aa r X i v : . [ m a t h . A T ] A ug A DOUBLE ( ∞ , -CATEGORICAL NERVE FOR DOUBLECATEGORIES LYNE MOSER
Abstract.
We construct a nerve from double categories into double ( ∞ , ∞ , Contents
1. Introduction 21.1. Outline 5Acknowledgments 52. Preliminaries on 2-dimensional categories 52.1. 2-categories, double categories, and their relations 62.2. Notions of equivalences in a double category 103. Model structures on 2Cat and DblCat 123.1. Lack’s model structure for 2-categories 133.2. Model structure for weakly horizontally invariant double categories 144. Model structures for ( ∞ , ∞ , ∞ , N is right Quillen 225.3. The nerve N is homotopically fully faithful 275.4. Level of fibrancy of nerves of double categories 296. Nerve of 2-categories 346.1. The nerve NH ≃ is right Quillen and homotopically fully faithful 346.2. 2Cat is right-induced from 2CSS along NH ≃ NH and NH ≃ H A , H ≃ A , and L ≃ A Introduction
Higher category theory aims to study more structured objects than categories. Whilecategories consist of objects and morphisms (or 1 -cells ) between objects, higher categoriesalso have higher cells. In this perspective, a 2-category is obtained by also adding 2-cellsbetween the morphisms. A 2-category can actually be seen as a category enriched in cat-egories – its morphisms and 2-cells between any pair of objects form a category. Anothertype of 2-dimensional categories is given by internal categories to categories, called doublecategories . Such a structure has two types of morphisms between objects – horizontal and vertical morphisms – and its 2-cells are squares . In particular, a 2-category A can beseen as a horizontal double category H A in which every vertical morphism is trivial; orequivalently, as an internal category to categories whose category of objects is discrete.Many aspects of 2-category theory benefit from a passage to double categories. Forexample, a good notion of limit for 2-categories is that of a 2 -limit , first introduced byAuderset [1] and Borceux-Kelly [5], and further developed by Street [31], Kelly [20, 21]and Lack [24]. As clingman and the author show in [9], a 2-limit cannot be characterizedas a 2-terminal object in the 2-category of cones, but a passage to double categories allowssuch a characterization by results of Grandis and Paré [12, 13]. Indeed, they show thatthe 2-limit of a 2-functor F is double terminal in the double category of cones over thecorresponding double functor H F .These notions of categories, 2-categories, and double categories are often too strict toaccommodate many examples that appear in nature. In the perspective of generalizingcategories, an ( ∞ , k -cells invertible for k >
1, where compositions are only defined,associative, and unital up to higher cells. Such a higher structure should be thought ofas a homotopical version of a category. Similarly to the strict case, we can then interpretan ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , H The existence of such a commutative diagram would show that aspects of the theoryof ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ -notions precise, the machinery used is often that of model categories ,introduced by Quillen in [28], and these ∞ -notions are then defined as the fibrant ob-jects of a given model structure. This is the approach we will be taking here. As amodel for ( ∞ , completeness condition and assures that no extra data have been added byconsidering a space of objects instead of a set of objects. There are many other modelsof ( ∞ , DOUBLE ( ∞ , of ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , h ∞ on bisimplicial spaces whose fibrant objects arethe 2-fold complete Segal spaces and the horizontally complete double ( ∞ , h ∞ , and this implies that theidentity functor id : 2CSS → DblCat h ∞ is a right Quillen functor, which we interpret asthe horizontal embedding of ( ∞ , ∞ , nerve – of 2-categories and double categories into their ∞ -analogue, we also need model structures in this stricter setting. In [22, 23], Lack endowsthe category 2Cat of 2-categories and 2-functors with a model structure in which the weakequivalences are the biequivalences and all 2-categories are fibrant. Several nerves thatfully embed this homotopy theory of 2-categories