibrations andweak equivalences defined in terms of homotopy with respect to a cylinder functor.
Definition 1.2.
Let C be a small category. A cylinder functor on C consists of an endofunctor I on the presheaf category Set C op , together with natural transformations ∂ , ∂ : id → I , σ : I → id ,such that: • ∂ and ∂ are sections of σ ; • For all X : C op → Set , the map ( ∂ X , ∂ X ) : X ⊔ X → IX is a monomorphism; • I preserves small colimits and monomorphisms; B. DOHERTY, K. KAPULKIN, Z. LINDSEY, C. SATTLER • For all monomorphisms j : X → Y in Set C op and all ε ∈ { , } , the following square is apullback: X ∂ ε (cid:15) (cid:15) f / / Y ∂ ε (cid:15) (cid:15) IX If / / IY In what follows, let C be a small category equipped with a cylinder functor I : Set C op → Set C op . Definition 1.3.
Let f, g : X → Y be maps of presheaves on C . An elementary homotopy from f to g is a map H : IX → Y such that H∂ = f, H∂ = g . A homotopy is a zig-zag of elementaryhomotopies. The set [ X, Y ] is the set of maps from X to Y modulo the relation of homotopy.It is easy to see that pre- and post-composition by a fixed map preserve the relation of homotopy;thus a map X → Y induces maps [ Z, X ] → [ Z, Y ] and [ Y, Z ] → [ X, Z ] for any Z . Definition 1.4. A cellular model for Set C op is a set M of monomorphisms in Set C op whose saturationis precisely the class of monomorphisms of Set C op .Let M be a cellular model for Set C op , and S a set of monomorphisms in Set C op . The set of morphisms Λ( S ) is defined by the following inductive construction. For a monomorphism X → Y in Set C op and ε ∈ { , } , let IX ∪ ε Y and IX ∪ ( Y ⊔ Y ) be defined by the following pushout squares: X / / ∂ ε (cid:15) (cid:15) Y (cid:15) (cid:15) (cid:15) (cid:15) IX / / IX ∪ ε Y ❴✤ X ⊔ X (cid:15) (cid:15) / / Y ⊔ Y (cid:15) (cid:15) IX / / IX ∪ ( Y ⊔ Y ) We now define a set of monomorphisms Λ( S ) by an inductive construction. We begin by setting: Λ ( S ) = S ∪ { IX ∪ ε Y → IY | X → Y ∈ M, ε ∈ { , }} Now, given Λ n ( S ) , we define: Λ n +1 ( S ) = { IX ∪ ( Y ⊔ Y ) → IY | X → Y ∈ Λ n ( S ) } Finally, we let Λ( S ) = S n ≥ Λ n ( S ) . We now define several distinguished classes of maps and objectsin Set C op . • A cofibration is a monomorphism; a trivial fibration is a map having the right lifting propertywith respect to the cofibrations. UBICAL MODELS OF ( ∞ , -CATEGORIES 5 • An anodyne map is a map in the saturation of Λ( S ) ; a naive fibration is a map having theright lifting property with respect to the anodyne maps. • A fibrant object is a presheaf X such that the map from X to the terminal presheaf is anaive fibration. • A weak equivalence is a map X → Y such that the induced map [ Y, Z ] → [ X, Z ] is abijection for any fibrant Z . • A trivial cofibration is a map which is both a cofibration and a weak equivalence; a fibration is a map having the right lifting property with respect to the trivial cofibrations. Theorem 1.5.
The classes above define a cofibrantly generated model structure on
Set C op , in whicha map between fibrant objects is a fibration if and only if it is a naive fibration.Proof. The existence of the model structure is [Cis06, Thm. 1.3.22]; the characterization of fibrantobjects is [Cis06, Thm. 1.3.36]. (cid:3)
Corollary 1.6.
The homotopy category of
Set C op with a model structure of Theorem 1.5 can bedescribed as follows: • its objects are the fibrant presheaves; • the maps from X to Y are given by [ X, Y ] . (cid:3) Example 1.7.
Let J denote the simplicial set depicted below: / / ❃❃❃❃❃❃❃❃ ❃❃❃❃❃❃❃❃ (cid:15) (cid:15) ❃❃❃❃❃❃❃❃ ❃❃❃❃❃❃❃❃ / / Taking the product with J defines a cylinder functor on sSet , with the natural transformations ∂ , ∂ given by taking the product with the endpoint inclusions { } ֒ → J, { } ֒ → J . ApplyingTheorem 1.5 with this cylinder functor, the cellular model M = { ∂ ∆ n → ∆ n | n ≥ } , and S = { Λ ni | n ≥ , < i < n } (the set of inner horn inclusions ), we obtain the Joyal model structure on sSet , characterized as follows: • Cofibrations are monomorphisms; • Fibrant objects are quasicategories , simplicial sets having fillers for all inner horns; • Fibrations between fibrant objects are characterized by the right lifting property with re-spect to the inner horn inclusions and the endpoint inclusions { ε } ֒ → J, ε ∈ { , } ; • Weak equivalences are maps X → Y inducing bijections [ Y, Z ] → [ X, Z ] for all quasicate-gories Z .For more on the Joyal model structure, see [Joy09]; for the details of its construction as a Cisinskimodel structure, see [Cis19, Sec. 3.3].Next we review a theorem which allows us to induce one model structure from another using anadjunction between their respective categories. B. DOHERTY, K. KAPULKIN, Z. LINDSEY, C. SATTLER
Definition 1.8.
Let F : C ⇋ D : U be an adjunction between model categories. The modelstructure on C is left induced by F if F preserves and reflects cofibrations and weak equivalences.Likewise, the model structure on D is right induced by U if U preserves and reflects weak equivalencesand fibrations. Remark 1.9.
Note that for a given adjunction C ⇋ D and a given model structure on D , theleft-induced model structure is unique, if one exists, since the definition determines the cofibrationsand weak equivalences of C . Likewise, for a given model structure on C , the right-induced modelstructure is unique, if one exists. Theorem 1.10 ([HKRS17, Thm. 2.2.1]) . Let F : C ⇄ D : U be an adjunction between locallypresentable categories such that D carries a cofibrantly generated model structure with all objectscofibrant. If, for every object X ∈ C , the co-diagonal map admits a factorization X ⊔ X i X −−→ IX p X −−→ X , such that F i X is a cofibration and F p X is a weak equivalence, then C admits a model structureleft-induced by F from that of D . (cid:3) Finally, we review some results which allow us to easily recognize Quillen adjunctions and Quillenequivalences.
Proposition 1.11 ([JT07, Prop. 7.15]) . Let F : C ⇄ D : U be an adjunction between modelcategories. If F preserves cofibrations and U preserves fibrations between fibrant objects, then theadjunction is Quillen. (cid:3) This statement has an immediate corollary, which we will apply in practice:
Corollary 1.12.
Let F : C → D be a left adjoint between model categories and suppose that fibra-tions between fibrant objects in C are characterized by right lifting against a class S . If F preservescofibrations and sends S to trivial cofibrations, then F is a left Quillen functor. (cid:3) Proposition 1.13 ([Hov99, Cor. 1.3.16]) . Let F : C ⇄ D : U be a Quillen adjunction betweenmodel categories. Then the following are equivalent.(i) F ⊣ U is a Quillen equivalence.(ii) F reflects weak equivalences between cofibrant objects and, for every fibrant Y , the derivedcounit F g U Y → Y is a weak equivalence.(iii) U reflects weak equivalences between fibrant objects and, for every cofibrant X , the derivedunit X → U ( F X ) ′ is a weak equivalence. Again, in practice we will often apply the following corollary:
Corollary 1.14.
Let F : C ⇄ D : U be a Quillen adjunction between model categories.(i) If U preserves and reflects weak equivalences, then the adjunction is a Quillen equivalenceif and only if, for all cofibrant X ∈ C , the unit X → U F X is a weak equivalence.(ii) If F preserves and reflects weak equivalences, then the adjunction is a Quillen equivalenceif and only if, for all fibrant Y ∈ D , the counit F U Y → Y is a weak equivalence. (cid:3) UBICAL MODELS OF ( ∞ , -CATEGORIES 7 The box category and cubical sets.
We begin by defining the box category (cid:3) . The objectsof (cid:3) are posets of the form [1] n and the maps are generated (inside the category of posets) undercomposition by the following three special classes: • faces ∂ ni,ε : [1] n − → [1] n for i = 1 , . . . , n and ε = 0 , given by: ∂ ni,ε ( x , x , . . . , x n − ) = ( x , x , . . . , x i − , ε, x i , . . . , x n − ) ; • degeneracies σ ni : [1] n → [1] n − for i = 1 , , . . . , n given by: σ ni ( x , x , . . . , x n ) = ( x , x , . . . , x i − , x i +1 , . . . , x n ) ; • connections γ ni : [1] n → [1] n − for i = 1 , , . . . , n − given by: γ ni ( x , x , . . . , x n ) = ( x , x , . . . , x i − , max { x i , x i +1 } , x i +2 , . . . , x n ) .These maps obey the following co-cubical identities : ∂ j,ε ∂ i,ε ′ = ∂ i +1 ,ε ′ ∂ j,ε for j ≤ i ; σ i σ j = σ j σ i +1 for j ≤ i ; σ j ∂ i,ε = ∂ i − ,ε σ j for j < i ; id for j = i ; ∂ i,ε σ j − for j > i ; γ j γ i = γ i γ j +1 for j ≥ i ; γ j ∂ i,ε = ∂ i − ,ε γ j for j < i − ; id for j = i − , i, ε = 0 ; ∂ i,ε σ i for j = i − , i, ε = 1 ; ∂ i,ε γ j − for j > i ; σ j γ i = γ i − σ j for j < i ; σ i σ i for j = i ; γ i σ j +1 for j > i . Theorem 1.15 ([GM03, Thm. 5.1]) . Every map in the category (cid:3) can be factored uniquely as acomposite ( ∂ k ,ε . . . ∂ k t ,ε t )( γ j . . . γ j s )( σ i . . . σ i r ) ,where i > . . . > i r ≥ , ≤ j < . . . < j s , and k > . . . > k t ≥ . (cid:3) Corollary 1.16. (cid:3) admits the structure of a Reedy category, in which: • deg([1] n ) = n ; • (cid:3) + is generated under composition by the face maps; • (cid:3) − is generated under composition by the degeneracy and connection maps. (cid:3) The category of cubical sets, i.e., contravariant functors (cid:3) op → Set will be denoted by cSet . We willwrite (cid:3) n for the representable cubical set, represented by [1] n . We adopt the convention of writingthe action of cubical operators on the right. For instance, the (1 , -face of an n -cube x : (cid:3) n → X will be denoted x∂ , .We write ∂ (cid:3) n → (cid:3) n for the maximal proper subobject of (cid:3) n , i.e., the union of all of its faces. Wewill refer to these as the n -box and the boundary of the n -box, respectively. The subobject of (cid:3) n given by the union of all faces except ∂ i,ε will be denoted ⊓ ni,ε and referred to as an ( i, ε ) open box. Definition 1.17.
The critical edge of (cid:3) n with respect to a face ∂ i,ε is the unique edge of (cid:3) n whichis adjacent to ∂ i,ε and which, together with ∂ i,ε , contains both of the vertices (0 , ..., and (1 , ..., . B. DOHERTY, K. KAPULKIN, Z. LINDSEY, C. SATTLER
More explicitly, the critical edge with respect to ∂ i,ε corresponds to the map f : [1] → [1] n given by f i = id [1] , f j = const − ε for j = i .The assignment ([1] m , [1] n ) → [1] m + n defines a functor (cid:3) × (cid:3) → (cid:3) . Postcomposing it with theYoneda embedding and left Kan extending, we obtain the geometric product functor (cid:3) × (cid:3) / / (cid:127) _ (cid:15) (cid:15) cSetcSet × cSet ⊗ ♦♦♦♦♦♦♦♦♦♦♦♦ The standard formula for left Kan extensions gives us the following formula for the geometricproduct: X ⊗ Y = colim x : (cid:3) m → Xy : (cid:3) n → Y (cid:3) m + n Note that the geometric product of cubical sets does not coincide with the cartesian product.However, the geometric product implements the correct homotopy type, and is better behavedthan the cartesian product – for instance, for m, n ≥ we have (cid:3) m ⊗ (cid:3) n = (cid:3) m + n . Furthermore,the geometric product is taken to the cartesian product by the geometric realization functor tospaces. Proposition 1.18.
The geometric product ⊗ defines a monoidal structure on the category of cubicalsets, with the unit given by (cid:3) . This monoidal structure is however not symmetric. Indeed, the existence of a symmetry naturaltransformation would in particular imply that there is a non-identity bijection [1] → [1] in (cid:3) .In particular, for any X, Y ∈ cSet , the unique maps from X and Y to (cid:3) induce maps π X : X ⊗ Y → X, π Y : X ⊗ Y → Y .Given a cubical set A , we form two non-isomorphic functors cSet → cSet : the left tensor − ⊗ A andthe right tensor A ⊗ − . As they are both co-continuous, they admit right adjoints and we write hom L ( A, − ) for the right adjoint of the left tensor − ⊗ A and hom R ( A, − ) for the right adjoint ofthe right tensor A ⊗ − . Thus the monoidal structure on cSet given by the geometric product isclosed, but non-symmetric.The standard construction of an arbitrary small colimit as a coequalizer of coproducts gives us thefollowing lemma about colimts in presheaf categories. Lemma 1.19.
Let C be a category and D a small diagram in Set C op . Then any map C ( − , c ) → colim D factors through some map in the colimit cone. (cid:3) This lemma allows us to describe the geometric product of cubical sets explicitly.
Proposition 1.20.
For
X, Y ∈ cSet , the geometric product X ⊗ Y admits the following description. • For k ≥ , the k -cubes of X ⊗ Y consist of all pairs ( x : (cid:3) m → X, y : (cid:3) n → Y ) such that m + n = k , subject to the identification ( xσ m +1 , y ) = ( x, yσ ) . UBICAL MODELS OF ( ∞ , -CATEGORIES 9 • For x : (cid:3) m → X, y : (cid:3) n → Y , the faces, degeneracies, and connections of the ( m + n ) -cube ( x, y ) are computed as follows: – ( x, y ) ∂ i,ǫ = ( ( x∂ i,ǫ , y ) 1 ≤ i ≤ m ( x, y∂ i − m,ǫ ) m + 1 ≤ i ≤ m + n – ( x, y ) σ i = ( ( xσ i , y ) 1 ≤ i ≤ n + 1( x, yσ i − m ) m + 1 ≤ i ≤ m + n + 1 – ( x, y ) γ i = ( ( xγ i , y ) 1 ≤ i ≤ m ( x, yγ i − m,ǫ ) m + 1 ≤ i ≤ m + n Proof.
We begin by noting that for every pair ( x : (cid:3) m → X, y : (cid:3) n → Y ) there is a corresponding ( m + n ) -cube ( x, y ) : (cid:3) m + n → X ⊗ Y given by the colimit cone. Next we will show that faces,degeneracies and connections of these cones are computed as described in the statement.For such an ( m + n ) -cube ( x, y ) , consider a face ( x, y ) ∂ i,ε for ≤ i ≤ m . We can express the facemap ∂ m + ni,ε as ∂ mi,ε ⊗ (cid:3) n ; thus ( x, y ) ∂ i,ε = ( x∂ i,ε , y ) by the naturality of the colimit cone. (cid:3) m − ⊗ (cid:3) n∂ i,ε ⊗ (cid:3) n / / ( x∂ i,ε ,y ) ' ' ❖❖❖❖❖❖❖❖❖❖❖ (cid:3) m ⊗ (cid:3) n ( x,y ) (cid:15) (cid:15) X ⊗ Y Likewise, for m + 1 ≤ i ≤ m + n we have ∂ m + ni,ε = (cid:3) m ⊗ ∂ ni − m,ε , implying ( x, y ) ∂ i,ε = ( x, y∂ i − m,ε ) .Similar proofs hold for degeneracies and connections. In particular, this implies that for any ( x, y ) we have ( xσ m +1 , y ) = ( x, yσ ) , as both are equal to ( x, y ) σ m +1 .To see that all cubes in X ⊗ Y are of this form, note that by Lemma 1.19, every cube of X ⊗ Y isequal ( x, y ) ψ for some such pair ( x, y ) and some map ψ in (cid:3) . We have shown that the set of cubesarising from pairs is closed under faces, degeneracies and connections; since these classes generateall maps in (cid:3) , this proves our claim.Finally, we must show that the cubes of X ⊗ Y are not subject to any additional identifications,beyond the identification ( xσ m +1 , y ) = ( x, yσ ) mentioned above. In other words, we must showthat for each k ≥ , ( X ⊗ Y ) k is the quotient of the set { ( x : (cid:3) m → X, y : (cid:3) n → Y ) | m + n = k } under the smallest equivalence relation ∼ such that ( x ′ σ m +1 , y ′ ) ∼ ( x ′ , y ′ σ ) for all x ′ : (cid:3) m ′ → X, y ′ : (cid:3) n ′ → Y such that m ′ + n ′ = k − .To that end, let x : (cid:3) m → X, y : (cid:3) n → Y, x ′ : (cid:3) m ′ → X, y ′ : (cid:3) n ′ → Y , such that m + n = m ′ + n ′ and ( x, y ) = ( x ′ , y ′ ) in ( X ⊗ Y ) . We compute the image of this cube under the map π X : X ⊗ Y → X . π X ( x, y ) = π X ( x ′ , y ) ∴ xσ m +1 σ m +2 ...σ m + n = x ′ σ m ′ +1 ...σ m + n (If m or m ′ is equal to 0, we interpret the corresponding string of degeneracies to be empty.) Wecan apply face maps to both sides of this equation to reduce the left-hand side to x . If m = m ′ then this gives the equation x = x ′ , and a similar calculation shows y = y ′ . Otherwise, we have x = x ′ σ m ′ +1 ...σ m . In this case, a similar calculation shows y ′ = yσ ...σ , where σ is applied m − m ′ times on the right-hand side of the equation. From this we can see that ( x, y ) ∼ ( x ′ , y ′ ) . Thus wesee that quotienting the set of pairs ( x, y ) of appropriate dimensions by ∼ does indeed suffice toobtain ( X ⊗ Y ) k . (cid:3) Corollary 1.21.
For cubical sets X and Y , we have ( X ⊗ Y ) ∼ = ( X × Y ) ∪ ( X × Y ) ( X × Y ) . (cid:3) The following lemma, which can be verified by simple computation, allows us to express boundaryinclusions and open box inclusions as pushout products with respect to this monoidal structure.
Lemma 1.22. (i) For m, n ≥ , we have ( ∂ (cid:3) m → (cid:3) m ) b ⊗ ( ∂ (cid:3) n → (cid:3) n ) = ( ∂ (cid:3) m + n → (cid:3) m + n ) .(ii) For ≤ i ≤ m and ε ∈ { , } , the open-box inclusion ⊓ ni,ε ֒ → (cid:3) n is the pushout product ( ∂ (cid:3) i − ֒ → (cid:3) i − ) b ⊗ ( { − ε } ֒ → (cid:3) ) b ⊗ ( ∂ (cid:3) m − i ֒ → (cid:3) m − i ) . The restriction of the nerve functor defines a functor (cid:3) → sSet ; taking the left Kan extension ofthis functor along the Yoneda embedding, we obtain the triangulation functor T : cSet → sSet . (cid:3) / / (cid:127) _ (cid:15) (cid:15) sSetcSet T rrrrrrrrrr The triangulation functor has a right adjoint U : sSet → cSet given by ( U X ) n = sSet ((∆ ) n , X ) .Intuitively, we think of triangulation as creating a simplicial set T X from a cubical set X bysubdividing the cubes of X into simplices.We now record two basic facts about triangulation. In the given references, these results are provenusing a different definition of the category (cid:3) , lacking connection maps, but the proofs apply equallywell to the cubical sets under consideration here. Proposition 1.23 ([Cis06, Ex. 8.4.24]) . The triangulation functor sends geometric products tocartesian products; that is, for cubical sets X and Y , there is a natural isomorphism T ( X ⊗ Y ) ∼ = T X × T Y . (cid:3) Corollary 1.24.
Triangulation preserves pushout products; that is, for maps f, g in cSet there isa natural isomorphism T ( f b ⊗ g ) ∼ = T f b × T g .Proof.
Immediate by Proposition 1.23 and the fact that T preserves colimits as a left adjoint. (cid:3) Proposition 1.25 ([Cis06, Lem. 8.4.29]) . The triangulation functor preserves monomorphisms. (cid:3)
UBICAL MODELS OF ( ∞ , -CATEGORIES 11 Homotopy theory of cubical sets.Lemma 1.26.
The boundary inclusions ∂ (cid:3) n → (cid:3) n form a cellular model for cSet .Proof. This follows from Corollary 1.16. (cid:3)
Definition 1.27.
A map of cubical sets is a
Kan fibration if it has the right lifting property withrespect to all open box fillings. A cubical set X is a cubical Kan complex if the map X → (cid:3) is aKan fibration.The functor (cid:3) ⊗− : cSet → cSet , together with the natural transformations ∂ , ⊗− , ∂ , ⊗− : id → (cid:3) ⊗ − , and π : (cid:3) ⊗ − → id , defines a cylinder functor on cSet in the sense of Definition 1.2. Thus,for any X, Y ∈ cSet we have a set [ X, Y ] of homotopy classes of maps from X to Y defined by thiscylinder functor. Theorem 1.28 (Cisinski) . The category cSet carries a cofibrantly generated model structure, re-ferred to as the Grothendieck model structure, in which • cofibrations are the monomorphisms; • weak equivalences are maps X → Y inducing bijections [ Y, Z ] → [ X, Z ] for all cubical Kancomplexes Z ; • fibrations are the Kan fibrations.Proof. The existence of the model structure and characterization of the cofibrations, weak equiva-lences, and fibrant objects follows from applying Theorem 1.5 with the cylinder functor I , cellularmodel M = { ∂ (cid:3) n → (cid:3) n | n ≥ } , and S = ∅ . The characterization of the fibrations is given in[Cis14, Thm. 1.7]. (cid:3) The canonical inclusion (cid:3) → Cat induces the adjoint pair τ : cSet ⇄ Cat : N (cid:3) via hom-out andthe left Kan extension. In particular, N (cid:3) ( C ) n = Cat ([1] n , C ) . The functor τ takes a cubical set X to its fundamental category , which is obtained as the quotient of the free category on the graph X ⇒ X modulo the relations: σ x = id x and gf = qp for every -cube • f / / p (cid:15) (cid:15) • g (cid:15) (cid:15) • q / / • Marked cubical sets.
To define marked cubical sets, we need to introduce a new category (cid:3) ♯ , a slight enlargement of (cid:3) . The category (cid:3) ♯ consists of objects of the form [1] n for n = 0 , , . . . and an object [1] e . The maps of (cid:3) ♯ are generated by the usual generating maps of (cid:3) along with ϕ : [1] → [1] e and ζ : [1] e → [1] subject to an additional identity ζϕ = σ . Proposition 1.29.