into the one of ( ∞ , ∞ -bicategories by Gagna,Harpaz, and Lanari in [11].In the double categorical case, several model structures for double categories are con-structed by Fiore, Paoli, and Pronk in [10], but the horizontal embedding of 2-categoriesdoes not induce a Quillen pair between Lack’s model structure and any of these modelstructures. Therefore, in [25], the author, Sarazola, and Verdugo define another modelstructure on the category DblCat of double categories and double functors in which theweak equivalence are the double biequivalences and all double categories are fibrant. In-deed, this model structure is constructed in such a way that the horizontal embedding H : 2Cat → DblCat creates Lack’s model structure, and the functor H is right Quillenand homotopically fully faithful. However, this model structure is not well-behaved withrespect to vertical composition; for example, the free double category on two compos-able vertical morphisms is not cofibrant, while the one on two composable horizontalmorphisms is.To remedy this failure, the author, Sarazola, and Verdugo construct in [26] a modelstructure on DblCat with the same weak equivalences, i.e., the double biequivalences,and an additional cofibration [0] ⊔ [0] → V [1]. The existence of this model structure wasindependently noticed by Campbell [6]. As we show in [26], this new model structureis Quillen equivalent to the previous one through the identity functor on DblCat. Themodification fixes the issue with respect to the vertical composition mentioned above,but the horizontal embedding H is not right Quillen anymore. Indeed, the fibrant doublecategories are now the weakly horizontally invariant ones (see Definition 2.2.7), and thehorizontal double category H A of a 2-category A does not generally satisfy this condition.Therefore a fibrant replacement of H needs to be considered instead and it is given bya more homotopical version H ≃ : 2Cat → DblCat of H sending a 2-category to a doublecategory whose underlying horizontal 2-category is still A , but whose vertical morphismsare given by the adjoint equivalences of A . This gives a right Quillen and homotopically LYNE MOSER fully faithful functor H ≃ : 2Cat → DblCat, where DblCat is endowed with the modelstructure for weakly horizontally invariant double categories.In this paper, we construct a nerve functor N : DblCat → sSet ∆ op × ∆ op , and we show inTheorems 5.2.8 and 5.3.1 that N is a right Quillen and homotopically fully faithful functorfrom DblCat to DblCat h ∞ . Theorem A.
There is a Quillen pair
DblCat DblCat h ∞ CN ⊥ between the model structure on DblCat for weakly horizontally invariant double categoriesand the model structure on sSet ∆ op × ∆ op for horizontally complete double ( ∞ , -categories.Moreover, the components ǫ A : CNA → A of the (derived) counit are double biequiva-lences, for all double categories A , and therefore the nerve functor N : DblCat → DblCat h ∞ is homotopically fully faithful. We then restrict the nerve functor N along the homotopical horizontal embedding H ≃ : 2Cat → DblCat and show in Theorems 6.1.1 and 6.1.3 that this gives a right Quillenand homotopically fully faithful functor from 2Cat to 2CSS. In the case of 2-categories,it is further true that their homotopy theory is created by NH ≃ in the sense that 2Cat isright-induced from 2CSS along NH ≃ . Theorem B.
The adjunction L ≃ CNH ≃ ⊥ is a Quillen pair between the model structure on for -categories and the modelstructure on sSet ∆ op × ∆ op for -fold complete Segal spaces, i.e., ( ∞ , -categories.Moreover, the components ǫ A : L ≃ CNH ≃ A → A of the (derived) counit are biequiv-alences, for all -categories A , and therefore the nerve functor NH ≃ : 2Cat → ishomotopically fully faithful.Furthermore, Lack’s model structure on is right-induced from along NH ≃ . This gives a commutative diagram of homotopically fully faithful right Quillen functors.2CatDblCat 2CSSDblCat h ∞ NH ≃ H ≃ N id However, we were hoping to find a nerve that is compatible with the horizontal em-bedding functor H , but the nerve NH A of a horizontal double category H A associated toa 2-category is not generally a double ( ∞ , NH ≃ A gives a fibrant replacementof NH A in 2CSS (or in DblCat h ∞ ). Theorem C.
There is a level-wise homotopy equivalence NH A → NH ≃ A , which exhibits NH ≃ A as a fibrant replacement of NH A in (or in DblCat h ∞ ), for every -category A . In particular, it follows from this result that we have a diagram of homotopically fullyfaithful right Quillen functors
DOUBLE ( ∞ , h ∞ NH ≃ H id ≃ QE N id ≃ filled with a natural transformation which is level-wise a weak equivalence. This gives theexpected compatibility of the nerve N with the horizontal embedding H .1.1. Outline.
In Section 2, we first recall the basic terminology for 2-categories anddouble categories, and describe several functors between the categories 2Cat and DblCat.We then introduce notions of horizontal equivalences and weakly horizontally invertiblesquares in a double category, which allows us to define weakly horizontally invariant double categories. In Section 3, we recall the main features of Lack’s model structureon 2Cat and of the model structure of [26] on DblCat. Then, in Section 4, we get tothe ∞ -setting and describe the model structures DblCat h ∞ and 2CSS for double ( ∞ , N : DblCat → DblCat h ∞ and show that it is right Quillen and homotopically fullyfaithful. By restricting along the functor H ≃ : 2Cat → DblCat, we show in Section 6that the nerve functor NH ≃ : 2Cat → NH ≃ . We then construct a level-wise homotopy equivalence NH A → NH ≃ A for any 2-category A , which exhibits NH ≃ A as a fibrant replacementof NH A .The aim of Appendix A is to prove some technical results about weakly horizontallyinvertible squares, which were recently introduced independently by the author, Sarazola,and Verdugo in [25], and by Grandis and Paré in [14]. In particular, we show thata horizontal pseudo-natural transformation is an equivalence if and only if each of itssquare components are weakly horizontally invertible squares. The aim of Appendix Bis to describe the lower simplices of the nerves NA , NH ≃ A , and NH A in order to giveintuition on what the nerve construction is doing to a double category or a 2-category. Inparticular, this allows us to better understand the difference between the nerves NH ≃ A ,and NH A and provides intuition on why the latter is not fibrant. Acknowledgments.
I am grateful to my advisor, Jérôme Scherer, for the many fruitfulconversations on the content of this paper and for a close reading of several drafts. Iwould also like to thank Martina Rovelli, Viktoriya Ozornova, Nima Rasekh, and MaruSarazola for their helpful answers to many of my questions. In particular, Martina Rovellisuggested using Theorem 5.2.1 and Maru Sarazola suggested a nice trick for the proof ofLemma A.2.1. Finally, I would like to thank Alexander Campbell for useful discussionsat an early stage of this project.During the realization of this work, the author was at the Mathematical Sciences Re-search Institute in Berkeley, California, during the Spring 2020 semester, and was sup-ported by the Swiss National Science Foundation under the project P1ELP2_188039.2.
Preliminaries on -dimensional categories There are two notions of strict 2-dimensional categories: 2-categories and double cat-egories. In Section 2.1, we first recall the definitions of the category 2Cat of 2-categoriesand 2-functors, of an equivalence in a 2-category, and of the Gray tensor product on 2Cat
LYNE MOSER [15]. We then introduce the category DblCat of double categories and double functors,and the horizontal embedding H : 2Cat → DblCat together with its left adjoint L , andits right adjoint H given by the underlying horizontal 2-category of a double category.The category of double categories also admits a Gray tensor product [4], which restrictsalong H to a tensoring functor of DblCat over 2Cat. This gives a 2Cat-enrichment onDblCat, whose hom-objects are given by the 2-categories of double functors, horizontalpseudo-natural transformations, and modifications; see Appendix A.3 for definitions. Wethen introduce two other functors between 2-categories and double categories. The func-tor V : DblCat → H ≃ : 2Cat → DblCat together with its left adjoint L ≃ .In Section 2.2, we define notions of weak invertibility in a double category A for horizon-tal morphisms and squares, which are defined as the equivalences in the 2-categories H A and V A , respectively. We then introduce weakly horizontally invariant double categories,which will be the fibrant objects of the model structure on DblCat we consider.2.1. 2 -categories, double categories, and their relations. Recall that a 2-category A consists of objects, morphisms f : A → B between objects, and 2-cells α : f ⇒ g betweenparallel morphisms. A 2-functor F : A → B consists of assignments on objects, on mor-phisms, and on 2-cells which preserve the 2-categorical structures strictly.