The category (cid:3) ♯ is a Reedy category with the Reedy structure defined as follows: • deg([1] ) = 0 , deg[1] = 1 , deg([1] e ) = 2 , and deg([1] n ) = n + 1 for n ≥ ; • ( (cid:3) ♯ ) + is generated by face maps and ϕ under composition; • ( (cid:3) ♯ ) − is generated by degeneracy maps, connections, and ζ under composition. (cid:3) A structurally marked cubical set is a contravariant functor X : (cid:3) op ♯ → Set and a morphism ofstructurally marked cubical sets is a natural transformation of such functors. We will write cSet ′′ for the category of structurally marked cubical sets. When working with the category of structurallymarked cubical sets, we will write X n for the value of X at [1] n and X e for the value of X at [1] e .Structurally marked cubical sets should be thought of as cubical sets with (possibly multiple) labelson their edges such that for each vertex x , the degenerate edge xσ has, in particular, a distinguishedlabel xζ .A marked cubical set is a structurally marked cubical set for which the map X e → X is a monomor-phism. We write cSet ′ for the category of marked cubical sets. Alternatively, we may view a markedcubical set as a pair ( X, W X ) consisting of a cubical set X together with a subset W X ⊆ X ofedges that includes all degenerate edges and a morphism of marked cubical sets is a map of cubicalsets that preserves marked edges.The functor taking a (structurally) marked cubical set to its underlying cubical set admits botha left and a right adjoint, given by the minimal and maximal marking respectively. The minimalmarking on a cubical set X , denoted X ♭ , marks exactly the degenerate edges, whereas the maximalmarking, denoted X ♯ , marks all edges of X . If considered as structurally marked cubical sets,the marked edges of X ♭ and X ♯ are marked exactly once. Altogether we obtain the followingadjunctions cSet ′ ( ′ ) / / cSet ( − ) ♯ j j ( − ) ♭ t t The notation cSet ′ ( ′ ) above indicates that the same constructions can be applied to both markedand structurally marked cubical sets. In the context of (structurally) marked cubical sets, we regarda cubical set with its minimal marking by default, writing X for X ♭ .There is moreover an inclusion cSet ′ → cSet ′′ . This inclusion admits a left adjoint taking X ∈ cSet ′′ to Im X given by (Im X ) n = X n and (Im X ) e = ϕ ∗ ( X e ) , i.e., the image of X e under ϕ ∗ = X ( ϕ ) .The inclusion is easily seen to not have a right adjoint, since it fails to preserve the pushout of (cid:3) → ( (cid:3) ) ♯ against itself.Altogether we obtain the following diagram:(*) cSet ′′ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ Im + + cSet ′ { { ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ? _ o o cSet ( − ) ♭ Z Z ( − ) ♯ j j ( − ) ♭ ( − ) ♯ E E A geometric product entirely analogous to that of Subsection 1.2 exists for structurally markedcubical sets. We extend (cid:3) × (cid:3) → cSet to (cid:3) ♯ × (cid:3) ♯ → cSet ′′ by taking [1] e ⊗ [ n ] to have (cid:3) n +1 as the underlying cubical set with edges of the form (0 , x , . . . , x n +1 ) < (1 , x , . . . , x n +1 ) uniquelymarked. Similarly, let [ n ] ⊗ [1] e have (cid:3) n +1 as its underlying cubical set, and marked edges those of UBICAL MODELS OF ( ∞ , -CATEGORIES 13 the form ( x , . . . , x n , < ( x , . . . , x n , . Finally, let [1] e ⊗ [1] e := ( (cid:3) ) ♯ . The left Kan extensionyields ⊗ : cSet ′′ × cSet ′′ → cSet ′′ .This geometric product admits a concrete description analogous to that of Proposition 1.20. Proposition 1.30.
For
X, Y ∈ cSet ′′ , the geometric product X ⊗ Y admits the following description. • The underlying cubical set of X ⊗ Y is the geometric product of the underlying cubical setsof X and Y . • ( X ⊗ Y ) e is the set of all pairs of the form ( x : ( (cid:3) ) ♯ → X, y : (cid:3) → Y ) or ( x : (cid:3) → X, y : ( (cid:3) ) ♯ → Y ) , subject to the identification ( xζ, y ) = ( x, yζ ) for x : (cid:3) → X, y : (cid:3) → Y . • Structure maps not arising from those of the underlying cubical set are computed as follows: – ( x, y ) ζ = ( xζ, y ) = ( x, yζ ) ; – ( x, y ) ϕ = ( xϕ, y ) ; – ( x, y ) ϕ = ( x, yϕ ) .Proof. To compute the underlying cubical set of X ⊗ Y , we analyze maps (cid:3) k → X ⊗ Y exactly asin the proof of Proposition 1.20.Now we consider maps ( (cid:3) ) ♯ → X ⊗ Y . First observe that for every pair of maps x : ( (cid:3) ) ♯ → X, y : (cid:3) → Y we have a map ( x, y ) : ( (cid:3) ) ♯ ∼ = ( (cid:3) ) ♯ ⊗ (cid:3) → X ⊗ Y in the colimit cone, and thesame holds for x : (cid:3) → X, y : ( (cid:3) ) ♯ → Y . Once again, the stated computations of structure mapsfollow from the naturality of the colimit cone.Now we will show that every map p : ( (cid:3) ) ♯ → X ⊗ Y has the form described above. By Lemma 1.19,for every such map we have a commuting diagram ( (cid:3) ) ♯ ψ / / p % % ❑❑❑❑❑❑❑❑❑❑ (cid:3) m ( e ) ⊗ (cid:3) n ( e ) (cid:15) (cid:15) X ⊗ Y where the map (cid:3) m ( e ) ⊗ (cid:3) n ( e ) → X ⊗ Y is part of the colimit cone.First note that if ψ factors through ζ , then p = ( x, y ) ζ for some x : (cid:3) → X, y : (cid:3) → Y . Thistakes care of the case (cid:3) m ( e ) = (cid:3) m , (cid:3) n ( e ) = (cid:3) n , since any map from ( (cid:3) ) ♯ into these objects factorsthrough ζ .Now assume ψ does not factor through ζ , implying that at least one of (cid:3) m ( e ) , (cid:3) n ( e ) is ( (cid:3) ) ♯ ; then (cid:3) m ( e ) ⊗ (cid:3) n ( e ) is either (cid:3) m ⊗ ( (cid:3) ) ♯ , ( (cid:3) ) ♯ ⊗ (cid:3) n , or ( (cid:3) ) ♯ . Since every map ( (cid:3) ) ♯ → ( (cid:3) ) ♯ factors througheither (cid:3) ⊗ ( (cid:3) ) ♯ or ( (cid:3) ) ♯ ⊗ (cid:3) , we need only consider the first two cases. If (cid:3) m ( e ) ⊗ (cid:3) n ( e ) = (cid:3) m ⊗ ( (cid:3) ) ♯ ,then ψ picks out the unique marking on an edge of the form ( x , . . . , x m , < ( x , . . . , x m , . Inother words, ψ factors through the map ( x , . . . , x m ) ⊗ ( (cid:3) ) ♯ : (cid:3) ⊗ ( (cid:3) ) ♯ → (cid:3) ⊗ ( (cid:3) ) ♯ . Thus wehave reduced the problem to the case (cid:3) m ( e ) = (cid:3) , (cid:3) n ( e ) = ( (cid:3) ) ♯ – but since the only endomorphismof ( (cid:3) ) ♯ which does not factor through ζ is the identity, this implies that p = ( x, y ) for some x : (cid:3) → X, y : ( (cid:3) ) ♯ → Y . In the case (cid:3) m ( e ) = ( (cid:3) ) ♯ , (cid:3) n ( e ) = (cid:3) n , a similar analysis shows p = ( x, y ) for some x : ( (cid:3) ) ♯ → X, y : (cid:3) → Y . To show that the elements of ( X ⊗ Y ) e are subject to no further identifications, consider two pairs ( x, y ) , ( x ′ , y ′ ) which are identified in ( X ⊗ Y ) e . Considering the image of the cube correspondingto these pairs under the projections π X , π Y , we see that x = x ′ , y = y ′ . A similar proof holds foridentified pairs of the form ( x, y ) , ( x ′ , y ′ ) . Finally, if ( x, y ) = ( x, y ) , then applying the projectionsshows x = xζ, y = yζ . (cid:3) Corollary 1.31.
For
X, Y ∈ cSet ′′ , ( X ⊗ Y ) e ∼ = ( X e × Y ) ∪ ( X × Y ) ( X × Y e ) . (cid:3) What Proposition 1.30 shows is that for x : (cid:3) → X, y : (cid:3) → Y , the set of markings on theedge ( x, y ) in X ⊗ Y is simply the set of markings on x in X , that the analogous result holds for x : (cid:3) → X, y : (cid:3) → Y , and that for a pair of vertices x, y the distinguished marking ( x, y ) ζ isidentified with both ( xζ, y ) and ( x, yζ ) . Remark 1.32.
This monoidal structure restricts to a monoidal product on the category cSet ′ ofmarked cubical sets. Corollary 1.33.
All functors in the diagram (*) are monoidal. (cid:3)
Finally, as in the case of cubical sets, given a marked cubical set A , we form two non-isomorphicfunctors cSet ′ ( ′ ) → cSet ′ ( ′ ) : the left tensor − ⊗ A and the right tensor A ⊗ − . As they are bothco-continuous, they admit right adjoints and we write hom L ( A, − ) for the right adjoint of the lefttensor − ⊗ A and hom R ( A, − ) for the right adjoint of the right tensor A ⊗ − .2. Model structure on marked cubical sets
The goal of this section is to construct a combinatorial model category structure on the category cSet ′ of marked cubical sets. One would like to that by applying the Cisinski theory, as describedin Subsection 1.1, but unfortunately cSet ′ is not a presheaf category. Although there exists ageneralization of Cisinski theory to a non-topos case (due to Olschok [Ols09]), we choose to constructthe model structure directly, using Jeff Smith’s Theorem 1.1 to obtain a better understanding of itas a result. It is also worth pointing out that our language (e.g., cellular model, cylinder functor)follows the conventions of Cisinski to make the analogy with the Cisinski machinery clear.2.1. Classes of maps.
To begin, we lay out the definitions of the classes of maps that will comprisethe model structure.The cofibrations are the monomorphisms. The trivial fibrations are the maps with the right liftingproperty with respect to the cofibrations.Using Lemma 1.26, one obtains:
Lemma 2.1.
The cofibrations are the saturation of the set consisting of the boundary inclusions ∂ (cid:3) n → (cid:3) n for n ≥ and the inclusion (cid:3) → ( (cid:3) ) ♯ . (cid:3) By Lemma 2.1, we have a cofibrantly generated weak factorization system (cofibrations, trivialfibrations).
Definition 2.2.
We introduce three classes of maps in cSet ′ . UBICAL MODELS OF ( ∞ , -CATEGORIES 15 (i) Let the special open box inclusions ι ni,ε be the marked cubical set maps whose underlyingcubical set maps are the open box inclusions ⊓ ni,ε → (cid:3) n , with the critical edge marked ineach (except for the domain of ι i,ε , i.e. (cid:3) , in which the critical edge is not present).(ii) Let K be the cubical set depicted as: • / / • (cid:15) (cid:15) •• • / / • Let K ′ be the marked simplicial set that has the middle edge in the above marked. Definethe saturation map to be the inclusion K ⊆ K ′ .(iii) For each of the four faces of the square, let the associated to that face bethe inclusion of (cid:3) with all but that face marked into ( (cid:3) ) ♯ .The anodyne maps are defined as the saturation of the set of maps consisting of the special openbox inclusions, the saturation map, and the 3-out-of-4 maps. The naive fibrations are those mapsthat have the right lifting property against anodyne maps. Call an object X of cSet ′ a markedcubical quasicategory if the map X → (cid:3) is a naive fibration. Remark 2.3.
Viewing marked cubical quasicategories as ( ∞ , -categories, the marked edges rep-resent equivalences. The generating anodyne maps have the following ( ∞ , -categorical meanings. • The n -dimensional special open box fillings for n ≥ correspond to composition of mapsand homotopies, analogous to filling inner and special horns in quasicategories. They alsoensure that every morphism presented by a marked edge has a left and right inverse, i.e.,is an equivalences. • The 1-dimensional special open box fillings, ι ,ε : (cid:3) → ( (cid:3) ) ♯ , are the inclusions of endpointsinto the marked interval; thus marked edges may be lifted along naive fibrations, analogousto the lifting of isomorphisms along isofibrations in 1-category theory. • The saturation map ensures that equivalences, having both left and right inverses, aremarked. • The 3-out-of-4 maps represent the principle that if three maps in a commuting square areequivalences, then so is the fourth. They encode a condition analogous to the two-out-of-three property.
Remark 2.4.
For n ≥ , the representable marked cubical set (cid:3) n is not a marked cubical quasi-category, as it lacks fillers for certain special open boxes. This stands in constrast to the case ofsimplicial sets, in which the representables ∆ n are quasicategories. Lemma 2.5.
Let X be a marked cubical quasicategory, and x : (cid:3) → X an edge of X . Then x ismarked if and only if it factors through the inclusion of the middle edge (cid:3) → K .Proof. The inclusions K → K ′ and ( (cid:3) ) ♯ → K ′ are both anodyne (the latter as a composite ofspecial open box fillings). The stated result thus follows from the fact that X → (cid:3) has the rightlifting property with respect to both of these maps. (cid:3) Lemma 2.6.
For a marked cubical set X to be a marked cubical quasicategory, it suffices for themap X → (cid:3) to have the right lifting property with respect to special open box fillings and thesaturation map.Proof. Assume that X has the right lifting property with respect to special open box inclusionsand the saturation map. The proof of Lemma 2.5 only requires lifting with respect to these maps,so the marked edges of X are precisely those which factor through K .To show that X → (cid:3) lifts against the 3-out-of-4 maps, we must show that, if three sides of a 2-cubein X are marked, then so is the fourth. Using the fact that the three marked sides factor through K , we can show that the fourth does as well by a simple exercise in filling three-dimensional specialopen boxes. Hence the fourth edge is also marked. (cid:3) Remark 2.7.
In view of Lemma 2.6, whether omitting the 3-out-of-4 maps as generators wouldchange the class of anodyne maps. To see that it is, observe that, using the small object argument,we can factor any three-out-of-four map as a composite of a map in the saturation of the specialopen box fillings and two-out-of-six map, followed by a map having the right lifting property withrespect to these maps. Examining the details of this construction, we can see that the second ofthese maps will not have the right lifting property with respect to the 3-out-of-4 maps. Thus the3-out-of-4 maps are not in the saturation of the other two classes of generating anodynes.One may further note that, without the 3-out-of-4 maps as generators, anodyne maps would notbe closed under pushout product with cofibrations. This makes them crucial for our development.
Definition 2.8.
Given a map f : X → Y of marked cubical sets, a naive fibrant replacement of f consists of a diagram as depicted below, with X and Y marked cubical quasicategories, ι X and ι Y anodyne, and f a naive fibration. X f / / ι X (cid:15) (cid:15) Y ι Y (cid:15) (cid:15) X f / / Y We have a cofibrantly generated weak factorization system (anodyne maps, naive fibrations). Thisinduces a functorial factorization of any map X → Y as X f / / η f anod ! ! ❈❈❈❈❈❈❈❈ YM f Qf n . f . = = ⑤⑤⑤⑤⑤⑤⑤⑤ where Q is an endofunctor on ( cSet ′ ) → sending objects to naive fibrations and η : Id → Q ispointwise anodyne. Where f is the unique map X → (cid:3) , we write η X for η f . Given f : X → Y ,we can use this factorization to obtain a canonical naive fibrant replacement of f : X f / / η ηY f (cid:15) (cid:15) Y η Y (cid:15) (cid:15) X Q ( η Y f ) / / Y . UBICAL MODELS OF ( ∞ , -CATEGORIES 17 We declare f to be a weak equivalence if Q ( η Y f ) is a trivial fibration. A trivial cofibration is amap that is a cofibration and weak equivalence, and a fibration is a map that has the right liftingproperty against trivial cofibrations.We now want to show that if Y is a marked cubical quasicategory, so is hom L ( X, Y ) . The followinglemma on pushout-products helps with the proof of this fact. Lemma 2.9.
The pushout product of two cofibrations is a cofibration. Furthermore, the pushoutproduct of an anodyne map and a cofibration is anodyne.Proof.
Since ⊗ preserves colimits in each variable and anodynes are stable under pushouts andtransfinite compositions, we can use induction on skeleta to show that if S → T is one of thegenerating cofibrations (resp. anodynes), then ( S → T ) ˆ ⊗ ( ∂ (cid:3) n → (cid:3) n ) and ( S → T ) ˆ ⊗ ( (cid:3) → ( (cid:3) ) ♯ ) are cofibrations (resp. anodyne). This will show that if i and j are cofibrations, and i is anodyne,then i b ⊗ j is anodyne; the proof for the case where j is anodyne is entirely analogous.Several cases can be taken care of by the following fact: If f : A → B is an inclusion which isa surjection on vertices and p : X → Y is an isomorphism of underlying cubical sets, then f b ⊗ p is an isomorphism. This follows because the pushout-product is an isomorphism of underlyingcubical sets, and so we need only consider what edges are marked. But the marked edges of ( B ⊗ Y ) e = ( B e × Y ) ∪ B × Y ( B × Y e ) , and since each map is a bijection on vertices, all of theseedges appear in ( B ⊗ X ) ∪ A ⊗ X ( A ⊗ Y ) .This claim, along with the fact that taking the pushout-product with ∅ → (cid:3) is the identity,handles all but the following pushout products: • ( ∂ (cid:3) m → (cid:3) m ) b ⊗ ( ∂ (cid:3) n → (cid:3) n ) : this is the map ∂ (cid:3) m + n → (cid:3) m + n . This completes the proofof the first statement, concerning the pushout product of two cofibrations; the remainingcases complete the second statement, concerning the pushout product of a cofibration andan anodyne map. • ι mi,ε b ⊗ ( ∂ (cid:3) n → (cid:3) n ) : the underlying cubical set map is the open box inclusion ⊓ m + ni,ε → (cid:3) m + n ,with edges in the codomain being marked if and only if they are present and marked in thedomain. The critical edge is marked, so this is anodyne as a pushout of a special open boxfilling. • ι i,ε b ⊗ ( ∂ (cid:3) → ( (cid:3) ) ♯ ) : this is the 3-out-of-4 map associated to the face (1 , − ε ) . (cid:3) Corollary 2.10. If f : A → B is a cofibration and g : X → Y is a naive fibration, then the pullbackexponential f ⊲ g : hom( A, Y ) → hom( A, X ) × hom( A,Y ) hom( B, Y ) (where hom may designate either hom L or hom R ) is a naive fibration. Furthermore, if f is anodyne or g is a trivial fibration, then f ⊲ g is a trivial fibration.In particular, if Y is a marked cubical quasicategory, then for any X , hom( X, Y ) is a marked cubicalquasicategory.Proof. Let i : C → D be anodyne; we wish to show that f ⊲ g has the right lifting property withrespect to i . By a standard duality, it suffices to show that g has the right lifting property withrespect to i ˆ ⊗ f . This map is anodyne by Lemma 2.9, so the first statement holds. For the second statement, we can apply the same result with i an arbitrary cofibration. Then g hasthe right lifting property with respect to i b ⊗ f , either because f , and hence also i b ⊗ f , are anodyne,or because i b ⊗ f is a cofibration and g is a trivial fibration.The third statement follows from the first by the fact that hom( X, Y ) → (cid:3) is the pullbackexponential of the cofibration ∅ → X with the naive fibration Y → (cid:3) . (cid:3) Homotopies.
Next we define the closely-related concepts of connected components in amarked cubical set, and homotopies of maps between cubical sets.
Definition 2.11.
For a marked cubical set X , let ∼ denote the relation on X , the set of verticesof X , given by x ∼ y if there is a marked edge from x to y in X . Let ∼ denote the smallestequivalence relation on X containing ∼ . Remark 2.12.
For x, y ∈ X , one can easily see that x ∼ y if and only if x and y are connectedby a zigzag of marked edges. Definition 2.13.
For a marked cubical set X , the set of connected components π ( X ) is X / ∼ .We may observe that the construction of π ( X ) is functorial, since maps of marked cubical setspreserve marked edges, and hence preserve the equivalence relation ∼ . Definition 2.14. An elementary left homotopy h : f ∼ g between maps f, g : A → B is a map h : ( (cid:3) ) ♯ ⊗ A → B such that h | { }⊗ A = f and h | { }⊗ A = g . Note that the elementary left homotopy h corresponds to an edge ( (cid:3) ) ♯ → hom L ( A, B ) between the vertices corresponding to f and g . A left homotopy between f and g is a zig-zag of elementary left homotopies.A left homotopy from f to g corresponds to a zig-zag of marked edges in hom L ( A, B ) and so mapsfrom A to B are left homotopic exactly if π ( f ) = π ( g ) , where the set of connected components istaken in hom L ( A, B ) . We write [ A, B ] for the set of left homotopy classes of maps A → B .These induce notions of elementary left homotopy equivalence and left homotopy equivalence . Eachof these notions has a “right” variant using A ⊗ ( (cid:3) ) ♯ and hom R ( A, B ) . Unless the potential forconfusion arises or a statement depends on the choice, we will drop the use of “left” and “right”. Lemma 2.15.
In a marked cubical quasicategory X , the relations ∼ and ∼ conicide.Proof. Using 2-dimensional open box fillers with certain edges degenerate, and the 3-out-of-4 prop-erty, we can reduce any zigzag of marked edges connecting x and y in X to a single marked edgefrom x to y . (cid:3) By adjointness, we obtain the following corollary.
Corollary 2.16. If f, g : A → B are homotopic and B a marked cubical quasicategory, then f and g are elementarily homotopic. Hence, between marked cubical quasicategories homotopy equivalencescoincide with elementary homotopy equivalences.Proof. By Corollary 2.10, hom(
A, B ) is a marked cubical quasicategory, and so ∼ is an equivalencerelation on hom( A, B ) by Lemma 2.15. Translating what this means for homotopies gives theresult. (cid:3) UBICAL MODELS OF ( ∞ , -CATEGORIES 19 Lemma 2.17. If f, g : X → Y are left homotopic, then for any Z , then the induced maps hom L ( Y, Z ) → hom L ( X, Z ) are right homotopic.Proof. We consider the case of elementary homotopies; the general result follows from this. Anelementary left homotopy f ∼ g is given by a map H : ( (cid:3) ) ♯ ⊗ X → Y . Pre-composition with H induces a map hom L ( Y, Z ) → hom L (( (cid:3) ) ♯ ⊗ X, Z ) . Under the adjunction defining hom L ,this corresponds to a map hom L ( Y, Z ) ⊗ ( (cid:3) ) ♯ ⊗ X → Z , which in turn corresponds to a map hom L ( Y, Z ) ⊗ ( (cid:3) ) ♯ → hom L ( X, Z ) . It is easy to see that this map defines an elementary righthomotopy between the pre-composition maps induced by f and g . (cid:3) Category theory in a marked cubical quasicategory.
Let X be a marked cubical qua-sicategory and x, y ∈ X . We will write X ( x, y ) for the subset of X consisting of -cubes f with f ∂ , = x and f ∂ , = y . Define an equivalence relation relation ∼ X on the set X ( x, y ) of edgesfrom x to y as follows: f ∼ X g if and only if there is a -cube in X of the form x f / / yx g / / y It is straightforward to verify that this is indeed an equivalence relation: reflexivity follows fromdegeneracies, whereas symmetry and transitivity are given by filling -dimensional open boxes.We now define three increasingly strong refinements of the concept of a homotopy equivalence. Definition 2.18.
Let f : X → Y be a map in cSet . Then: • f is a semi-adjoint equivalence if there exist g : Y → X and homotopies H : gf ∼ id X , K : f g ∼ id Y such that f H ∼ Kf as edges of hom( X, Y ) ; • f is a strong homotopy equivalence if there exist g, H, K as above with f H = Kf ; • a map g : Y → X is a strong deformation section of f if f g = id Y and there exists ahomotopy H : gf ∼ id X such that f H = id f .Our next goal will be two show the following: Lemma 2.19.
Let f : X → Y be a map of marked cubical quasicategories. The following areequivalent:(i) f is a homotopy equivalence;(ii) f is a semi-adjoint equivalence.Furthermore, if f is a naive fibration, then these are equivalent to:(iii) f is a strong homotopy equivalence. We will prove this by means of a 2-categorical argument.We now define the homotopy category Ho X of a marked cubical quasicategory X as follows: • the objects of Ho X are the -cubes of X ; • the morphisms from x to y in Ho X are the equivalence classes of edges X ( x, y ) / ∼ X ; • the identity map on x ∈ X is given by xσ ; • the composition of f : x → y and g : y → z is given by filling the open box x f / / y g (cid:15) (cid:15) x gf / / z Using standard arguments about open box fillings, one verifies the following lemma.
Lemma 2.20.
The above data define a category. (cid:3)
Lemma 2.21.
Let X be a marked cubical quasicategory. If there is a -cube of the form x f / / p (cid:15) (cid:15) y g (cid:15) (cid:15) z q / / w then gf = qp in Ho X .Proof. The desired homotopy appears as the top face of the following 3-cube: x / / (cid:15) (cid:15) ❆❆❆❆❆❆❆❆ ❆❆❆❆❆❆❆❆ y (cid:15) (cid:15) ❆❆❆❆❆❆❆❆ x qp / / qp (cid:15) (cid:15) wz / / ❇❇❇❇❇❇❇❇ w ❇❇❇❇❇❇❇❇ ❇❇❇❇❇❇❇❇ w w The remaining faces of the cube form a special open box in X , with critical edge wσ ; thus we canobtain the full cube, and in particular the top face, by filling this open box. The result then followsby the fact that composition in Ho X is well-defined. (cid:3) Lemma 2.22.
For a marked cubical quasicategory X , the categories Ho X and τ X are equivalent.Proof. There is a natural inclusion of Ho X → τ X , which is the identity on objects and takes a -cube f to a string of length consisting of f . This is clearly faithful and essentially surjective.To see that it is full, we simply fill in -dimensional open boxes with one degenerate edge to reducea sequence of arbitrary length to a sequence of length . (cid:3) UBICAL MODELS OF ( ∞ , -CATEGORIES 21 The assignment X → Ho X extends in a straightforward manner to a functor taking a markedcubical quasicategory to its homotopy category. Postcomposing this functor with core : Cat → Gpd ,we obtain a groupoid Ho ♯ X . Lemma 2.23.
The groupoid Ho ♯ X can be constructed directly as follows: • Objects are 0-cubes of X ; • Morphisms from x to y are equivalence classes of marked edges from x to y ; • Composition and identities are defined as in Ho X .Proof. Let X be a marked cubical quasicategory. It is easy to see that an edge f : (cid:3) → X isinvertible in Ho X if and only if it factors through the map (cid:3) → K which picks out the middleedge. Since the inclusions ( (cid:3) ) ♯ → K ′ and K → K ′ are anodyne, this holds if and only if f ismarked. (cid:3) Definition 2.24.
Define a strict 2-category Ho cSet ′ whose objects are the marked cubical quasi-categories and whose mapping category from X to Y is Ho cSet ′ ( X, Y ) := Ho hom L ( X, Y ) .This means the 1-morphisms are the usual 1-morphisms X → Y , and the 2-morphisms are maps X ⊗ (cid:3) → Y , modulo an equivalence relation. Denote the (vertical) composition in Ho hom L ( X, Y ) with ◦ . The (horizontal) composition Ho hom L ( Y, Z ) × Ho hom L ( X, Y ) → Ho hom L ( X, Z ) (which will be written by concatenation) is defined on objects by the usual composition. If H : Y ⊗ (cid:3) → Z and K : X ⊗ (cid:3) → Y are morphisms K : g → g ′ and H : f → f ′ , respectively, define themorphism KH : gf → g ′ f ′ by choosing a fill for the open box of hom L ( X, Z ) depicted by gf Kf / / g ′ f g ′ H (cid:15) (cid:15) gf KH / / g ′ f ′ where the top edge is induced by the composite X ⊗ (cid:3) → Y ⊗ (cid:3) → Z and the right edge by X ⊗ (cid:3) → Y → Z . The fact that the hom L ( X, Y ) are marked cubical quasicategories ensures thisdefines a well-defined, associative, unital, and functorial operation. For functoriality, note that themorphism X ⊗ (cid:3) ⊗ (cid:3) H ⊗ (cid:3) → Y ⊗ (cid:3) K → Z yields a 2-cube (cid:3) → hom L ( X, Z ) which can be depictedas gf Kf / / gH (cid:15) (cid:15) g ′ f g ′ H (cid:15) (cid:15) gf ′ Kf ′ / / g ′ f ′ and so by Lemma 2.21, we have ( g ′ H ) ◦ ( Kf ) = ( Kf ′ ) ◦ ( gH ) , which implies the interchange law. Definition 2.25.
Let Ho ♯ cSet ′ denote the maximal (2 , -category contained in Ho cSet ′ , i.e. the2-category whose objects are marked cubical sets, with Ho ♯ cSet ′ ( X, Y ) = Ho ♯ hom L ( X, Y ) , and the2-categorical operations induced by those of Ho cSet ′ .The Ho ♯ construction, together with the following general results about (2 , -categories, give usthe desired result about compatibility of homotopies. Lemma 2.26 (Undergraduate Lemma) . Let X be an object in a (2 , -category C , and let H : p ∼ id X be a morphism in C ( X, X ) . Then pH = Hp .Proof. By the interchange law, H ◦ ( pH ) = ( H id X ) ◦ ( pH ) = ( id X H ) ◦ ( Hp ) = H ◦ ( Hp ) . Since C ( X, X ) is a groupoid, we can cancel H . (cid:3) Lemma 2.27 (Graduate Lemma) . Let
X, Y be objects in a (2 , -category C , f : X ⇆ Y : g twomorphisms between them, and H : gf → id X and K : f g → id Y two 2-cells. Then there is a 2-cell K ′ : f g → id Y for which K ′ f = f H .Proof. Define K ′ := K ◦ ( f Hg ) ◦ ( Kf g ) − . Now, we compute: K ′ f = Kf ◦ ( f Hgf ) ◦ ( Kf gf ) − = Kf ◦ ( f gf H ) ◦ ( Kf gf ) − (by 2.26) = f H (by naturality/interchange) (cid:3) Proof of Lemma 2.19.
The implications ( iii ) ⇒ ( ii ) ⇒ ( i ) are clear. The implication ( i ) ⇒ ( ii ) follows from applying Lemma 2.27 to the (2 , -category Ho ♯ cSet ′ .Now let f be a naive fibration and a semi-adjoint equivalence. By Corollary 2.10, the map hom( X, X ) → hom( X, Y ) is a naive fibration. A simple exercise in 2-dimensional special openbox filling, using this fact and the definition of a semi-adjoint equivalence, shows that there existsa homotopy H ′ : gf ∼ id X such that f H ′ = Kf . (cid:3) Fibration category of marked cubical quasicategories.Lemma 2.28.
Every anodyne map between marked cubical quasicategories is a homotopy equiva-lence.Proof.
Now let f : X → Y be anodyne, with X and Y marked cubical quasicategories. We canobtain a retraction r : Y → X as a lift in the following diagram: X f (cid:15) (cid:15) X (cid:15) (cid:15) Y / / (cid:3) We can then obtain a left homotopy f r ∼ id Y as a lift in the following diagram: UBICAL MODELS OF ( ∞ , -CATEGORIES 23 ( ∂ (cid:3) ⊗ Y ) ∪ (( (cid:3) ) ♯ ⊗ X ) (cid:15) (cid:15) [[ fr, id Y ] ,fπ ] / / Y (cid:15) (cid:15) ( (cid:3) ) ♯ ⊗ Y / / (cid:3) The lift exists since the left-hand map is anodyne by Lemma 2.9.An analogous proof shows that f is a right homotopy equivalence. (cid:3) Lemma 2.29.
Let f : X → Y be a naive fibration. The following are equivalent:(i) f is a trivial fibration;(ii) f has a strong deformation section;(iii) f is a strong homotopy equivalence.Proof. If f : X → Y is a trivial fibration, then we can obtain a section g : Y → X as a lift of thefollowing diagram: ∅ / / (cid:15) (cid:15) X f (cid:15) (cid:15) Y Y
We can then obtain a left homotopy H : gf ∼ id X satisfying f H = id f as a lift in the followingdiagram: X ⊔ X (cid:15) (cid:15) (cid:15) (cid:15) [ sf, id X ] / / X f (cid:15) (cid:15) ( (cid:3) ) ♯ ⊗ X fπ X / / Y This shows ( i ) ⇒ ( ii ) and the implication ( ii ) ⇒ ( iii ) is trivial. To show that ( iii ) ⇒ ( i ) , we firstshow that ( iii ) implies the following condition:(iii)’ the canonical map ι , ⊲ f → f in ( cSet ′ ) → admits a section.To see ( iii ) ⇒ ( iii ) ′ , suppose f is a strong homotopy equivalence with homotopy inverse g : Y → X and homotopies H : gf ∼ id X , K : f g ∼ id Y satisfying f H = Kf . Then we have the followingcommuting diagram in cSet ′ : X / / f (cid:15) (cid:15) hom(( (cid:3) ) ♯ , X ) ι , ⊲f (cid:15) (cid:15) / / X f (cid:15) (cid:15) Y / / X × Y hom(( (cid:3) ) ♯ , Y ) / / Y The top-left map is the adjunct of H , while the bottom-left map is induced by g and the adjunctof K ; the right-hand square is as in the statement of condition ( iii ) ′ . It is easy to see that thecomposite square is simply the identity square on f . Finally, note that ι , ⊲ f is a trivial fibration by Corollary 2.10. Therefore, if the square given inthe statement of condition ( iii ) ′ has a section, then f is a trivial fibration as a retract of a trivialfibration. Thus ( iii ) ′ ⇒ ( i ) . (cid:3) Corollary 2.30.
A map f : X → Y between marked cubical quasicategories is a trivial fibrationexactly if it is a homotopy equivalence and a naive fibration.Proof. This follows from Lemmas 2.19 and 2.29, together with the fact that every trivial fibrationis a naive fibration since all anodyne maps are cofibrations. (cid:3)
Proposition 2.31.
The category of marked cubical quasicategories forms a fibration category, withnaive fibrations as the fibrations and homotopy equivalences as the weak equivalences.Proof.
The class of homotopy equivalences is closed under 2-out-of-3. Corollary 2.30 shows that themaps between marked cubical quasicategories which are naive fibrations and homotopy equivalencesare exactly the trivial fibrations; both fibrations and trivial fibrations are defined via a right liftingproperty, and hence they are stable under pullback. By Lemma 2.28, each anodyne map betweenmarked cubical quasicategories is a homotopy equivalence, and so the (anodyne, naive fibration)-factorization gives the factorization axiom. (cid:3)
Lemma 2.32.
Let f : X → Y be a map between marked cubical quasicategories. Then the followingconditions are equivalent:(i) f is a weak equivalence;(ii) f is a left homotopy equivalence;(iii) f is a right homotopy equivalence.Proof. Consider the canonical naive fibrant replacement of f used in the definition of the weakequivalences: X f / / ι X (cid:15) (cid:15) Y ι Y (cid:15) (cid:15) X f / / Y (here ι Y = η Y , f = Q ( η Y f ) , ι X = η η Y f ).By Lemma 2.28, ι X and ι Y are left homotopy equivalences. It is easy to show that left homotopyequivalences satisfy the two-out-of-three property, so f is a left homotopy equivalence if and only if f is one. By Corollary 2.30, f is a left homotopy equivalence if and only if it is a trivial fibration,i.e. if and only if f is a weak equivalence. So ( i ) ⇐⇒ ( ii ) ; an analogous argument shows ( i ) ⇐⇒ ( iii ) . (cid:3) Cofibration category of marked cubical sets.
Our next result shows that the definitionof the weak equivalences is not sensitive to the choice of naive fibrant replacement.
Lemma 2.33.
Let f : X → Y be a map of marked cubical sets. The following are equivalent:(i) f is a weak equivalence. UBICAL MODELS OF ( ∞ , -CATEGORIES 25 (ii) there exists a naive fibrant replacement of f by a trivial fibration;(iii) any naive fibrant replacement of f is a trivial fibration.Proof. ( i ) ⇒ ( ii ) and ( iii ) ⇒ ( i ) are immediate from the definition of the weak equivalences. Toprove ( ii ) ⇒ ( iii ) , consider a map f : X → Y having a naive fibrant replacement by a trivialfibration f : X → Y , and an arbitrary naive fibrant replacement f ′ : X ′ → Y ′ of f . As depictedbelow, let f ′′ : X ′′ → Y ′′ be a naive fibrant replacement of the induced map between the pushouts X ∪ X X ′ → Y ∪ Y Y ′ . X / / (cid:15) (cid:15) f & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ X ′ (cid:15) (cid:15) f ′ & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ Y / / (cid:15) (cid:15) Y ′ (cid:15) (cid:15) X / / f % % ▲▲▲▲▲▲▲▲▲▲▲▲▲ X ∪ X X ′ % % ▲▲▲▲▲▲▲▲▲▲ / / X ′′ f ′′ % % ▲▲▲▲▲▲▲▲▲▲▲▲ Y / / Y ∪ Y Y ′ / / Y ′′ The maps X → X ′′ , Y → Y ′′ , X ′ → X ′′ , Y ′ → Y ′′ are anodyne, as anodyne maps are closed underpushout and composition. Furthermore, f is a trivial fibration by assumption. Thus all of thesemaps are homotopy equivalences by Lemma 2.28 and Corollary 2.30. So we can apply the two-out-of-three property to see that f ′′ is a homotopy equivalence; applying it again, we see that f ′ is ahomotopy equivalence. Thus f ′ is a trivial fibration by Corollary 2.30. Since f ′ was arbitrary, wehave shown that f satisfies condition ( iii ) . (cid:3) Corollary 2.34.
Every anodyne map is a weak equivalence.Proof.
Let f : X → Y be anodyne. The following diagram gives a naive fibrant replacement of f : X f / / η Y f (cid:15) (cid:15) Y η Y (cid:15) (cid:15) Y Y
Since id Y is a trivial fibration, f is a weak equivalence by Lemma 2.33. (cid:3) Lemma 2.35.
The following are equivalent for a marked cubical map A → B :(i) A → B is a weak equivalence;(ii) for any marked cubical quasicategory X , the induced map hom( B, X ) → hom( A, X ) is ahomotopy equivalence; (iii) for any marked cubical quasicategory X , the induced map π (hom( B, X )) → π (hom( A, X )) is a bijection.Proof. First, suppose that A → B is a weak equivalence. Thus, there is a square A / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / B with A → A and B → B anodyne, and A → B a trivial fibration. By Corollary 2.30, A → B is aleft homotopy equivalence.Applying hom L ( − , X ) to the diagram above, we obtain a diagram in which all objects are markedcubical quasicategories by Corollary 2.10: hom L ( A, X ) hom L ( B, X ) o o hom L ( A, X ) O O hom L ( B, X ) o o O O The vertical maps are trivial fibrations by Corollary 2.10, hence homotopy equivalences by Corol-lary 2.30. By Lemma 2.17, the bottom horizontal map is a right homotopy equivalence, since A → B is a left homotopy equivalence. Hence so is the upper horizontal map by 2-out-of-3. Thuswe have proven ( i ) ⇒ ( ii ) .The implication ( ii ) ⇒ ( iii ) is clear, so it remains to show ( iii ) ⇒ ( i ) . For that, we first observe thatit suffices to consider A and B marked cubical quasicategories. To see this, consider the canonicalnaive fibrant replacement f : A → B of a map f : A → B . By definition, f is a weak equivalence ifand only if f is a trivial fibration; by Corollary 2.30 and Lemma 2.32, this holds if and only if f is aweak equivalence. Furthermore, the anodyne maps ι X , ι Y are weak equivalences by Corollary 2.34,and therefore satisfy condition ( iii ) ; hence f satisfies condition ( iii ) if and only if f does, by the2-out-of-3 property for bijections.Hence we can assume A and B are marked cubical quasicategories. Now take X := A and set g := ( π f ∗ ) − [id A ] . The verification that a representative of the class g ∈ π hom L ( B, A ) defines ahomotopy inverse of f is straightforward; thus f is a weak equivalence by Lemma 2.32. (cid:3) Corollary 2.36.
The weak equivalences satisfy the 2-out-of-6 property (and hence the 2-out-of-3property).Proof.
This is immediate from condition ( iii ) of Lemma 2.35. (cid:3) Corollary 2.37.
The endpoint inclusions (cid:3) → K are trivial cofibrations.Proof. The maps in question are clearly cofibrations. To see that they are weak equivalences,consider the following commuting diagram:
UBICAL MODELS OF ( ∞ , -CATEGORIES 27 (cid:3) / / (cid:15) (cid:15) K (cid:15) (cid:15) ( (cid:3) ) ♯ / / K ′ The left, right, and bottom maps are anodyne, hence weak equivalences by Corollary 2.34. Thusthe top map is a weak equivalence by Corollary 2.36. (cid:3)
Lemma 2.38.
Trivial fibrations are weak equivalences.Proof. If A → B is a trivial fibration, then it is a homotopy equivalence by Corollary 2.30. Hence hom( B, X ) → hom( A, X ) is a homotopy equivalence for all marked cubical quasicategories X byLemma 2.17, and hence A → B a weak equivalence by Lemma 2.35. (cid:3) Proposition 2.39.
The category of marked cubical sets forms a cofibration category with the aboveclasses of weak equivalences and cofibrations.Proof.
The class of weak equivalences is closed under 2-out-of-3 by Corollary 2.36. The categoryclearly has an initial object and pushouts. Cofibrations are the left class in a weak factorization sys-tem, hence stable under pushout. Using the characterization of weak equivalences given by item (ii)of Lemma 2.35, stability of cofibrations that are weak equivalences under pushout reduces to sta-bility of trivial fibrations under pullback. By Lemma 2.38, trivial fibrations are weak equivalences,so the (cofibration, trivial fibration)-factorization gives the factorization axiom. (cid:3)
Model structure for marked cubical quasicategories.Definition 2.40.
A marked cubical set is finite (resp. countable ) if it has only finitely (resp.countably) many non-degenerate cubes. The cardinality of a finite marked cubical set is its totalnumber of non-degenerate cubes, in all dimensions.
Lemma 2.41.
The trivial fibrations form an ω -accessible, ω -accessibly embedded subcategory of ( cSet ′ ) → .Proof. It suffices to show two things: that filtered colimits (and hence in particular ω -filteredcolimits) in cSet ′ preserve trivial fibrations, and that any trivial fibration can be expressed as an ω -filtered colimit in cSet ′ of trivial fibrations between countable marked cubical sets. The firststatement follows from the fact that the domains and codomains of the generating cofibrations arefinite.For the second statement, consider a trivial fibration f : X → Y . Let P denote the poset ofcountable subcomplexes of X ; note that we consider edges of subcomplexes of X to be markedif and only if they are marked in X . This category is ω -filtered since any countable union ofcountable subcomplexes is countable.Let i denote the inclusion P ֒ → cSet ′ ; the colimit of this diagram is X . The images under f of thecountable subcomplexes of X , with the natural inclusions, also define a diagram f i : P → cSet ′ .One can easily show that trivial fibrations are surjective on underlying cubical sets; thus everycube of Y appears in f S for some countable subcomplex S ⊆ X . So f i is a filtered diagram ofsubcomplexes of Y , in which the maps are inclusions and each cube of Y is contained in some object of the diagram, with every marked edge of Y being marked in some subcomplex in the diagram.From this, one can show that the colimit of f i is Y . The map f induces a natural transformationfrom i to f i , whose induced map on the colimits is f itself.However, it may not be the case that for every component of this natural transformation is a trivialfibration. Thus we will replace i by a different diagram, still having colimit X , with a naturaltransformation to f i which does satisfy this property. For each countable subcomplex S ⊆ X , wewill define a new countable subcomplex S ⊆ X , such that f S = f S , f | S : S → f S is a trivialfibration, and for S ′ ⊆ S , we have S ′ ⊆ S .We first define S for finite S , proceeding by induction on cardinality. For S = ∅ , we can simplyset S = ∅ . Now assume that we have defined S for | S | ≤ m , and consider a subcomplex S ofcardinality m + 1 . We will inductively define a family of subcomplexes S i for i ≥ , each countableand satisfying f S i = f S . Begin by setting S = S ∪ S S ′ ( S S ′ . Then S is countable, f S = f S , andfor S ′ ⊆ S we have S ′ ⊆ S .Now assume that we have defined S i for some i ≥ , and let D be the set of all diagrams D of theform: ∂ (cid:3) n ∂x D / / (cid:15) (cid:15) S i (cid:15) (cid:15) (cid:3) n y D / / f S Because S i and f S are countable, while ∂ (cid:3) n and (cid:3) n are finite for any given n , there are countablymany such diagrams. Because f is a trivial fibration, for each such diagram we may choose afiller in X , i.e. an n -cube x D : (cid:3) n → X whose boundary is ∂x D , such that f x D = y D . Let S i +1 = S i ∪ S D ∈D { x D } . Then S i +1 is still countable, since we have added at most countably manycubes to S i , and its image under f is still f S , since each x D was chosen to map to a specific y D ∈ f S .Now let S = S i ≥ S i . This is countable, its image is f S , and for any S ′ ⊆ S we have S ′ ⊆ S . Nowconsider a diagram: ∂ (cid:3) n ∂x / / (cid:15) (cid:15) S (cid:15) (cid:15) (cid:3) n y / / f S Because (cid:3) n is finite, the image of ∂x is contained in some finite subcomplex of S , hence in some S i , so it has a filler in S i +1 which maps to y . Furthermore, f | S has the right lifting property withrespect to the map (cid:3) → ( (cid:3) ) ♯ , i.e. an edge x : (cid:3) → S is marked if and only f x is marked, sincethis is true of edges in X . Thus f | S : S → f S is a trivial fibration.For a countably infinite S ⊆ X we let S = S S ′ , where the union is taken over all finite subcomplexes S ′ ⊆ S . Then f | S is the filtered colimit of the trivial fibrations f | S ′ , hence it is a trivial fibration. UBICAL MODELS OF ( ∞ , -CATEGORIES 29 The subcomplexes S with the natural inclusions define a diagram i : P → cSet ′ , and f induces anatural trivial fibration i = ⇒ f i . Observe that i is a filtered diagram of subcomplexes of X , inwhich the maps are inclusions and edges in the objects are marked if and only if they are markedin X ; furthermore, every cube of X is contained in some finite subcomplex S , and hence in S .From this we can deduce that the colimit of i is X , by the same argument we used to show thatthe colimit of f i is Y . The induced map between colimits is f ; thus we have expressed f as an ω -filtered colimit of trivial fibrations between countable marked cubical sets. (cid:3) Lemma 2.42.