Notation 2.1.1.
We denote by 2Cat the category of 2-categories and 2-functors.Since 2-categories have not only morphisms, but also 2-cells, a good notion of invert-ibility for a morphism in a 2-category is given by requiring that it has an inverse up toan invertible 2-cell, instead of strictly.
Definition 2.1.2. An equivalence f : A ≃ −→ B in a 2-category A is a tuple ( f, g, η, ǫ )consisting of morphisms f : A → B and g : B → A and invertible 2-cells η : id A ∼ = = ⇒ gf and ǫ : f g ∼ = = ⇒ id B in A . An equivalence ( f, g, η, ǫ ) is an adjoint equivalence if, moreover, the2-cells η and ǫ satisfy the following triangle identities. AB AB fgf ǫη AB f = B AB A g gfǫ η
B A g = Remark . Although we often denote an equivalence only by f , we always mean thatwe have the whole data ( f, g, η, ǫ ).Given an equivalence ( f, g, η, ǫ ), we can always modify the invertible 2-cell ǫ into aninvertible 2-cell ǫ ′ such that the new data ( f, g, η, ǫ ′ ) is an adjoint equivalence. Lemma 2.1.4.
Every equivalence in a -category can be promoted to an adjoint equiva-lence.Proof. See, for example, [30, Lemma 2.1.11]. (cid:3)
The category 2Cat admits a closed symmetric monoidal structure introduced by Grayin [15].
Definition 2.1.5.
Let I and A be 2-categories. We denote by [ I , A ] , ps the pseudo-hom -category of 2-functors I → A , pseudo-natural transformations, and modifications. Fora definition of these notions, see [19, Definition 4.2.1 and 4.4.1].
DOUBLE ( ∞ , Then the
Gray tensor product ⊗ : 2Cat × → I , A , and B , we have a bijection2Cat( I ⊗ B , A ) ∼ = 2Cat( B , [ I , A ] , ps )natural in I , A , and B . Notation 2.1.6.
Let i : I → A and i ′ : I ′ → A ′ be 2-functors. We denote by i (cid:3) ⊗ i ′ their pushout product i (cid:3) ⊗ i ′ : A ⊗ I ′ G I⊗ I ′ I ⊗ A ′ → A ⊗ A ′ . We are now ready to introduce the other type of 2-dimensional categories of interestin this paper: the double categories . While 2-categories are defined as categories enrichedover Cat, the category of categories and functors, double categories are defined as internalcategories to Cat. They therefore admit two categorical directions, called horizontal andvertical.
Definition 2.1.7.
A double category A consists of the following data:(i) a collection of objects A, B, . . . ,(ii) horizontal morphisms f : A → B with a horizontal identity id A : A → A for eachobject A ,(iii) vertical morphisms u : A A ′ with a vertical identity e A : A A for each object A ,(iv) squares α : ( u ff ′ v ) of the form A BA ′ B ′ ff ′ • u • vα with a vertical identity e f : ( e A ff e B ) for each horizontal morphism f : A → B and a horizontal identity id u : ( u id A id A ′ u ) for each vertical morphism u : A A ′ ,(v) an associative and unital horizontal composition law for horizontal morphisms,and squares along their vertical boundaries,(vi) an associative and unital vertical composition law for vertical morphisms, andsquares along their horizontal boundaries,such that horizontal and vertical compositions of squares satisfy the interchange law. Definition 2.1.8.
A double functor F : A → B consists of assignments on objects, onhorizontal morphisms, on vertical morphisms, and on squares, which are compatible withdomains and codomains and preserve all compositions and identities strictly. Notation 2.1.9.