The weak equivalences form an ω -accessible, ω -accessibly embedded subcategory of ( cSet ′ ) → .Proof. The (anodyne, naive fibration) factorization gives us a naive fibrant replacement functor F : ( cSet ′ ) → → ( cSet ′ ) → . By [Joy09, Prop. D.2.10], this functor is ω -accessible, since the domainsand codomains of the generating anodyne maps are all countable. By definition, the category ofweak equivalences we is given by the following pullback in Cat : we / / (cid:15) (cid:15) ❴✤ ( cSet ′ ) → F (cid:15) (cid:15) tfib / / / / ( cSet ′ ) → By Lemma 2.41, tfib is an ω -accessible category, and its embedding into ( cSet ′ ) → is an ω -accessiblefunctor. By [MP89, Thm. 5.1.6], the category of ω -accessible categories and ω -accesible functorshas finite limits, and these are computed in Cat . Thus we is ω -accessible, and its embedding into ( cSet ′ ) → is an ω -accessible functor. (cid:3) Theorem 2.43 (Analogue of model structure on marked simplicial sets) . The above classes of weakequivalences, cofibrations, and fibrations define a model structure on cSet ′ .Proof. We verify the assumptions of Theorem 1.1.The category of marked cubical sets is locally finitely presentable. Weak equivalences are an ω -accessibly embedded, ω -accessible subcategory of ( cSet ′ ) → by Lemma 2.42. Cofibrations have asmall set of generators by Lemma 2.1.Weak equivalences are closed under 2-out-of-3 and weak equivalences that are cofibrations are closedunder pushout by Proposition 2.39. Weak equivalences are closed under transfinite composition byLemma 2.42, implying that the same holds for trivial cofibrations. Every map lifting againstcofibrations is a weak equivalence by Lemma 2.38. (cid:3) We refer to the model structure constructed above as the cubical marked model structure . We willnow analyze this model structure, beginning with a strengthening of Lemma 2.9 and Corollary 2.10.
Lemma 2.44. If X → Y is a weak equivalence, then so is A ⊗ X → A ⊗ Y for any A ∈ cSet ′ .Proof. By the adjunction A ⊗ − ⊣ hom R ( A, − ) , for Z ∈ cSet ′ we have a natural isomorphism hom R ( A ⊗ X, Z ) ∼ = hom R ( X, hom R ( A, Z )) . Let Z be a marked cubical quasicategory; then we havea commuting diagram hom R ( A ⊗ Y, Z ) ∼ = (cid:15) (cid:15) / / hom R ( A ⊗ X, Z ) ∼ = (cid:15) (cid:15) hom R ( Y, hom R ( A, Z )) / / hom R ( X, hom R ( A, Z )) By Corollary 2.10, hom R ( A, Z ) is a marked cubical quasicategory, so the bottom map is a homotopyequivalence by Lemma 2.35. Hence the top map is a homotopy equivalence; thus we see that A ⊗ X → A ⊗ Y is a weak equivalence by Lemma 2.35. (cid:3) Lemma 2.45.
The pushout product of a cofibration and a weak equivalence is a weak equivalence.Proof.
Let i : A → B be a cofibration and f : X → Y a weak equivalence; we will show that i b ⊗ f is a weak equivalence (the case of f b ⊗ i is similar). Consider the diagram which defines i b ⊗ f : A ⊗ X / / (cid:15) (cid:15) B ⊗ X (cid:15) (cid:15) (cid:21) (cid:21) A ⊗ Y / / A ⊗ Y ∪ A ⊗ X B ⊗ X i b ⊗ f ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ B ⊗ Y The maps A ⊗ X → A ⊗ Y and B ⊗ X → B ⊗ Y are weak equivalences by Lemma 2.44. The map A ⊗ X → B ⊗ X is a cofibration by Lemma 2.9. The model structure is left proper, since all objectsare cofibrant; thus the map from B ⊗ X into the pushout is a weak equivalence. Hence i b ⊗ f is aweak equivalence by 2-out-of-3. (cid:3) Corollary 2.46.
Let i : A → B, j : A ′ → B ′ be cofibrations. If either i or j is trivial, then so is thepushout product i b ⊗ j .Proof. This is immediate from Lemmas 2.9 and 2.45. (cid:3)
Corollary 2.47. If i is a cofibration and f is a fibration, then the pullback exponential i ⊲ f is afibration, which is trivial if i or f is trivial. (cid:3) Corollary 2.48.
The category cSet ′ , equipped with the cubical marked model structure and thegeometric product, is a monoidal model category. (cid:3) Next we will characterize the fibrant objects, and fibrations between fibrant objects, of this modelstructure.
Proposition 2.49.
A map between marked cubical quasicategories is a fibration if and only if itis a naive fibration. In particular, the fibrant objects of the cubical marked model structure areprecisely the marked cubical quasicategories.Proof.
It is clear that every fibration is a naive fibration. Now let f : X → Y be a naive fibrationbetween marked cubical quasicategories, and i : A → B a trivial cofibration. We wish to show that f has the right lifting property with respect to i ; for this it suffices to show that i ⊲ f has the right UBICAL MODELS OF ( ∞ , -CATEGORIES 31 lifting property with respect to the map ∅ → (cid:3) . For this, in turn, it suffices to show that i ⊲ f isa trivial fibration.First, note that i ⊲ f is a naive fibration between marked cubical quasicategories by Corollary 2.10.Therefore, by Corollary 2.30, it is a trivial fibration if and only if it is a homotopy equivalence.Now consider the diagram which defines i ⊲ f : hom( B, X ) i⊲f * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ( ( + + hom( A, X ) × hom( A,Y ) hom( B, Y ) ❴✤ / / (cid:15) (cid:15) hom( A, X ) (cid:15) (cid:15) hom( B, Y ) / / hom( A, Y ) The maps hom(
B, X ) → hom( A, X ) and hom( B, Y ) → hom( A, Y ) are trivial fibrations by Corol-lary 2.47; the map from the pullback to hom( A, X ) is a trivial fibration as a pullback of a triv-ial fibration. Thus i ⊲ f is a weak equivalence by 2-out-of-3, hence a homotopy equivalence byLemma 2.32. (cid:3) Model structure on structurally marked cubical sets
The model structure on marked cubical sets described in the previous section resembles the Cisin-ski model structure on a presheaf category. In this section, we show that the category cSet ′′ ofstructurally marked cubical sets (see Subsection 1.4) admits a Cisinski model structure which rightinduces the model structure on marked cubical sets from the previous section via the embedding cSet ′ ֒ → cSet ′′ , and that the two are Quillen equivalent.Since cSet ′′ is a presheaf category, we may apply Theorem 1.5 in order to construct a modelstructure on this category. To do that, we first find a cellular model for cSet ′′ , i.e., a generating setof monomorphisms, using the Reedy category structure of (cid:3) ♯ , established in Proposition 1.29. Lemma 3.1.
The monomorphisms of cSet ′′ are the saturation of the set consisting of the boundaryinclusions ∂ (cid:3) n ֒ → (cid:3) n and the inclusion (cid:3) ֒ → ( (cid:3) ) ♯ . (cid:3) The functor ( (cid:3) ) ♯ ⊗ − : cSet ′′ → cSet ′′ , together with the natural transformations ∂ , ⊗ − , ∂ , ⊗− : id → ( (cid:3) ) ♯ ⊗ − , and π : ( (cid:3) ) ♯ ⊗ − → id , defines a cylinder functor on cSet ′′ in the sense ofDefinition 1.2.Thus we have a notion of homotopy defined in terms of this cylinder functor: an elementaryhomotopy f ∼ g : X → Y is a map H : ( (cid:3) ) ♯ ⊗ X → Y with H | { }⊗ X = f, H | { }⊗ X = g , and ahomotopy is a zigzag of elementary homotopies. In keeping with the notation of Subsection 1.1, wewill write [ X, Y ] for the set of homotopy classes of maps from X to Y . Lemma 3.2. (i) The cylinder functors in cSet ′ and cSet ′′ agree, i.e., the latter is the image of the formerunder the embedding cSet ′ → cSet ′′ . (ii) For marked cubical sets X and Y , the embedding cSet ′ → cSet ′′ induces a bijection [ X, Y ] cSet ′ → [ X, Y ] cSet ′′ ,where the subscript indicates which category the homotopy classes are taken in.Proof. Both of these statements follow easily from the fact that the embedding cSet ′ → cSet ′′ ismonoidal, established in Corollary 1.33. (cid:3) Let S be the set of maps in cSet ′′ consisting of the following maps: • the special open box inclusions, • the saturation map, and • the 3-out-of-4 maps. Definition 3.3.
A map of structurally marked cubical sets is anodyne if it is in the saturation of S .Note that the anodyne generators in cSet ′′ are precisely those of cSet ′ , embedded via cSet ′ → cSet ′′ . Remark 3.4.
It might seem natural to include the map ( (cid:3) ) ♯ → ( (cid:3) ) ♯ , the inclusion of themarked interval into the interval with two distinct markings, in S , so that adding a marking to analready-marked edge of a structurally marked cubical set would not change its homotopy type. Infact, however, this map is already anodyne, as it is a pushout of a 3-out-of-4 map.The following lemma shows that this definition of anodyne maps is consistent with that of Subsec-tion 1.1. Lemma 3.5.
The set Λ( S ) is contained in the saturation of S .Proof. The proof of Lemma 2.9 applies equally well in this context, showing that a pushout productof a monomorphism with a map in the saturation of S is again in the saturation of S . This impliesthat Λ ( S ) is contained in the saturation of S ; applying the same lemma inductively, we see thateach set Λ n ( S ) is contained in the saturation of S . (cid:3) Theorem 3.6.
The category cSet ′′ of structurally marked cubical sets carries a cofibrantly generatedmodel structure in which: • the cofibrations are the monomorphisms; • the fibrant objects, and fibrations between fibrant objects, are defined by the right liftingproperty with respect to the set of generating anodyne maps S ; • the weak equivalences are maps X → Y inducing bijections [ Y, Z ] → [ X, Z ] for all fibrantobjects Z .Proof. The existence of the model structure follows from Theorem 1.5. Lemma 3.5 shows that theset of generating anodyne maps is exactly S . (cid:3) The remainder of this section will be devoted to analyzing this model structure and its relationshipwith the model structure on marked cubical sets of Theorem 2.43. More precisely, we will prove:
UBICAL MODELS OF ( ∞ , -CATEGORIES 33 Theorem 3.7. (i) The adjunction cSet ′′ ⇄ cSet ′ is a Quillen equivalence.(ii) The cubical marked model structure is right induced from the model structure of Theorem 3.6by the embedding cSet ′ → cSet ′′ . Before proving this theorem, we establish a number of intermediate results.
Proposition 3.8.
The adjunction cSet ′′ ⇄ cSet ′ is a Quillen adjunction between the model struc-ture of Theorem 3.6 to the cubical marked model structure.Proof. By Corollary 1.12, it suffices to show that Im preserves monomorphisms and takes generatinganodynes to anodynes. Both of these statements are immediate. (cid:3) Lemma 3.9.
A map of structurally marked cubical sets f : X → Y is a trivial fibration if and onlyif the underlying map of cubical sets is a trivial fibration in model structure of Theorem 1.28 (i.e.,has the right lifting property with respect to monomorphisms) and, for each edge x of X , the mapfrom the set of markings of x to that of f x is surjective.Proof. By Lemma 3.1, f is a trivial fibration if and only if it has the right lifting property withrespect to all boundary inclusions and the inclusion of the interval into the marked interval. Havingthe right lifting property with respect to all boundary inclusions is equivalent to being a trivialfibration on underlying cubical sets; having the right lifting property with respect to the inclusionof the interval into the marked interval is equivalent to each map of marking sets being surjective. (cid:3) Corollary 3.10.
For all structurally marked cubical sets X , the adjunction unit X → Im X is atrivial fibration. (cid:3) Lemma 3.11.
The functor
Im : cSet ′′ → cSet ′ preserves fibrations and trivial fibrations.Proof. We will show that if p : X → Y is a fibration between structurally marked cubical sets, then Im p : Im X → Im Y is also a fibration. Given a trivial cofibration of marked cubical sets i : A ֒ → B with maps α : A → Im X and β : B → Im Y making the square commute, apply Corollary 3.10 to ∅ → A → Im X to get α ′ : A → X with α = u X α ′ and then again to the square A i (cid:15) (cid:15) pα ′ / / Y u Y (cid:15) (cid:15) B β / / Im Y to get β ′ : B → Y that fits into a square A i (cid:15) (cid:15) α ′ / / X p (cid:15) (cid:15) B β ′ / / Y whose lift L : B → X yields u X L : B → Im X which satisfies the equations (Im p ) u X L = u Y pL = u Y β ′ = β and u x Li = u x α ′ = α and so provides the lift. Thus Im p is a fibration.The proof for trivial fibrations is analogous. (cid:3) Lemma 3.12.
Let X be a structurally marked cubical set. Then X is fibrant if and only if Im X isfibrant (in the model structure of Theorem 3.6).Proof. If Im X is fibrant, then X is fibrant by Corollary 3.10. Conversely, if X is fibrant, then Im X is fibrant in cSet ′ by Lemma 3.11, hence also in cSet ′′ by Proposition 3.8. (cid:3) Proof of Theorem 3.7.
First, let us show that the right derived functor of the embedding cSet ′ → cSet ′′ is an equivalence. Since [ X, Y ] cSet ′ → [ X, Y ] cSet ′′ is bijective by Lemma 3.2, it is full andfaithful. For essential surjectivity, by Corollary 1.6, given fibrant X ∈ cSet ′′ , we need fibrant Y ∈ cSet ′ weakly equivalent to X in cSet ′′ . This is given by Lemmas 3.11 and 3.12.Now, let us show that cubical marked model structure is right induced. Since Im is a left Quillenequivalence and all objects are cofibrant, it preserves and reflects weak equivalences by Proposi-tion 1.13, hence so does the embedding. Since Im preserves fibrations, the embedding reflects them.Preservation of fibrations is part of Proposition 3.8. (cid:3) Joyal model structure on cubical sets
Recall the adjunction cSet ⇄ cSet ′ of Subsection 1.4, in which the left adjoint is the minimalmarking functor and the right adjoint is the forgetful functor. In this section we will use thisadjunction to induce a model structure on cSet from the model structure on cSet ′ of Theorem 2.43. Theorem 4.1 (Analogue of Joyal model structure) . The category cSet of cubical sets carries amodel structure in which: • the cofibrations are the monomorphisms, • the weak equivalences are created by the minimal marking functor, • the fibrations are right orthogonal to trivial cofibrations.Proof. Apply Theorem 1.10 to the adjunction cSet ⇄ cSet ′ and the cubical marked model structure,with the factorization X ⊔ X → K ⊗ X → X . That the minimal marking functor sends the firstof these maps to a cofibration is clear; that it sends the second to a weak equivalence follows fromCorollaries 2.37 and 2.46. (cid:3) We refer to the model structure constructed above as the cubical Joyal model structure . Proposition 4.2.
The adjunction cSet ⇄ cSet ′ is a Quillen equivalence.Proof. The minimal marking functor preserves and reflects weak equivalences by definition, thuswe may apply Corollary 1.14 and item (ii). Let X be a marked cubical quasicategory; abusingnotation slightly, let X ♭ denote the minimal marking of the underlying cubical set of X . We mustshow that the inclusion X ♭ → X is a weak equivalence. UBICAL MODELS OF ( ∞ , -CATEGORIES 35 The marked edges of X ♭ are precisely the degenerate edges; by Lemma 2.5, the marked edges of X are precisely those edges (cid:3) → X which factor through K . Thus X ♭ → X is a (possibly transfinite)composite of pushouts of saturation maps, hence a trivial cofibration. (cid:3) We define some terminology which will be used in the analysis of this model structure. • For n ≥ , ≤ i ≤ ε ∈ { , } , the ( i, ε ) inner open box , denoted b ⊓ ni,ε , is the quotient of anopen box with the critical edge quotiented to a point. The ( i, ε ) inner cube , denoted b (cid:3) ni,ε ,is defined similarly. • An inner fibration is a map having the right lifting property with respect to the inner openbox inclusions. • An isofibration is a map having the right lifting property with respect to the endpointinclusions (cid:3) ֒ → K . • A cubical quasicategory is a cubical set X such that the map X → (cid:3) is an inner fibration. • An equivalence in a cubical set X is an edge (cid:3) → X which factors through the inclusionof the middle edge (cid:3) → K . • For n ≥ , ≤ i ≤ n, ε ∈ { , } , a special open box in a cubical set X is a map ⊓ ni,ε → X which sends the critical edge to an equivalence.The concept of homotopy developed in Section 2 adapts naturally to this setting, using equivalencesin place of marked edges. Definition 4.3.
For a cubical set X , let ∼ denote the relation on X , the set of vertices of X ,given by x ∼ y if there is an equivalence from x to y in X . Let ∼ denote the smallest equivalencerelation on X containing ∼ . Remark 4.4.
For x, y ∈ X , one can easily see that x ∼ y if and only if x and y are connected bya zigzag of equivalences. Definition 4.5.
For a cubical set X , the set of connected components π ( X ) is X / ∼ . Definition 4.6. An elementary left homotopy h : f ∼ g between maps f, g : A → B is a map h : K ⊗ A → B such that h | { }⊗ A = f and h | { }⊗ A = g . Note that the elementary left homotopy h corresponds to an edge K → hom L ( A, B ) between the vertices corresponding to f and g . A lefthomotopy between f and g is a zig-zag of elementary left homotopies.A left homotopy from f to g corresponds to a zig-zag of equivalencs in hom L ( A, B ) and so mapsfrom A to B are left homotopic exactly if π ( f ) = π ( g ) , where the set of connected components istaken in hom L ( A, B ) .These induce notions of elementary left homotopy equivalence and left homotopy equivalence . Eachof these notions has a “right” variant using A ⊗ K and hom R ( A, B ) . As in Section 2, unless thepotential for confusion arises or a statement depends on the choice, we will drop the use of “left”and “right”. Definition 4.7.
Let X be a cubical set. The natural marking on X is a marked cubical set X ♮ whose underlying cubical set is X , with edges marked if and only if they are equivalences. It is easy to see that this defines a functor ( − ) ♮ : cSet → cSet ′ , as maps of cubical sets preserveequivalences.Many results about the cubical Joyal model structure follow easily from the corresponding resultsabout the cubical marked model structure. Lemma 4.8. If i, j are cofibrations in cSet , then the pushout product i b ⊗ j is a cofibration. Moreover,if either i or j is trivial then so is i b ⊗ j .Proof. This is immediate from Corollaries 1.33 and 2.46 and Lemma 2.9. (cid:3)
Corollary 4.9.
Let i, f be maps in cSet . If i is a cofibration and f is a fibration, then the pullbackexponential i ⊲ f is a fibration. (cid:3) Corollary 4.10.
The category cSet , equipped with the cubical Joyal model structure and the geo-metric product, is a monoidal model category. (cid:3)
Next we will characterize the fibrant objects, and fibrations between fibrant objects, in the cubicalJoyal model structure.
Lemma 4.11.
The inner open box inclusions b ⊓ ni,ε → b (cid:3) ni,ǫ , and the endpoint inclusions (cid:3) → K ,are trivial cofibrations.Proof. The minimal marking of an inner open box inclusion is a pushout of a special open boxinclusion in cSet ′ . The minimal marking of (cid:3) → K is a trivial cofibration by Corollary 2.37. (cid:3) Lemma 4.12.
Cubical quasicategories have fillers for special open boxes.Proof.
We only consider positive filling problems; the negative case is dual. We argue by inductionon the dimension of the filling problem. For a special open box of dimension 2, it is a simple exerciseto explicitly construct a filler by extending the given open box to an inner open box of dimension3.Now let X be a cubical quasicategory, and suppose that X has fillers for all special open boxes ofdimension less than n . Consider a filling problem in X of dimension n : ( ∂ (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ ∂ (cid:3) b ) / / (cid:15) (cid:15) (cid:15) (cid:15) X (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b We regard the codomain of the left map as a negative face of a larger cube via the map (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b / / / / (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b UBICAL MODELS OF ( ∞ , -CATEGORIES 37 and the domain as the corresponding subobject. The original filling problem then becomes a fillingproblem in X of the form ( ∂ (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ ∂ (cid:3) b ) → (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b where the critical edge is a b → a b . We will solve this problem by extending the given partial data to the whole of (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b . For n ≥ , let Γ n ⊆ (cid:3) n denote the union of the positive faces. We use degeneracies in the newdirection to fill (Γ a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ Γ b ) → (Γ a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ Γ b ) . Since the critical edge is an equivalence, we can fill the square(4.1) a b (cid:15) (cid:15) a b a b / / a b where the dotted edge is again an equivalence.In the following, we will indicate the filling direction of (generalized) open boxes by underlining theappropriate factor in the pushout monoidal product. What this means is that we can factor thegiven generalized open box inclusion as a series of open box fillings in different dimensions, each ofwhich fills in the specified direction. We now fill the generalized open box { a } ⊗ ( { } → (cid:3) ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { b } → (cid:3) b ) if a, b ≥ . Here, the critical edges are of the form uv w → uv w where u, v, w are certain verticesof (cid:3) a , (cid:3) , (cid:3) b , respectively. All of these edges are degenerate except for the bottom edge in (4.1),which is an equivalence. Moreover, this edge only appears as a critical edge in filling problemsof lower dimension. So we may indeed fill this generalized open box using fibrancy of X and theinduction hypothesis. Dually, we fill the generalized open box ( { a } → (cid:3) a ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { } → (cid:3) ) ⊗ { b } if a, b ≥ .We now fill the generalized open box ( { a } ∪ Γ a → ∂ (cid:3) a ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { } → (cid:3) ) b ⊗ ( ∂ (cid:3) b → (cid:3) b ) if a ≥ . Again, the critical edges are of the form as above and we may argue as before. Dually, wefill the generalized open box ( ∂ (cid:3) a → (cid:3) a ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { b } ∪ Γ b → ∂ (cid:3) b ) if b ≥ .At this stage, we have defined the cube on ( ∂ (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ ∂ (cid:3) b ) . We now fill the open box ( ∂ (cid:3) a → (cid:3) a ) b ⊗ ( { } → (cid:3) ) b ⊗ ( ∅ → { } ) b ⊗ ( ∂ (cid:3) b → (cid:3) b ) , noting that the critical edge a b → a b is degenerate. We then fill the open box ( ∂ (cid:3) a → (cid:3) a ) b ⊗ ( ∅ → { } ) b ⊗ ( { } → (cid:3) ) b ⊗ ( ∂ (cid:3) b → (cid:3) b ) , noting that the critical edge a b → a b is degenerate. We finally fill the open box ( ∂ (cid:3) a → (cid:3) a ) b ⊗ ( ∂ (cid:3) → (cid:3) ) b ⊗ ( { } → (cid:3) ) b ⊗ ( ∂ (cid:3) b → (cid:3) b ) , noting that the critical edge a b → a b is degenerate. This defines the entire cube. (cid:3) Lemma 4.13.
Inner fibrations between cubical quasicategories lift against special open box inclu-sions.Proof.
Again we only consider positive filling problems; the negative case is dual. Again we argueby induction on the dimension of the filling problem, with the case for dimension 2 being a simpleexercise in filling three-dimensional open boxes, entirely analogous to the base case of the previousproof. Consider a lifting problem ( ∂ (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ ∂ (cid:3) b ) / / (cid:15) (cid:15) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b / / Y where the right map is an inner fibration between cubical quasicategories. As before, we regard thecodomain of the left map as a negative face of a larger cube via the map (cid:3) a ⊗ (cid:3) ⊗ (cid:3) b / / / / (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b and the domain as the corresponding subobject H . The critical edge is once again a b → a b .Let H ′ be the union of H with the subobjects (Γ a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ { } ⊗ (cid:3) ⊗ (cid:3) b ) ∪ ( (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ Γ b ) and H ′′ be the union of H ′ with the square { a } ⊗ (cid:3) ⊗ (cid:3) ⊗ { b } . UBICAL MODELS OF ( ∞ , -CATEGORIES 39 We use degeneracies in the new direction to extend the map to X from H to H ′ : H / / (cid:127) _ (cid:15) (cid:15) X . H ′ > > Since the critical edge is special in X , we extend the map to X from H ′ to H ′′ by filling the square a b (cid:15) (cid:15) a b a b / / a b where the dotted edge is again special in X . Note that the map X → Y preserves special edges.We construct the dotted arrow in the diagram H / / (cid:15) (cid:15) (cid:15) (cid:15) H ′′ / / (cid:15) (cid:15) X (cid:15) (cid:15) (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b / / (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b / / Y by solving a filling problem ( (cid:3) a ⊗ (cid:3) ⊗ { } ⊗ (cid:3) b ) ∪ H ′′ / / (cid:15) (cid:15) Y (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b as follows: the left map factors as a finite composite of open box inclusions of the form ( ∂ (cid:3) a ′ → (cid:3) a ′ ) b ⊗ ( { } → (cid:3) ) b ⊗ ( { } → (cid:3) ) b ⊗ ( ∂ (cid:3) b ′ → (cid:3) b ′ ) where (cid:3) a ′ and (cid:3) b ′ are faces of (cid:3) a and (cid:3) b , respectively. All critical edges are of the form uv w → uv w where u, v, w are certain points of (cid:3) a , (cid:3) , (cid:3) b , respectively. All of these edges are degeneratein Y except for the bottom edge in (4.1), which is special. We can thus fill these open boxes usingfibrancy of Y and Lemma 4.12.It remains to construct a lift H ′′ / / (cid:15) (cid:15) X (cid:15) (cid:15) (cid:3) a ⊗ (cid:3) ⊗ (cid:3) ⊗ (cid:3) b / / Y ,which is done exactly as in the proof of Lemma 4.12 using that X → Y is a fibration. (cid:3) Lemma 4.14. If X is a cubical quasicategory, then X ♮ is a marked cubical quasicategory.Proof. Given a cubical quasicategory X , we have fillers for special open boxes in X by Lemma 4.12.This implies that X ♮ has fillers for special open boxes in the sense of Definition 2.2. Furthermore,the definition of the natural marking implies that X ♮ has the right lifting property with respect to the saturation map for any cubical set X . By Lemma 2.6, this suffices to show that X ♮ is a markedcubical quasicategory. (cid:3) Proposition 4.15.
The fibrant objects of the the cubical Joyal model structure are given by cubicalquasicategories. The fibrations between fibrant objects are characterized by lifting against inner openbox inclusions and endpoint inclusions (cid:3) ֒ → K .Proof. By Lemma 4.11, every fibrant object is a cubical quasicategory and every fibration is aninner isofibration.If X is a cubical quasicategory, then X ♮ is a marked cubical quasicategory by Lemma 4.14. Theforgetful functor cSet ′ → cSet preserves fibrant objects as a right Quillen adjoint, and the underlyingcubical set of X ♮ is X , thus X is fibrant.The case of fibrations between fibrant objects proceeds in an analogous way. Let f : X → Y bean inner isofibration between cubical quasicategories; we will show that f ♮ is a fibration in cSet ′ .Lifting against one-dimensional special open box inclusions follows from the isofibration property;lifting against higher-dimensional special open box inclusions follows from Lemma 4.13. Since X ♮ isa marked cubical quasicategory, f ♮ has the right lifting property with respect to the saturation and3-out-of-4 maps by Lemma 2.6 and the fact that these maps are epimorphisms in cSet ′ . Since X ♮ and Y ♮ are marked cubical quasicategories, this implies that f ♮ is a fibration by Proposition 2.49. (cid:3) Corollary 4.16.
Let f : X → Y be a map between cubical quasicategories. Then f is a weakequivalence if and only if it is a homotopy equivalence. (cid:3) Corollary 4.17.
Let
X, Y ∈ cSet , with Y a cubical quasicategory. Then hom( X, Y ) is a cubicalquasicategory.Proof. This follows from Corollary 4.9 and Proposition 4.15. (cid:3)
Our next goal will be to characterize the weak equivalences in the cubical Joyal model structure ina manner similar to Lemma 2.35.
Lemma 4.18.
For X ∈ cSet , we have a natural isomorphism π X ∼ = π X ♮ .Proof. It is clear that X and X ♮ have the same set of vertices. To see that the equivalence relationsdefining π X and π X ♮ coincide, observe that a pair of vertices are connected by a zigzag of markededges in X ♮ if and only if they are connected by a zigzag of equivalences in X . (cid:3) Lemma 4.19.
Let
X, Y ∈ cSet , and let Y ′ be a marked cubical set whose underlying cubical set is Y . The underlying cubical set of hom( X ♭ , Y ′ ) is hom( X, Y ) .Proof. The n -cubes in the underlying cubical set of hom( X ♭ , Y ′ ) are maps X ♭ ⊗ (cid:3) n ∼ = ( X ⊗ (cid:3) n ) ♭ → Y ′ (the isomorphism follows from Corollary 1.33). Under the adjunction cSet ⇄ cSet ′ , thesecorrespond to maps X ⊗ (cid:3) n → Y . (cid:3) Proposition 4.20.
The following are equivalent for a cubical map A → B :(i) A → B is a weak equivalence; UBICAL MODELS OF ( ∞ , -CATEGORIES 41 (ii) for any cubical quasicategory X , the induced map hom( B, X ) → hom( A, X ) is a homotopyequivalence;(iii) for any cubical quasicategory X , the induced map π (hom( B, X )) → π (hom( A, X )) is abijection.Proof. To see that (i) ⇒ (ii), let A → B be a weak equivalence in cSet , and X a marked cubicalquasicategory. Then X ♮ is a marked cubical quasicategory by Lemma 4.14, so hom( B ♭ , X ♮ ) → hom( A ♭ , X ♮ ) is a homotopy equivalence by Lemma 2.35. The underlying cubical set functorpreserves weak equivalences between fibrant objects by Ken Brown’s lemma, so hom( B, X ) → hom( A, X ) is a weak equivalence by Lemma 4.19. Hence it is a homotopy equivalence by Corollar-ies 4.16 and 4.17.The implication (ii) ⇒ (iii) is clear, so now we consider (iii) ⇒ (i). For this, let X be the underlyingcubical set of a marked cubical quasicategory X ′ , and note that by Lemma 4.18 and Lemma 4.19,we have the following commuting diagram in Set : π hom( B, X ) / / ∼ = (cid:15) (cid:15) π hom( A, X ) ∼ = (cid:15) (cid:15) π hom( B ♭ , X ′ ) / / π hom( A ♭ , X ′ ) Since the underlying cubical set functor preserves fibrant objects, X is a cubical quasicategory. Soif (iii) holds then the top map is an isomorphism, hence so is the bottom map. Thus A ♭ → B ♭ is aweak equivalence in cSet ′ by Lemma 2.35, meaning that A → B is a weak equivalence in cSet . (cid:3) We now state two straightforward properties of the cubical Joyal model structure.
Proposition 4.21. (i) The Grothendieck model structure on cSet of Theorem 1.28 is a localization of the cubicalJoyal model structure.(ii) The adjunction τ : cSet ⇄ Cat : N (cid:3) is a Quillen adjunction between the canonical modelstructure on Cat and the cubical Joyal model structure. (cid:3)
One of the advantages of working with cubical quasicategories as opposed to their simplicial ana-logues is a clean definition of a mapping space between two objects in a cubical quasicategory.
Definition 4.22.
Let x and x be -cubes in a cubical quasicategory X . The mapping space from x to x is the cubical set Map X ( x , x ) given by Map X ( x , x ) n = n (cid:3) n +1 s → X (cid:12)(cid:12) s∂ n +1 ,ε = x ε o ,with cubical operations given by those of X .There is a clear geometric intuition behind this definition, as the example below shows. Example 4.23.
Given a cubical quasicategory X and -cubes x , x : (cid:3) → X , we have that: • a -cube in Map X ( x , x ) is a -cube from x to x in X ; • a -cube in Map X ( x , x ) is a -cube in X of the form x / / x x / / x Proposition 4.24.
Given a cubical quasicategory X and -cubes x , x : (cid:3) n +1 → X , the mappingspace Map X ( x , x ) is a cubical Kan complex.Proof. By definition of
Map X ( x , x ) , a filler for ⊓ ni,ε → Map X ( x , x ) amounts to a filler for ⊓ n +1 i,ε → X where the critical edge is contained in the face fully degenerate on x − ε . (cid:3) We conclude this section with a proof of the following result, relating the cubical Joyal model struc-ture to the Joyal model structure on simplicial sets via the triangulation functor of Subsection 1.2.
Proposition 4.25.
The adjunction T : cSet ⇄ sSet : U is a Quillen adjunction between the cubicalJoyal model structure and the Joyal model structure on sSet . Lemma 4.26. T sends the endpoint inclusions (cid:3) → K to trivial cofibrations in the Joyal modelstructure.Proof. It is easy to see that
T K is the simplicial set depicted below: • / / (cid:31) (cid:31) ❅❅❅❅❅❅❅ • (cid:15) (cid:15) (cid:31) (cid:31) ❅❅❅❅❅❅❅ •• • / / • Let Z denote the simplicial set defined by the following pushout: Λ / / (cid:15) (cid:15) (cid:15) (cid:15) ∆ (cid:15) (cid:15) ∆ / / Z ❴✤ The map ∆ → Z is a trivial cofibration, as a pushout of an inner horn inclusion; thus Z iscontractible. We have a pair of cofibrations Z ֒ → T K , picking out the bottom-left and top-rightsimplices in the illustration above; the induced map Z ⊔ Z → T K is a cofibration since these twosimplices have no faces in common. We obtain J as a quotient of T K by contracting each of thesetwo simplices to a point; in other words, we have the following pushout diagram: Z ⊔ Z (cid:15) (cid:15) / / / / T K (cid:15) (cid:15) ∆ ⊔ ∆ / / J ❴✤ The left map is a weak equivalence, since coproducts preserve weak equivalences in the Joyal modelstructure. Thus
T K → J is a weak equivalence, as a pushout of a weak equivalence along acofibration. The composite of ∆ → T K with this quotient map is an endpoint inclusion ∆ → J ,hence a weak equivalence; thus ∆ → T K is a weak equivalence by 2-out-of-3. (cid:3)
UBICAL MODELS OF ( ∞ , -CATEGORIES 43 Lemma 4.27. U preserves fibrations between fibrant objects.Proof. Let X → Y be a fibration between quasicategories. To show that U X → U Y is a fibration, itsuffices, by Proposition 4.15, to show that it has the right lifting property with respect to endpointinclusions into K and inner open box fillings. The former property follows from Lemma 4.26. Forthe latter, consider a diagram of the form: T b ⊓ ni,ε / / (cid:15) (cid:15) X (cid:15) (cid:15) T b (cid:3) ni,ε / / Y To obtain a lift in such a diagram, it suffices to obtain a lift in a diagram T ⊓ ni,ε / / (cid:15) (cid:15) X (cid:15) (cid:15) (∆ ) n / / Y where the image of the critical edge in X is degenerate.By Lemma 1.22, Corollary 1.24, and the symmetry of the cartesian product in sSet , the trian-gulation of an open box inclusion is the pushout product ( T ∂ (cid:3) m − ֒ → (∆ ) m − ) ˆ × ( { ε } ֒ → ∆ ) .Since T ∂ (cid:3) m − ֒ → (∆ ) m − is a monomorphism of simplicial sets, it can be written as a compos-ite of boundary fillings; since pushout products commute with composition, we can thus rewrite T ⊓ mi,ε ֒ → (∆ ) m as a composite of pushouts of maps of the form ( ∂ ∆ n → ∆ n ) ˆ × ( { ε } ֒ → ∆ ) . Thus,to obtain a lift in the diagram above, it suffices to find a lift in each of the induced diagrams (∆ n × { ε } ) ∪ ( ∂ ∆ n × ∆ ) / / (cid:15) (cid:15) X (cid:15) (cid:15) ∆ n × ∆ / / Y The left-hand map in the diagram above can be explicitly written as a composite of horn fillings.Each of these horn-fillings will be inner except for the last one to be filled, but the critical edge ofthis horn will be mapped to the critical edge of ⊓ mi,ε by the relevant inclusion ∆ n × ∆ → (∆ ) n .Thus its image in X is degenerate, so the horn is special. Hence a lift exists for each of these hornfillings. (cid:3) Proof of Proposition 4.25.
This follows from Corollary 1.12, together with Proposition 1.25 and Lemma 4.27. (cid:3)
Corollary 4.28.
The triangulation functor preserves weak equivalences.Proof.
Since all cubical sets are cofibrant, this is immediate from Proposition 4.25 and Ken Brown’slemma. (cid:3) Comparison with the Joyal model structure
In this section we prove our main theorem:
Theorem 5.1.
The adjunction T : cSet ⇄ sSet : U is a Quillen equivalence between the cubicalJoyal model structure on cSet and the Joyal model structure on sSet . Throughout this section, sSet and cSet will be equipped with the Joyal and cubical Joyal modelstructures, respectively, unless otherwise noted.Due to the difficulty of working directly with the triangulation functor, we first establish a secondQuillen adjunction Q : sSet ⇄ cSet : R ; this adjunction was previously studied in [KLW19], buthere we will construct it using a general theory of cones in cubical sets. Using this theory of cones,we will prove that Q ⊣ R is a Quillen equivalence, and that the left derived functor of Q is aninverse to that of T .5.1. Cones in cubical sets.
Before we can define the adjunction Q ⊣ R , we must first introducethe concept of a cone in a cubical set. We will also prove various lemmas about such cones, whichwill be of use later on in showing that Q ⊣ R is a Quillen equivalence. Definition 5.2.
For m, n ≥ , the standard ( m, n ) -cone C m,n is a cubical set given by the followinginductive construction. For a given m , let C m, = (cid:3) m . Then for each n ≥ , C m,n is the pushoutof the inclusion ∂ , ⊗ C m,n − : C m,n − ∼ = (cid:3) ⊗ C m,n − ֒ → (cid:3) ⊗ C m,n − along the unique map C m,n − → (cid:3) . C m,n − (cid:127) _ (cid:15) (cid:15) / / (cid:3) (cid:127) _ (cid:15) (cid:15) (cid:3) ⊗ C m,n − / / C m,n ❴✤ Definition 5.3.
For m, n ≥ , an ( m, n ) -cone in a cubical set X is a map C m,n → X .Observe that each cone C m,n → X corresponds to a unique ( m + n ) -cube of X by pre-compositionwith the quotient map (cid:3) m + n → C m,n . Thus we will also use the term “ ( m, n ) -cone” to refer to a map (cid:3) m + n → X which factors through this quotient map. In particular, when we refer to the ( i, ε ) faceof a cone x , this means the ( i, ε ) face of the corresponding cube: (cid:3) m + n − ∂ i,ε −−→ (cid:3) m + n → C m,n x −→ X .For m, n, k ≥ , recall that (cid:3) m + nk is the set of maps [1] k → [1] m + n in the box category (cid:3) ; thus wemay write such a k -cube f as ( f , ..., f m + n ) where each f i is a map [1] k → [1] . This allows us todescribe C m,n explicitly as a quotient of (cid:3) m + n . Lemma 5.4.
For all m, n ≥ , C m,n is the quotient of (cid:3) m + n obtained by identifying two k -cubes f, g if there exists j with ≤ j ≤ n such that f i = g i for i ≤ j and f j = g j = const (the constantmap [1] k → [1] with value 1).Proof. We fix m and proceed by induction on n . For the base case n = 0 , there cannot exist any j satisfying the given criteria, thus no identifications are to be made; and indeed we have C m, = (cid:3) m by definition.Now suppose that the given description holds for C m,n , and let q denote the quotient map (cid:3) m + n → C m,n . Then because the functor (cid:3) ⊗ − preserves colimits, (cid:3) ⊗ C m,n is a quotient of (cid:3) m + n with quotient map (cid:3) ⊗ q . From this description we see that (cid:3) ⊗ C m,n is obtained from (cid:3) m + n by identifying two k -cubes f, g whenever f = g and the cubes ( f , ..., f n +1 ) and ( g , ..., g n +1 ) are UBICAL MODELS OF ( ∞ , -CATEGORIES 45 identified in C m,n . In other words, we obtain (cid:3) ⊗ C m,n from (cid:3) m + n by identifying f and g ifthere exists j with ≤ j ≤ n + 1 such that f i = g i for all i ≤ j and f j = g j = const . Takingthe pushout of the inclusion ∂ , ⊗ C m,n : C m,n ֒ → (cid:3) ⊗ C m,n along the unique map C m,n → (cid:3) ,we then see that C m,n +1 is the quotient of (cid:3) ⊗ C m,n obtained by identifying cubes f, g whenever f = g = const . Thus the description holds for C m,n +1 . (cid:3) Corollary 5.5.
For all n ≥ , C ,n ∼ = C ,n − . (cid:3) Using the characterization of cones given above, we can show that any face of a given cone is a coneof a specified degree.
Lemma 5.6.
For i ≤ n , the image of the composite map (cid:3) m + n − ∂ i, −−→ (cid:3) m + n → C m,n is isomorphicto C m,n − . For i ≥ n + 1 , ε ∈ { , } , the image of the composite map (cid:3) m + n − ∂ i,ε −−→ (cid:3) m + n → C m,n is isomorphic to C m − ,n .Proof. First consider the composite map (cid:3) m + n − ∂ i, −−→ (cid:3) m + n → C m,n . Let f = ( f , ..., f m + n − ) denote a k -cube of (cid:3) m + n − , as in the proof of Lemma 5.4. We denote the image of this cube under ∂ i, by f ′ = ( f ′ , ..., f ′ m + n − ) , where f ′ j = f j for j < i , f ′ i = const , and f ′ j = f j − for j > i . ByLemma 5.4, given two k -cubes f and g in (cid:3) m + n − , their images under ∂ i, will be identified in thequotient C m,n if and only if there exists j ≤ n such that f ′ l = g ′ l for l ≤ j and f ′ j = g ′ j = const – inother words, if there exists j ≤ n − such that f l = g l for l ≤ j and f j = g j = const . The desiredisomorphism thus follows from Lemma 5.4.The analysis of ∂ i,ε where i ≥ n + 1 , ε ∈ { , } is similar, except that in that case we have f ′ j = f j for all j ≤ i . Thus we conclude that the images of f and g in the quotient C m,n are equal if andonly if there exists j ≤ n such that f l = g l for l ≤ j and f j = g j = const . (cid:3) Lemma 5.7.
Let x be an ( m, n ) -cone in a cubical set X . Then: • If n ≥ , then for i ≤ n , x∂ i, is an ( m, n − -cone; • If m ≥ , then for i ≥ n + 1 , x∂ i, is an ( m − , n ) -cone; • If m ≥ , then for all i , x∂ i, is an ( m − , n ) -cone.Proof. First consider n ≥ , i ≤ n . By Lemma 5.6, the image of (cid:3) m + n +1 under the compositemap above will be isomorphic to C m,n − ; thus the composite map factors through C m,n − , givinga commuting diagram as shown below: (cid:3) m + n − (cid:15) (cid:15) (cid:31) (cid:127) ∂ i, / / (cid:3) m + n (cid:15) (cid:15) C m,n − (cid:31) (cid:127) / / C m,n Now, for an ( m + n ) -cube x ∈ X m + n to be an ( m, n ) -cone means precisely that the correspondingmap x : (cid:3) m + n → X factors through C m,n . So the face x∂ i, can be written as (cid:3) m + n − ∂ i, −−→ (cid:3) m + n → C m,n x −→ X ; by the diagram above we can rewrite this as (cid:3) m + n − → C m,n − → C m,n x −→ X . So x∂ i, factors through C m,n − , meaning that it is an ( m, n − -cone. A similar argument shows that for m ≥ , i ≥ n +1 , the composite map (cid:3) m + n − ∂ i,ε −−→ (cid:3) m + n → C m,n will factor through C m − ,n , implying that x∂ i,ε is an ( m − , n ) -cone for any ( m, n ) -cone x .Finally, let m ≥ , i ≤ n and consider the composite (cid:3) m + n − ∂ i, −−→ (cid:3) m + n → C m,n . As above, welet f denote an arbitrary k -cube of (cid:3) m + n − and let f ′ denote its image under ∂ i, ; then once againwe have f ′ j = f j for j ≤ i − , but now f ′ i = const . So let f and g be two k -cubes of (cid:3) m + n − , andsuppose that there exists j ≤ n such that f l = g l for l ≤ j and f j = g j = const . Then there exists j ′ ≤ n such that f ′ l = g ′ l for l ≤ j ′ and f ′ j ′ = g ′ j ′ = const : if j < i then j ′ = j , while if j ≥ i then j ′ = i . So f ′ and g ′ are identified in C m,n . Thus the composite map factors through C m − ,n , sofor any ( m, n ) -cone x , x∂ i, is an ( m − , n ) -cone. (cid:3) Remark 5.8.
In contrast to Lemma 5.6, for i ≤ n the image of (cid:3) m + n − ∂ i, −−→ (cid:3) m + n → C m,n isnot isomorphic to C m − ,n . For instance, when i = 1 this image is isomorphic to (cid:3) .In some cases it will be more convenient to characterize cones in a cubical set by a set of conditionson their faces. By a direct analysis of the cubes of C m,n , or by an inductive argument similar tothat used in the proof of Lemma 5.4, we have the following characterization of ( m, n ) -cones in X . Lemma 5.9.
For m, n with n ≥ , and X ∈ cSet , a cube x : (cid:3) m + n → X is an ( m, n ) -cone if andonly if for all i such that ≤ i ≤ n we have x∂ i, = x∂ m + n, ∂ m + n − , ...∂ i +1 , ∂ i, σ i σ i +1 ...σ m + n − σ m + n − .(In the case m = 0 , i = n we interpret this statement as the tautology x∂ n, = x∂ n, ). (cid:3) Corollary 5.10. If x : (cid:3) m + n → X is an ( m, n ) -cone, then x is also an ( m + k, n − k ) -cone for all k ≤ n . (cid:3) This characterization allows us to prove some technical lemmas concerning faces and degeneraciesof cones.
Lemma 5.11.
Let X be a cubical set, and let x : C m,n → X be an ( m, n ) -cone in X . Then:(i) for i ≥ n + 1 , xσ i is an ( m + 1 , n ) -cone;(ii) if n ≥ then for i ≤ n , xγ i is an ( m, n + 1) -cone;(iii) for i ≥ n + 1 , xγ i is an ( m + 1 , n ) -cone;Proof. We will prove item (i); the remaining proofs are similar.We will show that xσ i satisfies the conditions of Lemma 5.9 for ( m + 1 , n ) . For j ≤ n , we have j < i . Using this fact and the cubical identities, we can compute: ( xσ i ) ∂ j, = x∂ j, σ i − = x∂ m + n, ...∂ j +1 , ∂ j, σ j ...σ m + n − σ i − = x∂ m + n, ...∂ j +1 , ∂ j, σ j ...σ m + n = xσ i ∂ i, ∂ m + n, ...∂ j +1 , ∂ j, σ j ...σ m + n = ( xσ i ) ∂ m + n +1 , ...∂ j +1 , ∂ j, σ j ...σ m + n UBICAL MODELS OF ( ∞ , -CATEGORIES 47 Thus xσ i is an ( m + 1 , n ) -cone.The proof of (2) requires separate computations for the cases ≤ j < i , j = i, j = i + 1 , and i + 1 < j ≤ n + 1 , while the proof of (3) is essentially identical to the above. (cid:3) Lemma 5.12.