We denote by DblCat the category of double categories and doublefunctors.In particular, a 2-category can be seen as an internal category to Cat where the categoryof objects is discrete. This gives an embedding of 2Cat into DblCat which associates toa 2-category its corresponding horizontal double category.
Definition 2.1.10.
We define the horizontal embedding functor H : 2Cat → DblCat.It sends a 2-category A to the double category H A with the same objects as A , whosehorizontal morphisms are the morphisms of A , with only trivial vertical morphisms, andwhose squares LYNE MOSER
A BA ′ B ′ ff ′ • • α are given by the 2-cells α : f ⇒ f ′ of A . Compositions are induced by the ones in A .The functor H sends a 2-functor F : A → B to the double functor H F : H A → H B which acts as F does on the corresponding data.The functor H admits both adjoints. Its right adjoint extracts from a double categoryits underlying horizontal 2-category, which forgets about the vertical direction, while itsleft adjoint acts on a double category by squashing the vertical direction. Definition 2.1.11.
We define the functor H : DblCat → A to its underlying horizontal -category H A with the same objects as A , whosemorphisms are the horizontal morphisms of A , and whose 2-cells α : f ⇒ f ′ are given bythe squares in A of the form A BA ′ B ′ . ff ′ • • α Compositions are induced by the ones in A .The functor H sends a double functor F : A → B to the 2-functor H F : H A → H B which acts as F does on the corresponding data. Remark . Since H : 2Cat → DblCat is a functor between locally presentable cate-gories which preserves limits and colimits, it admits both a left and a right adjoint by theAdjoint Functor Theorem. Its right adjoint is given by the functor H : DblCat → L its left adjoint.2Cat DblCat . L H H ⊥⊥ The functor L : DblCat → A to a 2-category L A whoseobjects are equivalence classes of objects in A under the following relation: two objectsare identified if and only if they are related by a zig-zag of vertical morphisms. Themorphisms of L A are then generated by the horizontal morphisms of A , and the 2-cellsof L A are generated by the squares of A .Since 2-categories can also be embedded vertically into double categories, there areanalogous functors for the vertical direction. However, in this paper, a 2-category isalways seen as a horizontal double category, unless specified otherwise. Remark . Similarly, there is a functor V : 2Cat → DblCat sending a 2-category A to the double category V A with the same objects as A , only trivial horizontal morphisms,vertical morphisms the morphisms of A , and squares given by the 2-cells of A . Thisfunctor also admits both adjoints, and its right adjoint V : DblCat → underlying vertical category .As in the 2-categorical case, the category DblCat also admits a closed symmetricmonoidal structure given by a Gray tensor product introduced by Böhm in [4], with hom DOUBLE ( ∞ , double categories having horizontal and vertical morphisms the horizontal and vertical pseudo -natural transformations. Definition 2.1.14.
Let I and A be double categories. We define the pseudo-hom dou-ble category [ I , A ] ps to be the double category of double functors I → A , horizontalpseudo-natural transformations, vertical pseudo-natural transformations, and modifica-tions. See [4, §2.2] or [12, §3.8] for more details.By [4, §3], the Gray tensor product ⊗ G : DblCat × DblCat → DblCat endows thecategory DblCat with a closed symmetric monoidal structure with respect to these pseudo-homs. More explicitly, for all double categories I , A , and B , we have a bijectionDblCat( I ⊗ G B , A ) ∼ = DblCat( B , [ I , A ] ps )natural in I , A , and B .In this paper, we are interested in the underlying horizontal 2-categories of thesepseudo-hom double categories. This gives a tensored and cotensored 2Cat-enrichmenton DblCat with tensoring functor obtained by restricting the Gray tensor product fordouble categories defined above along the horizontal embedding H in one of the variables. Definition 2.1.15.