For m ≥ , n ≥ , let x be an ( m + n − -cube in a cubical set X . If xγ n is an ( m, n ) -cone, then it is also an ( m − , n + 1) -cone.Proof. By Lemma 5.7, xγ n ∂ n +1 , = x is an ( m − , n ) -cone. Therefore, xγ n is an ( m − , n + 1) -coneby Lemma 5.11. (cid:3) We will also have use for the following result, which shows that the standard cones contain manyinner open boxes.
Lemma 5.13.
For n ≥ , ≤ i ≤ m + n , the quotient map (cid:3) m + n → C m,n sends the critical edgewith respect to the face ∂ i, to a degenerate edge.Proof. The critical edge in question corresponds to the function f : [1] → [1] m + n with f i = id [1] , f j = const for j = i . In particular, f = const , so f is equivalent, under the equivalence relationof Lemma 5.4, to the map [1] → [1] m + n which is constant at (1 , ..., . (cid:3) Theorem 1.15 gives us the following:
Proposition 5.14.
Given a cubical set X , for any cube x : (cid:3) n → X there exist unique (possiblyempty) sequences a < ... < a p , b < ... < b q and a unique non-degenerate cube y : (cid:3) n − p − q → X such that x = yγ b ...γ b q σ a ...σ a p . (cid:3) Definition 5.15.
For x : (cid:3) n → X , the expression given by Proposition 5.14 is the standard form of x .For brevity, we will often say that the standard form of a cube x is zf , or “ends with f ", where f is some map in (cid:3) ; this is understood to mean that f is the rightmost map in the standard form of x . For instance, if the standard form of x is zσ a p , then z = yγ b ...γ b q σ a ...σ a p − in the notation ofProposition 5.14.We now prove a lemma regarding the standard forms of cones. Lemma 5.16.
Let m ≥ , and let x : C m,n → X be a degenerate ( m, n ) -cone whose standard formis yγ b ...γ b q σ a ...σ a p , where the string σ a ...σ a p is non-empty. Then a p ≥ n + 1 .Proof. For n = 0 this is trivial, so assume n ≥ . Towards a contradiction, suppose that a p ≤ n ,and let z = yγ b ...γ b q σ a ...σ a p − , so that zσ a p = x . Taking the ( a p , faces of both sides of thisequation, and applying Lemma 5.9, we see that: z = x∂ m + n, ...∂ a p +1 , ∂ a p , σ a p ...σ m + n − ∴ zσ a p = x∂ m + n, ...∂ a p +1 , ∂ a p , σ a p ...σ m + n − σ a p ∴ x = x∂ m + n, ...∂ a p +1 , ∂ a p , σ a p ...σ m + n In the last step, we have repeatedly used the co-cubical identity σ j σ i = σ i σ j +1 for i ≤ j to rearrangethe string σ a p ...σ m + n − , σ a p into one whose indices are in strictly increasing order. (We can dothis because, by our assumption on m , m + n − ≥ n ≥ i .) Now let y ′ σ a ′ ...σ a ′ p ′ γ b ′ ...γ b ′ q ′ be thestandard form of x∂ m + n, ...∂ a p +1 , ∂ a p , ; then we have: x = y ′ σ a ′ ...σ a ′ p ′ γ b ′ ...γ b ′ q ′ σ a p ...σ m + n We can apply further co-cubical identities to re-order the maps on the right-hand side of thisequation, obtaining a standard form for x in which the rightmost degeneracy map has index greaterthan or equal to m + n . But as the standard form of x is unique, this contradicts our assumptionthat a p ≤ n . (cid:3) We will also require some lemmas concerning subcomplexes of C m,n consisting of specified faces. Definition 5.17.
For m, n ≥ , k ≤ n , B m,n,k is the subcomplex of C m,n consisting of the imagesof the faces ∂ , through ∂ k, , as well as all all faces ∂ i, , under the quotient map (cid:3) m + n → C m,n .In order to characterize maps out of B m,n,k , we will need to prove a couple of lemmas concerningthe faces of C m,n . Lemma 5.18.
For m, n ≥ , ≤ i < i ≤ m + n, ε , ε ∈ { , } , where i j ≥ n + 1 if ε j = 1 , theintersection of the images of the faces ∂ i ,ε and ∂ i ,ε of (cid:3) m + n under the quotient map (cid:3) m + n → C m,n is exactly the image of the face ∂ i ,ε ∂ i ,ε = ∂ i ,ε ∂ i − ,ε .Proof. That the intersection of the images of ∂ i ,ε and ∂ i ,ε contains the image of ∂ i ,ε ∂ i ,ε isclear, as this face is the intersection of ∂ i ,ε and ∂ i ,ε in (cid:3) m + n . Now we will verify the oppositecontainment, using description of C m,n from Lemma 5.4.To this end, consider a map f : [1] k → [1] m + n such that the equivalence class [ f ] ∈ C m,nk is containedin the images of faces ( i , ε ) and ( i , ε ) . We will construct f ′ : [1] k → [1] m + n such that f ∼ f ′ and f ′ is contained in the intersection of faces ( i , ε ) and ( i , ε ) , thereby showing that [ f ] = [ f ′ ] is contained in the image of this intersection under the quotient map.Since f is in the image of face ( i , ε ) , f ∼ g for some g : [1] k → [1] m + n such that g i = const ε .Therefore, at least one of the following holds:(i) f i = const ε ;(ii) f j = g j = const for some j ≤ min( i − , n ) .If (ii) holds, then f is equivalent to any f ′ such that f ′ l = f l for l ≤ j ; in particular, we can choosesuch an f ′ satisfying f ′ i = const ε , f ′ i = const ε .Now suppose that (i) holds, but (ii) does not. Then because f is in the image of face ( i , ε ) , f ∼ h for some h : [1] k → [1] m + n such that h i = const ε . Therefore, at least one of the following holds:(i) f i = const ε ;(ii) f j = h j = const for some i + 1 ≤ j ≤ min( i − , n ) . UBICAL MODELS OF ( ∞ , -CATEGORIES 49 In case (i), we have f i = const ε , f i = const ε , so we can simply choose f ′ = f . In case (ii), f isequivalent to any f ′ such that f ′ l = f l for l ≤ j (which implies f ′ i = const ε ); in particular, we canchoose such an f ′ satisfying f ′ i = const ε . (cid:3) Lemma 5.19.
For i ≤ n , the image of the face ∂ i, under the quotient map (cid:3) m + n → C m,n iscontained in the image of ∂ m + n, .Proof. Let f : [1] k → [1] m + n be a k -cube of (cid:3) m + n which factors through ∂ i, . Then f i = const .Thus f is equivalent to any f ′ : [1] k → [1] m + n such that f ′ j = f j for all j ≤ i ; in particular, we maychoose such an f ′ with f ′ m + n = const . So f ′ factors through ∂ m + n, ; thus [ f ] = [ f ′ ] is containedin the image of ∂ m + n, under the quotient map. (cid:3) Lemma 5.20.
For a cubical set X , a map x : B m,n,n → X is determined by a set of ( m, n − -cones x i, : C m,n − → X for ≤ i ≤ n and a set of ( m − , n ) -cones x i, for n + 1 ≤ i ≤ m + n such that for all i < i , ε , ε ∈ { , } , x i ,ε ∂ i ,ε = x i ,ε ∂ i − ,ε , with x i,ε being the image of ∂ i,ε under x . (cid:3) Proof.
To define a map x : B m,n,k → X , it suffices to assign the values of x on the faces [ ∂ i,ε ] of C m,n for which i ≤ k or ε = 1 , provided that these choices are consistent on the intersections of faces.By Lemma 5.19, it suffices to consider only those faces for which i ≤ k, ε = 0 or i ≥ n + 1 , ε = 1 .These faces are isomorphic to C m,n − or C m − ,n , respectively, by Lemma 5.6. By Lemma 5.18,to show that these choices are consistent on the intersections of faces, it suffices to show that theysatisfy the co-cubical identity for composites of face maps. (cid:3) Proposition 5.21.
For all m, n ≥ , n ≤ k ≤ m + n − , the inclusion B m,n,k ֒ → C m,n is a trivialcofibration.Proof. We proceed by induction on m . In the base case m = 1 , the only relevant value of k is k = n .The only face of C ,n which is missing from B ,n,n is [ ∂ n +1 , ] , so the inclusion B ,n,n ֒ → C ,n is an ( n + 1 , -open box filling. By Lemma 5.13, the critical edge for this open box filling is degenerate,hence an equivalence in B ,n,n , so the inclusion is a trivial cofibration.Now let m ≥ , and suppose the statement holds for m − . For n ≤ k ≤ m + n − , consider theintersection of the ( k + 1 , face of C m,n , [ ∂ k, ] , with the subcomplex B m,n,k . By Lemma 5.18 andLemma 5.19, this intersection consists of faces (1 , through ( k, and (1 , through ( m + n − , of [ ∂ k +1 , ] . By Lemma 5.6, it is thus isomorphic to B m − ,n,k .Thus we can express B m,n,k +1 as the following pushout: B m − ,n,k (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) B m,n,k (cid:127) _ (cid:15) (cid:15) C m − ,n (cid:31) (cid:127) / / B m,n,k +1 ❴✤ By the induction hypothesis, B m,n,k ֒ → C m − ,n is a trivial cofibration, since n ≤ k ≤ m + n − . Thus B m,n,k ֒ → B m,n,k +1 is a trivial cofibration, as a pushout of a trivial cofibration. From this we can seethat for any n ≤ k ≤ m + n − , the composite inclusion B m,n,k ֒ → B m,n,k +1 ֒ → ... ֒ → B m,n,m + n − is a trivial cofibration. Thus it suffices to prove that B m,n,m + n − ֒ → C m,n is a trivial cofibration. Here, as in the base case,the subcomplex B m,n,m + n − is only missing the face [ ∂ m + n, ] , so the inclusion is an ( m + n, -openbox filling. The critical edge of this open box is degenerate by Lemma 5.13, so the inclusion isindeed a trivial cofibration.Thus we see that the inclusion B m,n,k ֒ → C m,n is a trivial cofibration for any m, n, k satisfying theconstraints given in the statement. (cid:3) Q ⊣ R . For n ≥ , let Q n = C ,n . These objects were previously studied in [KLW19], in whichthey were described as quotients of (cid:3) n under a certain equivalence relation; this relation is preciselythat of Lemma 5.4 in the case m = 0 . We begin by recalling some of the theory developed in thatpaper. Proposition 5.22 ([KLW19, Prop. 2.3]) . The assignment [ n ] → Q n extends to a cosimplicial object Q • : ∆ → cSet , in which each simplicial structure map Q m → Q n is given by a map (cid:3) m → (cid:3) n which descends to a map between the quotients. The correspondence is as follows:a map Q n − → Q n th face st face nd face · · · j th face · · · n th faceis induced by a map (cid:3) n − → (cid:3) n ∂ n, ∂ n, ∂ n − , · · · ∂ n − j +1 , · · · ∂ , a map Q n → Q n − th deg. st deg. nd deg. · · · j th deg. · · · ( n − st deg.is induced by a map (cid:3) n → (cid:3) n − σ n γ n − γ n − · · · γ n − j · · · γ (cid:3) Taking the left Kan extension of this cosimplicial object along the Yoneda embedding, we obtain afunctor Q : sSet → cSet . ∆ / / (cid:127) _ (cid:15) (cid:15) cSetsSet Q rrrrrrrrrr This functor has a right adjoint R : cSet → sSet , given by ( R X ) n = cSet ( Q n , X ) . Lemma 5.23 ([KLW19, Lem. 4.2]) . For any X ∈ cSet , the counit Q R X → X is a monomorphism. (cid:3) This lemma shows that for any cubical set X , Q R X is a subcomplex of X . Specifically, it is thesubcomplex whose non-degenerate n -cubes, for each n , are those which factor through Q n – in otherwords, they are the non-degenerate (0 , n ) -cones in X . Remark 5.24.
Viewing sSet as the slice category sSet ↓ ∆ and cSet as the functor category cSet [0] , the adjunction Q ⊣ R coincides with the cubical straightening-unstraightening adjunctiondeveloped in [KV18].Our next goal is to show the following: Proposition 5.25.
The adjunction Q ⊣ R is Quillen. UBICAL MODELS OF ( ∞ , -CATEGORIES 51 To prove this, we will show that this adjunction satisfies the hypotheses of Corollary 1.12. We beginwith a simple lemma relating the interval objects in the two model structures of interest.
Lemma 5.26. QJ ∼ = K . (cid:3) Lemma 5.27 ([KLW19, Lem. 4.5]) . Q preserves monomorphisms. (cid:3) Lemma 5.28.
The image under Q of an inner horn inclusion Λ ni ⊆ ∆ n is a trivial cofibration.Proof. Because Q preserves colimits, Q Λ ni is the subcomplex of Q n consisting of the images of themaps Q∂ j : Q n − → Q n for which j = i . By Proposition 5.22 we can see that this subcomplex isthe image of ⊓ nn − i +1 , under the quotient map (cid:3) n → Q n . We thus have the following commutingsquare: ⊓ nn − i +1 , / / (cid:15) (cid:15) Q Λ ni (cid:15) (cid:15) (cid:3) n / / Q n ❴✤ It is easy to see that this square is a pushout. Furthermore, the critical edge of the open box ⊓ nn − i +1 , → Q Λ ni is degenerate by Lemma 5.13. Thus Q Λ ni ֒ → Q n is a trivial cofibration, as aninner open box filling. (cid:3) Proof of Proposition 5.25.
By Lemma 5.27, Q preserves cofibrations. By Lemma 5.28, the imageunder Q of an inner-horn inclusion is a trivial cofibration. The image under Q of an endpointinclusion ∆ → J is an endpoint inclusion (cid:3) → K , hence a trivial cofibration by Lemma 4.11.Thus the adjunction is Quillen by Corollary 1.12. (cid:3) Corollary 5.29. Q preserves weak equivalences.Proof. Since all simplicial sets are cofibrant in the Quillen model structure, this follows from Propo-sition 5.25 and Ken Brown’s lemma. (cid:3)
Next we will concern ourselves with the relationship between Q and the triangulation functor. Ourgoal will be to prove the following: Proposition 5.30. Q reflects weak equivalences. To do this, we will develop a natural weak equivalence
T Q = ⇒ id sSet . Definition 5.31.
For n ≥ , let a = ( a , ..., a n ) , where a i ∈ { , } , be an object of [1] n . Then F ( a ) = 0 if a i = 0 for all i ; otherwise, F ( a ) = n − i + 1 , where i is minimal such that a i = 1 . Proposition 5.32.
For all n, F defines a poset map [1] n → [ n ] .Proof. Let a ≤ b be objects of [1] n . If a i = 0 for all i , then F ( a ) = 0 . Otherwise, let i be minimalsuch that a i = 1 , and let j be the minimal such value for b . Then b i = 1 as well, so j ≤ i . In eithercase, we see that F ( a ) ≤ F ( b ) . (cid:3) Proposition 5.33.
For all n , F induces a map of simplicial sets (∆ ) n → T Q n . Proof.
First, observe that by applying the nerve functor N : Cat → sSet , we get an induced map N F : (∆ ) n → ∆ n .The simplicial set T Q n is a quotient of T (cid:3) n = (∆ ) n . Specifically, since N is fully faithful, wemay regard n -simplices ∆ n → (∆ ) n as poset maps [ n ] → [1] n . Then by an argument analogous tothe proof of Lemma 5.4, using the fact that T preserves colimits and sends geometric products tocartesian products, T Q n is obtained by identifying two such maps f, g if there exists i such that f j = g j for j ≤ i and f i = g i = const . N F then acts on such maps by post-composition with F .Since F depends only on the position of the first in an object of [1] n , it is clear that maps whichare identified in T Q n agree after post-composition with F . Thus N F factors through the quotient
T Q n . (cid:3) Let ¯ F : T Q n → ∆ n denote the map constructed above. Then we can show: Lemma 5.34.
The maps ¯ F : T Q n → ∆ n form a natural transformation of co-simplicial objects in sSet . That is, for any map φ : [ m ] → [ n ] in ∆ , we have a commuting diagram: T Q m T Qφ / / ¯ F (cid:15) (cid:15) T Q n ¯ F (cid:15) (cid:15) ∆ m φ / / ∆ n Proof.
It suffices to show that this holds for the generating morphisms of ∆ , namely the co-face andco-degeneracy maps. For each such map φ : [ m ] → [ n ] we have a corresponding map φ ′ : [1] m → [1] n in (cid:3) , as described in Proposition 5.22: • For ∂ : [ n − → [ n ] , ∂ ′ = ∂ n, ; • For i ≥ , ∂ i : [ n − → [ n ] , ∂ ′ i = ∂ n − i +1 , ; • For σ : [ n ] → [ n − , σ ′ = σ n ; • For σ i : [ n ] → [ n − , σ ′ i = γ n − i .For every such φ we have a commuting diagram in cSet , where the vertical maps (cid:3) m → Q m arethe quotient maps:(5.1) (cid:3) m (cid:15) (cid:15) φ ′ / / (cid:3) n (cid:15) (cid:15) Q m Qφ / / Q n Furthermore, by direct computation we have commuting diagrams in
Cat : UBICAL MODELS OF ( ∞ , -CATEGORIES 53 (5.2) [1] mF (cid:15) (cid:15) φ ′ / / [1] nF (cid:15) (cid:15) [ m ] φ / / [ n ] Now consider the following diagram in sSet : (∆ ) m T φ ′ / / (cid:15) (cid:15) (∆ ) n (cid:15) (cid:15) T Q m T Qφ / / ¯ F (cid:15) (cid:15) T Q n ¯ F (cid:15) (cid:15) ∆ m φ / / ∆ n The top square commutes, as it is obtained by applying T to diagram (1); the outer rectangle alsocommutes, as it is obtained by applying N to diagram (2). We wish to show that the bottom squarecommutes, i.e. that φ ◦ ¯ F = ¯ F ◦ T Qφ ; since the quotient map (∆ ) m → T Q m is an epimorphism,we can show the desired equality by pre-composing with this map and performing a simple diagramchase. (cid:3) Corollary 5.35. ¯ F extends to a natural transformation ¯ F : T Q = ⇒ id sSet .Proof. Let X be an arbitrary simplicial set. Recall that X = colim ∆ n → X ∆ n ; since T and Q bothpreserve colimits, we have T QX = colim ∆ n → X T Q n . Thus, by Lemma 5.34, we obtain an induced mapon the colimits ¯ F : T QX → X , natural in X . (cid:3) Proposition 5.36.
For every simplicial set X , the map ¯ F : T QX → X is a weak equivalence.Proof. We begin by proving the statement for the case where X is m -skeletal for some m ≥ ,proceeding by induction on m . For m = 0 , m = 1 , the map in question is an isomorphism.Now let m ≥ , and suppose that the statement holds for any ( m − -skeletal X . Then in particular,it holds for any horn Λ mi . For any < i < n , consider the following commuting diagram: T Q Λ mi (cid:31) (cid:127) ∼ / / ∼ (cid:15) (cid:15) T Q m (cid:15) (cid:15) Λ mi (cid:31) (cid:127) ∼ / / ∆ m The left-hand map is a weak equivalence by the induction hypothesis; the bottom map is a trivialcofibration as an inner horn inclusion; and the top map is a trivial cofibration by Proposition 5.25and Proposition 4.25. Thus ¯ F : T Q m → ∆ m is a weak equivalence by the two-out-of-three property.Extending this result to an arbitrary m -skeletal simplicial set X is a straightforward application ofthe gluing lemma, using the fact that both T and Q preserve colimits. Now let X be an arbitrary simplicial set; then ¯ F is a weak equivalence on the n -skeleton of X for each dimension n . Thus ¯ F : T QX → X is a weak equivalence, using the fact that sequentialcolimits of cofibrations preserve weak equivalences. (cid:3) Proof of Proposition 5.30.
Let f : X → Y be a map of simplicial sets, such that Qf is a weakequivalence. We have a commuting diagram: T QX
T Qf / / ¯ F (cid:15) (cid:15) T QY ¯ F (cid:15) (cid:15) X f / / Y The top horizontal map is a weak equivalence by Proposition 1.25, as are the vertical maps byProposition 5.36. Thus f is a weak equivalence by the 2-out-of-3 property. (cid:3) The counit of Q ⊣ R . We have shown that the adjunction Q ⊣ R satisfies the hypothesesof Corollary 1.14, item (ii). To show that it is a Quillen equivalence, therefore, we must prove thefollowing: Theorem 5.37.
For any cubical quasicategory X , the counit ε : Q R X ֒ → X is a trivial cofibration. Throughout this section, fix a cubical quasicategory X ; we will build X from Q R X via successiveinner open-box fillings, thereby showing that the inclusion of Q R X into X is a trivial cofibration.We will do this via a three-stage induction: X will be constructed as the sequential colimit ofsubcomplexes X m , m ≥ , each of which will be constructed as the sequential colimit of a sequenceof subcomplexes X m,n , n ≥ − , each of which will be constructed by a kind of induction on skeleta.We begin by establishing the induction hypothesis which the subcomplexes X m must satisfy: Definition 5.38.