Let I and A be double categories. We define the pseudo-hom -category H [ I , A ] ps to be the 2-category of double functors I → A , horizontal pseudo-natural transformations, and modifications; see Definitions A.3.1 and A.3.2.Then the Gray tensor product ⊗ G : DblCat × DblCat → DblCat restricts to a tensoringfunctor ⊗ := DblCat × id × H −−−→ DblCat × DblCat ⊗ G −−→ DblCatwith respect to these pseudo-homs. More explicitly, for every pair of double categories I and A , and every 2-category B , we have a bijectionDblCat( I ⊗ B , A ) ∼ = 2Cat( B , H [ I , A ] ps )natural in I , A , and B . See [25, Proposition 7.6]. Notation 2.1.16.
Given a double functor I : I → A in DblCat and a 2-functor i : I → A in 2Cat, we denote by I (cid:3) ⊗ i their pushout product I (cid:3) ⊗ i : A ⊗ I G I ⊗I I ⊗ A → A ⊗ A . Before introducing another functor from DblCat to 2Cat which first appears in [25,Definition 2.12] and extracts from a double category a 2-category of vertical morphisms,squares, and 2-cells as below, we first settle the following notations.
Notation 2.1.17.
We denote by [ n ] the category given by the poset { < < . . . < n } ,for n ≥
0. It can be thought of as the free category on n composable morphisms. Inparticular, the category [0] is the terminal category, and the category [1] is the freecategory { → } on a morphism. Definition 2.1.18.
We define the functor V : DblCat → V := H [ V [1] , − ] ps : DblCat −→ . More explicitly, it sends a double category A to the 2-category V A whose objects are thevertical morphisms of A , and whose morphisms are the squares of A . A 2-cell in V A between parallel morphisms α : ( u ff ′ v ) and β : ( u gg ′ v ) consists of squares σ : ( e A fg e B )and σ ′ : ( e A ′ f ′ g ′ e B ′ ) satisfying the following pasting in A . A BA BA ′ B ′ • • fg • u • vg ′ σβ A BA ′ = B ′ A ′ B ′ • • ff ′ • u • vg ′ ασ ′ As we will see later in the paper, the horizontal embedding is sometimes not homotopi-cally good enough to embed 2-categories into double categories, and we therefore intro-duce for a 2-category A a double category H ≃ A whose underlying horizontal 2-categoryis still A , but where the vertical direction sees all the adjoint equivalences of A , insteadof just the identities. Definition 2.1.19.
We define the functor H ≃ : 2Cat → DblCat. It sends a 2-category A to the double category H ≃ A with the same objects as A , whose horizontal morphismsare the morphisms of A , whose vertical morphisms are the adjoint equivalences of A , andwhose squares A BA ′ B ′ ff ′ u = ( u, u ′ , η u , ǫ u ) ≃ v = ( v, v ′ , η v , ǫ v ) ≃ α are given by the 2-cells α : vf ⇒ f ′ u in A . Compositions are induced by the ones in A .The functor H ≃ sends a 2-functor F : A → B to the double functor H ≃ F : H ≃ A → H ≃ B which acts as F does on the corresponding data.The functor H ≃ is not a left adjoint, since it does not preserve colimits; see [26, Remark3.10]. However, it admits a left adjoint, which we describe below. Remark . The functor H ≃ preserves all limits, and therefore it admits a left adjoint,denoted by L ≃ , by the Adjoint Functor Theorem.2Cat DblCat L ≃ H ≃ ⊥ The functor L ≃ : DblCat → A to the 2-category L ≃ A withthe same objects as A , and whose morphisms are generated by a morphism for eachhorizontal morphism in A and by an adjoint equivalence for each vertical morphism in A .The 2-cells are generated by the squares of A . See [26, Proposition 3.4].2.2. Notions of equivalences in a double category.
As for 2-categories, a goodnotion of invertibility for a horizontal morphism in a double category is not given by thatof an isomorphism, but rather by a weaker notion. Indeed, a double category has anunderlying horizontal 2-category which contains all horizontal morphisms, and which wecan use to define the notion of horizontal equivalences . Let us fix a double category A . Definition 2.2.1.
A horizontal morphism f : A → B in A is a horizontal equivalence if f is an equivalence in the 2-category H A . In other words, f is a horizontal equivalenceif we have the data ( f, g, η, ǫ ) of horizontal morphisms f : A → B and g : B → A in A andvertically invertible squares η and ǫ in A as depicted below. DOUBLE ( ∞ , A AA B A f g • • η ∼ = B A BB B g f • • ǫ ∼ = Analogously, we also have a notion of horizontal adjoint equivalence . Definition 2.2.2.
A horizontal morphism f : A → B in A is a horizontal adjointequivalence if f is an adjoint equivalence in the 2-category H A . In other words, f is a horizontal adjoint equivalence if it is a horizontal equivalence ( f, g, η, ǫ ) in A and,moreover, the squares η and ǫ satisfy the following triangle identities. A AA B A BB f g ff • • • η ∼ = e f A B B f • • • e f ǫ ∼ = A BA B ff • • e f = A AA B ABB f ggg • • • e g η ∼ = B B A g • • • ǫ ∼ = e g B AB A gg • • e g =Since these horizontal equivalences are actually equivalences of a 2-category, we canapply Lemma 2.1.4 to obtain the following result. Lemma 2.2.3.
Every horizontal equivalence in a double category A can be promoted toa horizontal adjoint equivalence. In Definition 2.1.18, we described a 2-category V A whose objects and morphisms arevertical morphisms and squares of A . By looking at the equivalences in this 2-category,we get a notion of weak invertibility for squares. Definition 2.2.4.
A square α : ( u ff ′ v ) in A is weakly horizontally invertible if α isan equivalence in the 2-category V A . In other words, α is weakly horizontally invertibleif we have the data of a square β : ( u gg ′ v ) in A together with vertically invertible squares η , η ′ , ǫ , and ǫ ′ satisfying the following pasting equalities. A AA B A f g • • η ∼ = A ′ B ′ A ′ f ′ g ′ • u • v • uα β A AA ′ A ′ = • u • u id u A ′ B ′ A ′ f ′ g ′ • • η ′ ∼ = B A BB B g f • • ǫ ∼ = B ′ B ′ • v • v id v B A BB ′ = A ′ B ′ g fg ′ f ′ • v • u • vβ α B ′ B ′ • • ǫ ′ ∼ = Note that the data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ) are horizontal equivalences in A , and wecall β a weak inverse of α with respect to the horizontal equivalence data ( f, g, η, ǫ ) and( f ′ , g ′ , η ′ , ǫ ′ ).By applying Lemma 2.1.4 to the equivalences of the 2-category V A , we obtain thefollowing result. Lemma 2.2.5.
Every weakly horizontally invertible square in a double category A can bepromoted to a weakly horizontally invertible square whose horizontal equivalence data arehorizontal adjoint equivalences.Proof. A square α : ( u ff ′ v ) is an adjoint equivalence in the 2-category V A if and only if itis weakly horizontally invertible in A and the horizontal equivalence data ( f, g, η, ǫ ) and( f ′ , g ′ , η ′ , ǫ ′ ) are horizontal adjoint equivalences. Then the result follows from Lemma 2.1.4. (cid:3) Remark . If the horizontal equivalence data of a weakly horizontally invertible squareare horizontal adjoint equivalences, we call it horizontal adjoint equivalence data .With this terminology settled, we are now ready to define what will be the fibrantdouble categories in the considered model structure on DblCat.
Definition 2.2.7.