A subcomplex X m ⊆ X satisfies the induction hypothesis on base dimension for m if:(1) For ≤ m ′ ≤ m and n ≥ , all ( m ′ , n ) -cones of X are contained in X m ;(2) For i ≥ , every non-degenerate cube (cid:3) m + i → X m is an ( m, i ) -cone;(3) For every ( m ′ , n ) -cone of X with m ′ ≤ m, n ≥ , X m contains an ( m ′ , n + 1) -cone θ m ′ ,n ( x ) ,satisfying the following identities:(a) For i ≤ n , θ m ′ ,n ( x ) ∂ i, = θ m ′ ,n − ( x∂ i, ) ;(b) θ m ′ ,n ( x ) ∂ n +1 , = x ;(c) For i ≥ n + 2 , θ m ′ ,n ( x ) ∂ i, = θ m ′ − ,n ( x∂ i − , ) ;(d) If xσ i is an ( m ′ , n ) -cone for i ≥ n + 1 , then θ m ′ ,n ( xσ i ) = θ m ′ − ,n ( x ) σ i +1 ;(e) If xγ i is an ( m ′ , n ) -cone for i ≤ n − , then θ m ′ ,n ( xγ i ) = θ m ′ ,n − ( x ) γ i ;(f) If xγ i is an ( m ′ , n ) -cone for i ≥ n + 1 , then θ m ′ ,n ( xγ i ) = θ m ′ − ,n ( x ) γ i +1 ;(g) θ m ′ ,n ( θ m ′ ,n − ( x )) = θ m ′ ,n − ( x ) γ n ; UBICAL MODELS OF ( ∞ , -CATEGORIES 55 (h) For m ′ ≥ , if x is an ( m ′ − , n + 1) -cone, then θ m ′ ,n ( x ) = xγ n +1 ;(4) If m ≥ then we have a trivial cofibration X m − ֒ → X m .Now we establish our base case: Proposition 5.39. Q R X satisfies the induction hypothesis on base dimension for m = 1 .Proof. The non-degenerate n -cubes of Q R X are the cubes of X which factor through Q n , i.e. thenon-degenerate (0 , n ) -cones of X , and for n ≥ these are also its non-degenerate (1 , n − -conesby Corollary 5.5. Thus X satisfies Definition 5.38, item (1); item (2) follows by Corollary 5.10.Now we must define the functions θ ,n and θ ,n for all n and show that they satisfy the identities ofitem (3). For a (0 , n ) -cone x ∈ X n , we set θ ,n ( x ) = xσ n +1 ; this is a (1 , n + 1) -cone by Lemma 5.11.The hypotheses of item (c), item (d) and item (f) are vacuous here, as there are no cubical structuremaps satisfying the given constraints on their indices; item (h) similarly does not apply in this case.The remaining identities follow easily from the cubical identities: • For item (a), let i ≤ n . Then θ ,n ( x ) ∂ i, = xσ n +1 ∂ i, = x∂ i, σ n = θ ,n − ( x ) σ n . • For item (b), we have θ ,n ( x ) ∂ n +1 , = xγ n +1 ∂ n +1 , = x . • For item (e), let ≤ i ≤ n − . Then θ ,n ( xγ i ) = xγ i σ n +1 = xσ n γ i = θ ,n − ( x ) γ i . • For item (g), we have θ ,n +1 ( θ ,n ( x )) = xσ n +1 σ n +2 = xσ n +1 γ n +1 = θ ,n ( x ) γ n +1 .Next we define θ ,n . Because every (1 , n ) -cone is a (0 , n + 1) -cone, we must have θ ,n ( x ) = xγ n +1 inorder to satisfy item (h). This is indeed a (1 , n + 1) -cone by Lemma 5.11. The hypothesis of item (f)is still vacuous in this case, as there are no connection maps γ i : [1] n → [1] n − with i ≥ n + 1 . Onceagain, we can verify the remaining identities of item (3) using the cubical identities: • For item (a), let i ≤ n . Then θ ,n ( x ) ∂ i, = xγ n +1 ∂ i, = x∂ i, γ n = θ ,n − ( x∂ i, ) . • For item (b), we have θ ,n ( x ) ∂ n +1 , = xγ n +1 ∂ n +1 , = x . • For item (c), we need only consider the case m ′ = 1 , i = n + 2 . For this case we have θ ,n ( x ) ∂ n +2 , = xγ n +1 ∂ n +2 , = x∂ n +1 , σ n +1 = θ ,n ( x∂ n +1 , ) . • For item (d), the only relevant degeneracy is σ n +1 , and we have θ ,n ( xσ n +1 ) = xσ n +1 γ n +1 = xσ n +1 σ n +2 = θ ,n ( x ) σ n +2 . • For item (e), let ≤ i ≤ n − . Then θ ,n ( xγ i ) = xγ i γ n +1 = xγ n γ i = θ ,n − ( x ) γ i . • For item (g), we have θ ,n +1 ( θ ,n ( x )) = xγ n +1 γ n +2 = xγ n +1 γ n +1 = θ ,n ( x ) γ n +1 . (cid:3) In view of Proposition 5.39, let X = Q R X . Now fix m ≥ , and assume that we have defined X m − satisfying the induction hypothesis on base dimension for some m ≥ . We will define X m by a further induction; we now establish our induction hypothesis and base case for this inductiveconstruction. Definition 5.40.
For n ≥ − , a subcomplex X m,n ⊆ X satisfies the induction hypothesis on degree if: (1) Given ( m ′ , n ′ ) where either ≤ m ′ < m or m ′ = m and ≤ n ′ ≤ n , all ( m ′ , n ′ ) -cones of X are contained in X m,n ;(2) For i ≥ , every non-degenerate cube (cid:3) m + i → X m,n is an ( m, i ) -cone;(3) For every ( m ′ , n ′ ) -cone in X with either ≤ m ′ < m or m ′ = m and ≤ n ′ ≤ n , X m,n contains an ( m ′ , n ′ + 1) -cone θ m ′ ,n ′ ( x ) satisfying the identities of Definition 5.38, item (3);(4) Any ( m, n ′ ) -cone in X m,n with n ′ > n is either degenerate, an ( m − , n ′ + 1) -cone, or θ m,n ′ − ( x ) for some ( m, n ′ − -cone x ;(5) If n ≥ then there is a trivial cofibration X m,n − ֒ → X m,n . Proposition 5.41. X m − satisfies the induction hypothesis on degree for n = − .Proof. This follows immediately from the induction hypothesis on base dimension. (cid:3)
In view of Proposition 5.41, let X m, − = X m − . Now fix n ≥ and assume that we have defined X m,n − satisfying the induction hypothesis on degree for n − . Lemma 5.42. X m,n − contains all degenerate ( m, n ) -cones of X .Proof. If y = xσ i or y = xγ i is an ( m, n ) -cone, then x = y∂ i, is an ( m, n − -cone by Lemma 5.7and Corollary 5.10. Thus x is in X m,n − , and therefore so is y . (cid:3) Before we can definine X m,n , we must prove a lemma involving cones in X m,n − of the form θ m ′ , ( x ) . Lemma 5.43.
Let m ′ ≥ , x : (cid:3) m ′ → X m,n − , and ≤ i ≤ m +1 . The image under θ m ′ , ( x ) : (cid:3) m ′ +1 → X m,n − of the critical edge with respect to the face ∂ i, is degenerate.Proof. For i ≥ , this follows from Lemma 5.13, since θ m ′ , ( x ) is an ( m ′ , -cone. For i = 1 , weproceed by induction on m ′ . For m ′ = 0 , we have θ , ( x ) = xσ ; so θ , ( x ) is a degeneracy of avertex, thus its unique edge is degenerate.Now let m ′ ≥ , and suppose that the statement holds for m ′ − . The edge in question may be writ-ten as θ m ′ , ( x ) ∂ m ′ +1 , ...∂ , ∂ , . By Definition 5.38, item (c), this is equal to θ m ′ − , ( x∂ m ′ , ) ∂ m ′ , ...∂ , ,which is degenerate by the induction hypothesis. (cid:3) Our next step will be to define X m,n by a further (transfinite) induction. To that end, let T be theset of all ( m, n ) -cones in X which are not contained in X m,n − . Lemma 5.44.
The set T consists of those non-degenerate ( m, n ) -cones which are not ( m − , n +1) -cones and are not equal to θ m,n − ( x ) for any ( m, n − -cone x .Proof. By Lemma 5.42, every cone in T is non-degenerate, while all ( m − , n + 1) -cones of x and allcones of the form θ m,n − ( x ) are contained in X m,n − by item (1) and item (3) of Definition 5.38.That these are the only ( m, n ) -cones of X contained in X m,n − follows from Definition 5.38, item (4). (cid:3) UBICAL MODELS OF ( ∞ , -CATEGORIES 57 We now impose an arbitrary well-ordering on T , indexing its elements as x t for t < κ , for a suitableordinal κ . Similarly to a typical proof by induction on skeleta, we will build X m,n from X m,n − bya series of open-box fillings. Proposition 5.45.
For each ordinal number t ≤ κ , there exists a subcomplex X m,n,t whose cubesare exactly those of X m,n − , plus all cubes x t ′ for t ′ < t and an ( m, n + 1) -cone θ m,n ( x t ′ ) satisfyingthe identities of item (a) through item (c) of Definition 5.38 for every such x t ′ . Furthermore, foreach t < κ , the inclusion X m,n,t ֒ → X m,n,t +1 is a trivial cofibration.Proof. We begin by setting X m,n, = X m,n − . Now suppose that we have constructed X m,n,t , andconsider the ( m, n ) -cone x t . For each i ≤ n , the face x t ∂ i, is an ( m, n − -cone by Lemma 5.7;thus X m,n − contains an ( m, n ) -cone θ m,n − ( x t ∂ i, ) . Similarly, for each i ≥ n + 2 , the face x t ∂ i − , is an ( m − , n ) -cone, and so X m,n − contains an ( m − , n + 1) -cone θ m − ,n ( x t ∂ i − , ) , and thesecones satisfy the identities of Definition 5.38, item (a) through item (c). Using Lemma 5.20, we willdefine a map y : B m,n +1 ,n +1 → X m,n,t with y i, = θ m,n − ( x t ∂ i, ) for ≤ i ≤ n , y n +1 , = x t , and y i, = θ m − ,n ( x t ∂ i − , ) for i ≥ n + 2 .To show that we can define such a map, we must verify that our choices of y i,ε satisfy the cubicalidentity for composing face maps.For i < i ≤ n, ε = ε = 0 , we have: y i , ∂ i , = θ m,n − ( d i , x t ) ∂ i , = θ m,n − ( x t ∂ j, ∂ i, )= θ m,n − ( x t ∂ i, ∂ j − , )= θ m,n − ( x t ∂ i, ) ∂ j − , = y i, ∂ j − , For i < i = n + 1 , we have: y n +1 , ∂ i , = x t ∂ i, = θ m,n − ( x t ∂ i, ) ∂ n, = y i , ∂ n, For n + 1 = i < i we have: y i , ∂ n +1 , = θ m − ,n ( x t ∂ i − , ) ∂ n +1 , = x t ∂ i − , = y n +1 , ∂ i − , Finally, for n + 2 ≥ i < i , we have: y i , ∂ i , = θ m − ,n ( x t ∂ i − , ) ∂ i , = θ m − ,n ( x t ∂ i − , ∂ i − , )= θ m − ,n ( x t ∂ i − , ∂ i − , )= θ m − ,n ( x t ∂ i − , ) ∂ i − , = y i , ∂ i − , Thus the ( n + 1) -tuple y does indeed define a map B m,n +1 ,n +1 → X . Now consider the followingcommuting diagram: B m,n +1 ,n +1 y / / (cid:127) _ ∼ (cid:15) (cid:15) X (cid:15) (cid:15) C m,n +1 / / (cid:3) The left-hand map is a trivial cofibration by Proposition 5.21, while the right-hand map is a fibrationby assumption. Thus there exists a lift of this diagram, i.e. an ( m, n + 1) -cone θ m,n ( x t ) : C m,n +1 → X such that for i ≤ n, θ m,n ( x t ) ∂ i, = θ m,n − ( x t ∂ i, ) , θ m,n ( x t ) ∂ n +1 , = x t , and for i ≥ n +2 , θ m,n ( x t ) ∂ i, = θ m − ,n ( x t ∂ i − , ) . So each face θ m,n ( x t ) ∂ i, is in X m,n − ⊆ X m,n,t , while θ m,n ( x t ) ∂ n +1 , is not contained in X m,n,t by assumption. Furthermore, all other faces of θ m,n ( x t ) are ( m − , n ) -cones by Lemma 5.7, hence they are in X m − ,n ⊆ X m,n,t by the induction hypoth-esis on base dimension. Thus the restriction of the cube θ m,n ( x t ) : (cid:3) m + n +1 → X to the openbox ⊓ m + n +1 n +1 , defines a map ⊓ m + n +1 n +1 , → X m,n,t . Furthermore, the critical edge of this open box isdegenerate – for n = 0 this follows from Lemma 5.43, while for n ≥ it follows from Lemma 5.13.So we may define X m,n,t +1 by the following pushout diagram: b ⊓ m + n +1 n +1 , θ m,n ( x t ) | ⊓ m + n +1 n +1 , / / (cid:127) _ (cid:15) (cid:15) X m,n,t (cid:127) _ (cid:15) (cid:15) b (cid:3) m + n +1 θ m,n ( x t ) / / X m,n,t +1 ❴✤ Then X m,n,t +1 contains X m,n,t , plus x t and θ m,n ( x t ) , and no other non-degenerate cubes. Fur-thermore, the inclusion X m,n,t ֒ → X m,n,t +1 is a trivial cofibration, as a pushout of an inner openbox inclusion.For a limit ordinal t , we define X m,n,t to be the sequential colimit of the inclusions X m,n, ֒ → ... ֒ → X m,n,t ′ ֒ → ... for t ′ < t . (cid:3) We define X m,n to be the cubical set X m,n,κ constructed in Proposition 5.45. Our next task isto show that X m,n satisfies the induction hypothesis on degree. Verifying condition item (3) ofDefinition 5.40 will take the most work, so we begin with the other conditions. Proposition 5.46. X m,n satisfies item (1), item (2), item (4), and item (5) of Definition 5.40. UBICAL MODELS OF ( ∞ , -CATEGORIES 59 Proof.
For item (1), we first observe that any ( m ′ , n ′ ) -cone with m ′ < m or m ′ = m, n ′ < n iscontained in X m,n − ⊆ X m,n . Furthermore, each ( m, n ) -cone of X which is not in X m,n − is equalto x t for some t < κ , and is thus contained in X m,nt +1 ⊆ X m,n . The condition of item (2) holds forthe cubes of X m,n − by the induction hypothesis on degree, while the only new non-degeneratecubes added in the construction of X m,n are the ( m, n ) -cones x t and the ( m, n + 1) -cones θ m,n ( x t ) .For item (4), we can again apply the induction hypothesis for the cubes of X m,n − , and observethat the only non-degenerate ( m, n ′ ) -cones with n ′ > n which we have added in X m,n are those ofthe form θ m,n ( x t ) . Finally, for item (5), the inclusion X m,n − ֒ → X m,n is a trivial cofibration sinceit is the sequential colimit of the trivial cofibrations X m,n,t ֒ → X m,n,t +1 . (cid:3) Now we consider item (3). By the induction hypothesis on n , for every ( m ′ , n ′ ) -cone x in X m,n with m ′ < m or m ′ = m, n ′ < n there is an ( m ′ , n ′ + 1) -cone θ m ′ ,n ′ ( x ) in X m,n − satisfying thenecessary identities; thus we only need to define θ m,n and show that it satisfies these identities aswell. Definition 5.47.
Let x be an ( m, n ) -cone of X . Then θ m,n ( x ) : (cid:3) m + n +1 → X m,n is defined asfollows:(1) If the standard form of x is zσ a p for some a p ≥ n + 1 , then θ m,n ( x ) = θ m − ,n ( z ) σ a p +1 ;(2) If the standard form of x is zγ b q for some b q ≤ n − , then θ m,n ( x ) = θ m,n − ( z ) γ b q ;(3) If the standard form of x is zγ b q for some b q ≥ n + 1 , then θ m,n ( x ) = θ m − ,n ( z ) γ b q +1 ;(4) If x is an ( m − , n + 1) -cone not covered under any of cases (1) through (3), then θ m,n ( x ) = xγ n +1 ;(5) If x = θ m,n − ( x ′ ) for some x ′ : C m,n − → X and x is not covered under any of cases (1)through (4) then θ m,n ( x ) = xγ n ;(6) If x ∈ T , then θ m,n ( x ) is as constructed in Proposition 5.45. Proposition 5.48.
Definition 5.47 defines a function θ m,n : cSet ( C m,n , X ) → cSet ( C m,n +1 , X ) .Proof. There are two things we need to show: first, that each of the constructions of Definition 5.47produces an ( m, n + 1) -cone; second, that at least one of cases (1) through (6) applies to every ( m, n ) -cone of X .That the construction of case (6) produces an ( m, n ) -cone follows from Proposition 5.45; the othercases follow from Lemma 5.11 and Corollary 5.10. To see that every ( m, n ) -cone of X falls underone of cases (1) through (6), we first consider degenerate cones. Those whose standard forms endwith any map other than γ n fall under one of cases (1) through (3) (for those whose standard formsend with degeneracy maps, this follows from Lemma 5.16). Those whose standard forms end with γ n fall under case (4) by Lemma 5.12. Every non-degenerate cone falls under one of cases (4), (5)or (6) by Lemma 5.44. (cid:3) The proof that θ m,n satisfies the identities of Definition 5.38, item (3) involves many elaborate caseanalyses; for brevity, these calculations have been relegated to appendix A. Corollary 5.49. X m,n satisfies the induction hypothesis on degree for n . Proof.
All criteria of Definition 5.40 are proven in Proposition 5.46, except for item (3), which isproven in Proposition A.1 through Proposition A.5. (cid:3)
Now, given X m,n satisfying the induction hypothesis for all n ≥ − , we let X m be the colimit ofthe sequence of inclusions: X m − = X m, − ֒ → X m, ֒ → ... ֒ → X m,n ֒ → ... Proposition 5.50. X m satisfies the induction hypothesis on base dimension.Proof. For item (1) of Definition 5.38, we may first note that all ( m ′ , n ) -cones for m ′ < m arecontained in X m − ⊆ X m by the induction hypothesis on X m − . Furthermore, if x is an ( m, n ) -cone of X for some n ≥ , then x is contained in X m,n ⊆ X m . Thus X m contains all ( m ′ , n ) -cones of X for m ′ ≤ m, n ≥ . Since every cube of X m is contained in some X m,n , item (2)and item (3) follow immediately from the corresponding conditions in the induction hypothesis ondegree. Finally, by the induction hypothesis on degree, each map X m,n − ֒ → X m,n for n ≥ is atrivial cofibration, hence the sequential colimit X m − ֒ → X m is a trivial cofibration as well. Thus X m satisfies item (4). (cid:3) So for every m ≥ we can construct a subcomplex X m ⊆ X satisfying the induction hypothesis.By considering the union of all these subcomplexes, we can prove Theorem 5.37. Proof of Theorem 5.37.
Consider the sequence of inclusions Q R X = X ֒ → X ֒ → ... ֒ → X m ֒ → ... The colimit of this diagram is the union of all the subcomplexes X m . But since every cube (cid:3) m → X is contained in X m (as an ( m, -cone), this colimit is X itself. Because each map in the diagramis a trivial cofibration, the colimit map Q R X ֒ → X is a trivial cofibration as well. (cid:3) Theorem 5.51.
The adjunction Q : sSet ⇄ cSet : R is a Quillen equivalence.Proof. The adjunction is Quillen by Proposition 5.25. Q preserves and reflects the weak equivalencesof the Quillen model structure on sSet by Corollary 5.29 and Proposition 5.30. Thus Q ⊣ R satisfiesthe hypotheses of Corollary 1.14, item (ii) and we can apply Theorem 5.37 to conclude that it is aQuillen equivalence. (cid:3) Proof of Theorem 5.1.
First note that, because all objects in both cSet and sSet are cofibrant, theleft derived functor L ( T Q ) is the composite of the left derived functors LT and LQ , while the leftderived functor of the identity is the identity; this can easily be seen from [Hov99, Def. 1.36]. ByCorollary 5.35, we have a natural weak equivalence T Q ⇒ id sSet . In the homotopy category HosSet ,this natural weak equivalence becomes a natural isomorphism LT ◦ LQ ∼ = id HosSet . By Theorem 5.51, LQ is an equivalence of categories, thus LT is an equivalence of categories as well. The adjunction T ⊣ U is Quillen by Proposition 4.25, so this implies it is a Quillen equivalence. (cid:3) UBICAL MODELS OF ( ∞ , -CATEGORIES 61 The proofs in this section can easily be adapted to show that Q ⊣ R is a Quillen equivalence betweenthe standard model structures for ∞ -groupoids on sSet and cSet . (This result was essentially statedas [KLW19, Prop. 5.3], but the proof supplied there only shows that Q and R form a Quillenadjunction.) Theorem 5.52.
The adjunction Q : sSet ⇄ cSet : R is a Quillen equivalence between the Quillenmodel structure on sSet and the Grothendieck model structure on cSet .Proof. Proposition 4.25 and Proposition 5.25 both have natural analogues, showing that T ⊣ U and Q ⊣ R are Quillen adjunctions between these model structures (implying in particular that Q preserves weak equivalences). Since every weak equivalence in the Joyal model structure is also aweak equivalence in the Quillen model structure, F is a natural weak equivalence in the Quillenmodel structure as well. Thus the proof of Proposition 5.30 adapts to show that Q reflects the weakequivalences of the Quillen model structure. Corollary 1.14, item (ii) and Theorem 5.37 then implythe analogue of Theorem 5.51, since every cubical Kan complex is a cubical quasicategory and everyweak equivalence in the cubical Joyal model structure is a weak equivalence in the Grothendieckmodel structure. (cid:3) The proof of Theorem 5.1 can then be adapted to obtain a new proof that T ⊣ U is a Quillenequivalence between the Grothendieck and Quillen model structures, as was previously shown in[Cis06, Prop. 8.4.30]. Appendix A. Verification of identities on θ Here we prove that the construction θ m,n of Definition 5.47 satisfies all of the necessary identities.We begin with the identities involving faces. Proposition A.1. θ m,n satisfies the identities of Definition 5.38, item (a) and item (b); that is,for i ≤ n , θ m,n ( x ) ∂ i, = θ m,n − ( x∂ i, ) , while θ m,n ( x ) ∂ n +1 , = x .Proof. We will prove this via a case analysis, based on the six cases of Definition 5.47. First,let x = zσ a p in standard form, for a p ≥ n + 1 . By the induction hypotheses, for m ′ < m or m ′ = m, n ′ < n , θ m ′ ,n ′ satisfies all the identities of Definition 5.38, item (3) (in future computationswe will often use this assumption without comment). So for i ≤ n we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) σ a p +1 ∂ i, = θ m − ,n ( z ) ∂ i, σ a p = θ m − ,n − ( z∂ i, ) σ a p = θ m,n − ( z∂ i, σ a p − )= θ m,n − ( zσ a p ∂ i, )= θ m,n − ( x∂ i, ) And for i = n + 1 we have: θ m,n ( x ) ∂ n +1 , = θ m − ,n ( z ) σ a p +1 ∂ n +1 , = θ m − ,n ( z ) ∂ n +1 , σ a p = zσ a p = x Now suppose that the standard form of x is zγ b q , where b q ≤ n − . Note that we must have b q ≥ ,so this case can only occur when n ≥ . Now for i ≤ b q − we have: θ m,n ( x ) ∂ i, = θ m,n − ( z ) γ b q ∂ i, = θ m,n − ( z ) ∂ i, γ b q − = θ m,n − ( z∂ i, ) γ b q − = θ m,n − ( z∂ i, γ b q − )= θ m,n − ( zγ b q ∂ i, )= θ m,n − ( x∂ i, ) For i = b q or i = b q + 1 we have: θ m,n ( x ) ∂ i, = θ m,n − ( z ) γ b q ∂ i, = θ m,n − ( z )= θ m,n − ( zγ b q ∂ i, )= θ m,n − ( x∂ i, ) For b q + 2 ≤ i ≤ n we have: θ m,n ( x ) ∂ i, = θ m,n − ( z ) γ b q ∂ i, = θ m,n − ( z ) ∂ i − , γ b q = θ m,n − ( z∂ i − , ) γ b q = θ m,n − ( z∂ i − , γ b q )= θ m,n − ( zγ b q ∂ i, )= θ m,n − ( x∂ i, ) And for i = n + 1 we have: UBICAL MODELS OF ( ∞ , -CATEGORIES 63 θ m,n ( x ) ∂ n +1 , = θ m,n − ( z ) γ b q ∂ n +1 , = θ m,n − ( z ) ∂ n, γ b q = zγ b q = x Next we consider the case where the standard form of x is zγ b q , b q ≥ n + 1 . Then for i ≤ n we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) γ b q +1 ∂ i, = θ m − ,n ( z ) ∂ i, γ b q = θ m − ,n − ( z∂ i, ) γ b q = θ m,n − ( z∂ i, γ b q − )= θ m,n − ( zγ b q ∂ i, )= θ m,n − ( x∂ i, ) And for i = n + 1 we have: θ m,n ( x ) ∂ n +1 , = θ m − ,n ( z ) γ b q +1 ∂ n +1 , = θ m − ,n ( z ) ∂ n +1 , γ b q = zγ b q = x Next, we consider case (4) of Definition 5.47: let x be an ( m − , n + 1) -cone not falling under anyof cases (1)-(3). By Lemma 5.7, every face x∂ i, for i ≤ n is an ( m − , n ) -cone, and therefore θ m,n − ( x∂ i, ) = x∂ i, γ n by the induction hypothesis. Now, for i ≤ n , we can compute: θ m,n ( x ) ∂ i, = xγ n +1 ∂ i, = x∂ i, γ n = θ m,n − ( x∂ i, ) And θ m,n ( x ) ∂ n +1 , = xγ n +1 ∂ n +1 , = x .Next, we consider case (5): consider an ( m, n ) -cone θ m,n − ( x ′ ) not falling under any of cases (1)through (4). Then for i ≤ n − we have: θ m,n ( θ m,n − ( x ′ )) ∂ i, = θ m,n − ( x ′ ) γ n ∂ i, = θ m,n − ( x ′ ) ∂ i, γ n − = θ m,n − ( x ′ ∂ i, ) γ n − = θ m,n − ( θ m,n − ( x ′ ∂ i, ))= θ m,n − ( θ m,n − ( x ′ ) ∂ i, ) For i = n we have: θ m,n ( θ m,n − ( x )) ∂ n, = θ m,n − ( x ′ ) γ n ∂ n, = θ m,n − ( x ′ )= θ m,n − ( θ m,n − ( x ′ ) ∂ n, ) And for i = n + 1 we have θ m,n ( θ m,n − ( x ′ )) ∂ n +1 , = θ m,n − ( x ′ ) γ n ∂ n +1 , = θ m,n − ( x ′ ) .Finally, we consider case (6). Let x ∈ T ; then the identities hold by Proposition 5.45. (cid:3) Proposition A.2. θ m,n satisfies the identity of Definition 5.38, item (c); that is, for all x : C m,n → X m,n , i ≥ n + 2 , we have θ m,n ( x ) ∂ i, = θ m − ,n ( x∂ i − , ) .Proof. Throughout the proof, we fix i ≥ n +2 . First we consider case (1) of Definition 5.47. Supposethat the standard form of x is zσ a p , for some a p ≥ n + 1 . Here we must consider various casesbased on a comparison of i with a p . First suppose that i ≤ a p ; note that this implies a p ≥ n + 2 .Then we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) σ a p +1 ∂ i, = θ m − ,n ( z ) ∂ i, σ a p = θ m − ,n ( z∂ i − , ) σ a p = θ m − ,n ( z∂ i − , σ a p − )= θ m − ,n ( zσ a p ∂ i − , )= θ m − ,n ( x∂ i − , ) To obtain the fourth equality, we have used the identity of item (d) and the fact that a p − ≥ n + 1 .Next suppose that i = a p + 1 ; then we have: UBICAL MODELS OF ( ∞ , -CATEGORIES 65 θ m,n ( x ) ∂ a p +1 , = θ m − ,n ( z ) σ a p +1 ∂ a p +1 , = θ m − ,n ( z )= θ m − ,n ( zσ a p ∂ a p , )= θ m − ,n ( x∂ a p , ) Finally, suppose i ≥ a p + 2 ; note that this implies a p ≥ n + 3 . Then we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) σ a p +1 ∂ i, = θ m − ,n ( z ) ∂ i − , σ a p +1 = θ m − ,n ( z∂ i − , ) σ a p +1 = θ m − ,n ( z∂ i − , σ a p )= θ m − ,n ( zσ a p ∂ i − , )= θ m − ,n ( x∂ i − , ) Next we consider case (2): suppose that x = zγ b q in standard form, where b q ≤ n − . Then i ≥ b q + 3 , and we have: θ m,n ( x ) ∂ i, = θ m,n − ( z ) γ b q ∂ i, = θ m,n − ( z ) ∂ i − , γ b q = θ m − ,n − ( z∂ i − , ) γ b q = θ m − ,n ( z∂ i − , γ b q )= θ m − ,n ( zγ b q ∂ i − , )= θ m − ,n ( x∂ i − , ) Next we consider case (2): suppose that x = γ b q z in standard form, where b q ≥ n + 1 . Once again,we must perform a case analysis based on a comparison of i with b q . First suppose that i ≤ b q ,implying b q ≥ n + 2 . Then we can compute: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) γ b q +1 ∂ i, = θ m − ,n ( z ) ∂ i, γ b q = θ m − ,n ( z∂ i − , ) γ b q = θ m − ,n ( z∂ i − , γ b q − )= θ m − ,n ( zγ b q ∂ i − , )= θ m − ,n ( x∂ i − , ) Next suppose that i = b q + 1 or b q + 2 ; then we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) γ b q +1 ∂ i, = θ m − ,n ( z ) ∂ b q +1 , σ b q +1 = θ m − ,n ( z∂ b q , ) σ b q +1 = θ m − ,n ( z∂ b q , σ b q )= θ m − ,n ( zγ b q ∂ i − , )= θ m − ,n ( x∂ i − , ) To obtain the third equality, we used the identity of item (c) for θ m − ,n and the assumption that b q ≥ n + 1 . Finally, suppose i ≥ b q + 3 , implying i ≥ n + 4 . Then we have: θ m,n ( x ) ∂ i, = θ m − ,n ( z ) γ b q +1 ∂ i, = θ m − ,n ( z ) ∂ i − , γ b q +1 = θ m − ,n ( z∂ i − , ) γ b q +1 = θ m − ,n ( z∂ i − , γ b q )= θ m − ,n ( zγ b q ∂ i − , )= θ m − ,n ( x∂ i − , ) Next we consider case (4): let x be an ( m − , n + 1) -cone not covered under any of cases(1) through (3). Then x∂ i − , is an ( m − , n + 1) -cone by Lemma 5.7, so θ m − ,n ( x∂ i − , ) = x∂ i − , γ n +1 by the identity of item (h) for θ m − ,n . Furthermore, note that by Lemma 5.9, x∂ n +1 , = x∂ m + n +1 ...∂ n +1 , σ n +1 ...σ m + n . Using the co-cubical identities, we can rewrite this as x∂ m + n +1 ...∂ n +1 , σ n +1 ...σ n +1 . Then for i = n + 2 , we can compute: UBICAL MODELS OF ( ∞ , -CATEGORIES 67 θ m,n ( x ) ∂ n +2 , = xγ n +1 ∂ n +2 , = x∂ n +1 , σ n +1 = x∂ m + n +1 ...∂ n +1 , σ n +1 ...σ n +1 σ n +1 = x∂ m + n +1 ...∂ n +1 , σ n +1 ...σ n +1 γ n +1 = x∂ n +1 , γ n +1 While for i ≥ n + 3 , we have: θ m,n ( x ) ∂ i, = xγ n +1 ∂ i, = x∂ i − , γ n +1 = θ m − ,n ( x∂ i − , ) Next we consider case (5). Let x ′ : C m,n − → X m,n , and consider θ m,n ( θ m,n − ( x ′ )) . Then we cancompute: θ m,n ( θ m,n − ( x ′ )) ∂ i, = θ m,n − ( x ′ ) γ n ∂ i, = θ m,n − ( x ′ ) ∂ i − , γ n = θ m − ,n − ( x ′ ∂ i − , ) γ n = θ m − ,n ( θ m − ,n − ( x ′ ∂ i − , ))= θ m − ,n ( θ m,n − ( x ′ ) ∂ i − , ) Finally, in case (6), the identity holds by Proposition 5.45. (cid:3)
Next we consider the identities involving degeneracies and connections.
Proposition A.3. θ m,n satisfies the identities of Definition 5.38, item (d), item (e), and item (f ).That is: • If xσ i is an ( m, n ) -cone for i ≥ n + 1 , then θ m,n ( xσ i ) = θ m − ,n ( x ) σ i +1 ; • If xγ i is an ( m, n ) -cone for i ≤ n − , then θ m,n ( xγ i ) = θ m,n − ( x ) γ i ; • If xγ i is an ( m, n ) -cone for i ≥ n + 1 , then θ m,n ( xγ i ) = θ m − ,n ( x ) γ i +1 .Proof. For each identity, we will perform a case analysis based on the standard form of x . Foritem (d), consider an ( m, n ) -cube xσ i , where i ≥ n +1 and the standard form of x is yγ b ...γ b q σ a ...σ a p .If the string of degeneracy maps in the standard form of x is empty, or a p < i , then the standard form of xσ i ends with σ i , so θ m,n ( xσ i ) = θ m − ,n ( x ) σ i +1 by definition. So suppose that a p ≥ i .Then: θ m,n ( xσ i ) = θ m,n ( yγ b ...γ b q σ a ...σ a p σ i )= θ m,n ( yγ b ...γ b q σ a ...σ a p − σ i σ a p +1 ) By assumption, all the indices a , ..., a p − , are less than a p . Rearranging the expression on theright-hand side of the equation into standard form using the co-cubical identities will not increaseany of these indices by more than 1, so the rightmost map in the standard form of xσ i , i.e. thedegeneracy map with the highest index, is σ a p +1 . Therefore, we can compute: θ m,n ( yγ b ...γ b q σ a ...σ a p − σ i σ a p +1 ) = θ m − ,n ( yγ b ...γ b q σ a ...σ a p − σ i ) σ a p +2 = θ m − ,n ( yγ b ...γ b q σ a ...σ a p − ) σ i +1 σ a p +2 = θ m − ,n ( yγ b ...γ b q σ a ...σ a p − ) σ a p +1 σ i +1 = θ m − ,n ( yγ b ...γ b q σ a ...σ a p − σ a p ) σ i +1 = θ m − ,n ( x ) σ i +1 So θ m,n satisfies the identity of item (d).Next we will verify the identity of item (f). Consider an ( m, n ) -cube xγ i , where i ≥ n + 1 and thestandard form of x is as above. If this standard form contains no degeneracy maps, and b q < i or x is non-degenerate, then the standard form of xγ i ends with γ i , so the identity holds by definition.The remaining possibilities for the standard form of x can be divided into various cases. First,suppose that the string of degeneracy maps in the standard form of x is non-empty, i.e. x = zσ a p in standard form. By Lemma 5.7, x = xγ i ∂ i, is an ( m − , n ) -cone, so a p ≥ n + 1 by Lemma 5.16.Now we must break this into further cases based on a comparison between i and a p . If i < a p then,using the co-cubical identities, item (d) for θ m,n , and item (f) for θ m − ,n , we can compute: θ m,n ( xγ i ) = θ m,n ( zσ a p γ i )= θ m,n ( zσ a p γ i )= θ m,n ( zγ i σ a p +1 )= θ m − ,n ( zγ i ) σ a p +2 = θ m − ,n ( z ) γ i +1 σ a p +2 = θ m − ,n ( z ) σ a p +1 γ i +1 = θ m − ,n ( zσ a p ) γ i +1 = θ m − ,n ( x ) γ i +1 UBICAL MODELS OF ( ∞ , -CATEGORIES 69 Next we consider the case i = a p : θ m,n ( xγ a p ) = θ m,n ( zσ a p γ a p )= θ m,n ( zσ a p σ a p +1 )= θ m − ,n ( zσ a p ) σ a p +2 = θ m − ,n ( z ) σ a p +1 σ a p +2 = θ m − ,n ( z ) σ a p +1 γ a p +1 = θ m − ,n ( zσ a p ) γ a p +1 = θ m − ,n ( x ) γ a p +1 Now we consider the case i > a p . Note that this implies i ≥ n + 2 , so i − ≥ n + 1 . Thus we cancompute: θ m,n ( xγ i ) = θ m,n ( zσ a p γ i )= θ m,n ( zγ i − σ a p )= θ m − ,n ( zγ i − ) σ a p +1 = θ m − ,n ( z ) γ i σ a p +1 = θ m − ,n ( z ) σ a p +1 γ i +1 = θ m − ,n ( zσ a p ) γ i +1 = θ m − ,n ( x ) γ i +1 Next we will verify item (f) in the case where the standard form of x contains no degeneracy maps,and i ≤ b q . In this case we can compute: θ m,n ( xγ i ) = θ m,n ( yγ b ...γ b q γ i )= θ m,n ( yγ b ...γ b q − γ i γ b q +1 ) Similarly to what we saw when verifying item (d), the indices b , ..., b q − are all strictly less than b q .So after we have rearranged the expression on the right-hand side of this equation into standard formby repeatedly applying the identity γ k γ j = γ j γ k +1 for j ≤ k , the leftmost map in the expressionwill still be γ b q +1 . Thus we can apply the definition of θ m,n to compute: θ m,n ( yγ b ...γ b q − γ i γ b q +1 ) = θ m − ,n ( yγ b ...γ b q − γ i ) γ b q +2 = θ m − ,n ( yγ b ...γ b q − ) γ i +1 γ b q +2 = θ m − ,n ( yγ b ...γ b q − ) γ b q +1 γ i +1 = θ m − ,n ( yγ b ...γ b q ) γ i +1 = θ m − ,n ( x ) γ i +1 Thus θ m,n satisfies item (f).Finally we will verify item (e). Consider an ( m, n ) -cube xγ i , where i ≤ n − and the standard formof x is as above. Once again, we must consider several possible cases based on the standard formof x . As with item (f), if the standard form of x contains no degeneracy maps, and b q < i or x isnon-degenerate, then γ i is the rightmost map in the standard form of xγ i , and the identity holdsby definition. Once again, the remaining cases will require computation.As above, we begin with the case where the string of degeneracy maps in the standard form of x is non-empty. By Lemma 5.7, x = xγ i ∂ i, is an ( m, n − -cone, so a p ≥ n by Lemma 5.16. Then,using the co-cubical identities, item (d) for θ m,n , and item (e) for θ m − ,n , we can compute: θ m,n ( xγ i ) = θ m,n ( zσ a p γ i )= θ m,n ( zγ i σ a p +1 )= θ m − ,n ( zγ i ) σ a p +2 = θ m − ,n − ( z ) γ i σ a p +2 = θ m − ,n − ( z ) σ a p +1 γ i = θ m,n − ( zσ a p ) γ i = θ m,n − ( x ) γ i Next we consider the cases in which the standard form of x contains no degeneracy maps; first,suppose b q ≥ n . Then, using the cubical identities, item (f) for θ m,n , and item (e) for θ m − ,n , wecan compute: UBICAL MODELS OF ( ∞ , -CATEGORIES 71 θ m,n ( xγ i ) = θ m,n ( zγ b q γ i )= θ m,n ( zγ i γ b q +1 )= θ m − ,n ( zγ i ) γ b q +2 = θ m − ,n − ( z ) γ i γ b q +2 = θ m − ,n − ( z ) γ b q +1 γ i = θ m,n − ( γ b q z ) γ i = θ m,n − ( x ) γ i Next we consider the case b q = n − . Here we can compute: xγ i = yγ b ...γ b q − γ n − γ i = yγ b ...γ b q − γ i γ n As we have seen in previous cases, all of the coefficients b , ..., b q − are strictly less than n − . Soafter rearranging this expression into standard form, the rightmost map will still be γ n . Thus xγ i belongs to case (4), so: θ m,n ( xγ i ) = xγ i γ n +1 = xγ n γ i By Lemma 5.7, x = xγ i ∂ i is an ( m, n − -cone, so the fact that b q = n − implies that x alsobelongs to case (4). Thus xγ n = θ m,n − ( x ) , so item (e) is satisfied in this case.Finally, we consider the case i ≤ b q ≤ n − . Here we can compute: θ m,n ( xγ i ) = θ m,n ( yγ b ...γ b q γ i )= θ m,n ( yγ b ...γ b q − γ i γ b q +1 ) As we have done in previous computations, we may observe that all the indices b , ..., b q − are strictlyless than b q , so once the expression on the right-hand side of the equation has been rearranged intostandard form, its rightmost map will still be γ b q +1 . By assumption, b q + 1 ≤ n − , so using theco-cubical identities, the definition of θ m,n , and item (e) for θ m,n − , we can compute: θ m,n ( yγ b ...γ b q − γ i γ b q +1 ) = θ m,n − ( yγ b ...γ b q − γ i ) γ b q +1 = θ m,n − ( yγ b ...γ b q − ) γ i γ b q +1 = θ m,n − ( yγ b ...γ b q − ) γ b q γ i = θ m,n − ( yγ b ...γ b q ) γ i = θ m,n − ( x ) γ i Thus θ m,n satisfies item (e). (cid:3) Proposition A.4. If n ≥ then θ m,n satisfies the identity of Definition 5.38, item (g). That is,for any x : C m,n − → X , θ m,n ( θ m,n − ( x )) = θ m,n − ( x ) γ n .Proof. We proceed by a case analysis on x , based on the cases of Definition 5.47. In our computa-tions, we will freely use the identities for θ m,n which we have already proven. First suppose that x = zσ a p in standard form, for some a p ≥ n . Then we can compute: θ m,n ( θ m,n − ( x )) = θ m,n ( θ m,n − ( zσ a p ))= θ m,n ( θ m − ,n − ( z ) σ a p +1 )= θ m − ,n ( θ m − ,n − ( z )) σ a p +2 = θ m − ,n − ( z ) γ n σ a p +2 = θ m − ,n − ( z ) σ a p +1 γ n = θ m,n − ( zσ a p ) γ n = θ m,n − ( x ) γ n Next let the standard form of x be zγ b q where b q ≤ n − . Then we can compute: θ m,n ( θ m,n − ( x )) = θ m,n ( θ m,n − ( zγ b q ))= θ m,n ( θ m,n − ( z ) γ b q )= θ m,n − ( θ m,n − ( z )) γ b q = θ m,n − ( z ) γ n − γ b q = θ m,n − ( z ) γ b q γ n = θ m,n − ( zγ b q ) γ n = θ m,n − ( x ) γ n Now let the standard form of x be zγ b q where b q ≥ n . Then we can compute: UBICAL MODELS OF ( ∞ , -CATEGORIES 73 θ m,n ( θ m,n − ( x )) = θ m,n ( θ m,n − ( zγ b q ))= θ m,n ( θ m − ,n − ( z ) γ b q +1 )= θ m − ,n ( θ m − ,n − ( z )) γ b q +2 = θ m − ,n − ( z ) γ n γ b q +2 = θ m − ,n − ( z ) γ b q +1 γ n = θ m,n − ( zγ b q ) γ n = θ m,n − ( x ) γ n Next, we consider case (4): suppose that x is an ( m − , n ) -cone not falling under any of cases (1)through (3). Then θ m,n − ( x ) = xγ n . The assumption that x does not belong to any of cases (1)through (3) implies that either it is non-degenerate, or its standard form ends with γ n − . Eitherway, the standard form of xγ n ends with γ n , so it falls under case (4) by Lemma 5.12. Thus we cancompute: θ m,n ( θ m,n − ( x )) = θ m,n ( xγ n )= xγ n γ n +1 = xγ n γ n = θ m,n − ( x ) γ n Finally, case (5) consists of all cubes of the form θ m,n − ( x ) not falling under any of the previouscases, and in this case identity (f) holds by definition. (cid:3) Proposition A.5. θ m,n satisfies the identity of Definition 5.38, item (h). That is, if x is an ( m − , n + 1) -cone, then θ m,n ( x ) = xγ n +1 .Proof. As in previous proofs, we proceed via case analysis on x , based on the cases of Definition 5.47.First suppose that x is an ( m − , n + 1) -cone whose standard form is zσ a p . By Lemma 5.16, a p ≥ n + 2 . Therefore, by Lemma 5.7, x∂ a p = z is an ( m − , n + 1) -cone, so θ m − ,n ( z ) = zγ n +1 byitem (h) for θ m − ,n . Thus we can compute: θ m,n ( x ) = θ m − ,n ( z ) σ a p +1 = zγ n +1 σ a p +1 = zσ a p γ n +1 = xγ n +1 Now let x be an ( m − , n + 1) -cone whose standard form is zγ b q , b q ≤ n − . Then by Lemma 5.7, x∂ b q = z is an ( m − , n ) -cone. So by item (h) for θ m,n − , we have θ m,n − ( z ) = zγ n . Thus we cancompute: θ m,n ( x ) = θ m,n − ( z ) γ b q = zγ n γ b q = zγ b q γ n +1 = xγ n +1 Next let x be an ( m − , n + 1) -cone whose standard form is zγ b q , where b q ≥ n + 1 . Then byLemma 5.7, x∂ b q +1 = z is an ( m − , n + 1) -cone, so θ m − ,n ( z ) = zγ n +1 by item (h). Thus we cancompute: θ m,n ( x ) = θ m − ,n ( z ) γ b q +1 = zγ n +1 γ b q +1 = zγ b q γ n +1 = xγ n +1 Finally, case (4) consists of all ( m − , n + 1) -cones not falling under any of the previous cases, andin this case the identity of item (h) holds by definition. (cid:3) References [Bar10] Clark Barwick,
On left and right model categories and left and right Bousfield localizations , Homology,Homotopy and Applications, Vol. 12 (2010), No. 2, pp.245-320 (2010), no. 2, 245–320.[CCHM18] Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mörtberg, Cubical type theory: a constructiveinterpretation of the univalence axiom , 21st International Conference on Types for Proofs and Pro-grams, LIPIcs. Leibniz Int. Proc. Inform., vol. 69, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern,2018, pp. Art. No. 5, 34.[Cis06] Denis-Charles Cisinski,
Les préfaisceaux comme modèles des types d’homotopie , Astérisque (2006),no. 308, xxiv+390.[Cis14] ,
Univalent universes for elegant models of homotopy types , Preprint, 2014.[Cis19] ,
Higher categories and homotopical algebra , Cambridge Studiesin Advanced Mathematics, Cambridge University Press, Cambridge, 2019, .[GM03] Marco Grandis and Luca Mauri,
Cubical sets and their site , Theory and Applications of Categories (2003), no. 8, 185–211.[HKRS17] Kathryn Hess, Magdalena K¸edziorek, Emily Riehl, and Brooke Shipley, A necessary and sufficient con-dition for induced model structures , J. Topol. (2017), no. 2, 324–369, doi:10.1112/topo.12011 .[Hov99] Mark Hovey, Model categories , Mathematical Surveys and Monographs, vol. 63, American MathematicalSociety, Providence, RI, 1999.[Jar06] J. F. Jardine,
Categorical homotopy theory , Homology Homotopy Appl. (2006), no. 1, 71–144, http://projecteuclid.org/euclid.hha/1140012467 . UBICAL MODELS OF ( ∞ , -CATEGORIES 75 [Joy09] André Joyal, The theory of quasi-categories and its applications , Vol. II of course notes fromSimplicial Methods in Higher Categories, Centra de Recerca Matemàtica, Barcelona, 2008, 2009, http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf .[JT07] André Joyal and Myles Tierney,
Quasi-categories vs Segal spaces , Categories in algebra, geometry andmathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 277–326, doi:10.1090/conm/431/08278 .[KLW19] Krzysztof Kapulkin, Zachery Lindsey, and Liang Ze Wong,
A co-reflection of cubical sets into simplicialsets with applications to model structures , New York J. Math. (2019), 627–641.[KV18] Krzysztof Kapulkin and Vladimir Voevodsky, Cubical approach to straightening , Preprint., 2018.[Mal09] Georges Maltsiniotis,
La catégorie cubique avec connexions est une catégorie test stricte , HomologyHomotopy Appl. (2009), no. 2, 309–326, http://projecteuclid.org/euclid.hha/1296138523 .[MP89] Michael Makkai and Robert Paré, Accessible categories: the foundations of categorical model the-ory , Contemporary Mathematics, vol. 104, American Mathematical Society, Providence, RI, 1989, doi:10.1090/conm/104 , https://doi.org/10.1090/conm/104 .[Ols09] Marc Olschok, On constructions of left determined model structures , Ph.D. thesis, Masaryk University,2009.[Qui67] Daniel G. Quillen,