A double category A is weakly horizontally invariant if for everypair of horizontal equivalences f : A ≃ −→ B and f ′ : A ′ ≃ −→ B ′ and every vertical mor-phism v : B B ′ in A , there is a vertical morphism u : A A ′ together with a weaklyhorizontally invertible square in A as depicted below. A BA ′ B ′ f ≃ f ′ ≃ • u • v ≃ Model structures on and
DblCatThe category 2Cat admits a model structure in which the weak equivalences are thebiequivalences, constructed by Lack in [22, 23]. In [25], we constructed a model structureon DblCat right-induced from two copies of Lack’s model structure along the functor( H , V ) : DblCat → × H then creates Lack’smodel structure from this model structure on DblCat. However, as mentioned in theintroduction, this model structure is not well-behaved with respect to composition ofvertical morphisms in a double category. Therefore, in [26], we constructed anothermodel structure obtained by adding the inclusion [0] ⊔ [0] → V [1] to the set of generatingcofibrations and keeping the same weak equivalences. With this new model structure, thefunctor H is not right Quillen anymore, but the functor H ≃ fulfills this role. While in thefirst model structure, all double categories were fibrant, the fibrant double categories ofthis new model structure are precisely the weakly horizontally invariant double categories. DOUBLE ( ∞ , In Section 3.1, we recall the main features of Lack’s model structure and, in Section 3.2,those of the model structure of [26]. In particular, we describe the generating (trivial)cofibrations and characterize the cofibrations of these model structures since these de-scriptions will be used to prove that the left adjoint to the double ( ∞ , Lack’s model structure for -categories. Let us first recall the definition of abiequivalence between 2-categories.
Definition 3.1.1.
A 2-functor F : A → B is a biequivalence if it is(i) bi-essentially surjective on objects, i.e., for every object B ∈ B , there is an object A ∈ A together with an equivalence B ≃ −→ F A in B ,(ii) essentially full on morphisms, i.e., for every morphism g : F A → F C in B , thereis a morphism f : A → C in A together with an invertible 2-cell g ∼ = F f in B ,(iii) fully faithful on 2-cells, i.e., for every 2-cell β : F f ⇒ F f ′ in B , there is a unique2-cell α : f ⇒ f ′ in A such that β = F α .We introduce two sets of 2-functors which correspond to sets of generating cofibrationsand generating trivial cofibrations for Lack’s model structure on 2Cat.
Notation 3.1.2.
We denote by I the set containing the following 2-functors:(i) the unique map i : ∅ → [0],(ii) the inclusion i : [0] ⊔ [0] → [1],(iii) the inclusion i : δC → C , where C is the free 2-category on a 2-cell, and δC is itssub-2-category containing the boundary of the 2-cell, i.e., it is free on two parallelmorphisms,(iv) the 2-functor i : C → C sending the two non-trivial 2-cells of C to the non-trivial 2-cell of C , where C is the free 2-category on two parallel 2-cells.We denote by J the set containing the following 2-functors:(i) the inclusion j : [0] → E adj , where the 2-category E adj is the “free-living adjointequivalence”,(ii) the inclusion j : [1] → C inv , where the 2-category C inv is the “free-living invertible2-cell”.We state the main features of Lack’s model structure. Theorem 3.1.3.
There is a cofibrantly generated model structure on , in which theweak equivalences are the biequivalences, and sets of generating cofibrations and generatingtrivial cofibrations are given by I and J , respectively. In particular, every -category isfibrant.Moreover, the model structure is monoidal with respect to the Gray tensor prod-uct ⊗ .Proof. The existence of the model structure is given in [23, Theorem 4] (which is a slightlymodified version of [22, Theorem 3.3]). The sets of generating (trivial) cofibrations aredescribed at the beginning of [22, §3], and the monoidality is the content of [22, Theo-rem 7.5]. (cid:3)
Remark . In particular, the model structure on 2Cat being monoidal with respectto ⊗ implies that the pushout-product i (cid:3) ⊗ i ′ (see Notation 2.1.6) of two cofibrations i and i ′ in 2Cat is a cofibration in 2Cat, which is trivial if i or i ′ is a biequivalence.The following results provide characterizations of cofibrations and of cofibrant objectsin 2Cat. We denote by U : 2Cat → Cat the functor sending a 2-category to its underlyingcategory.
Proposition 3.1.5. A -functor F : A → B is a cofibration in if and only if(i) it is injective on objects and faithful on morphisms, and(ii) the underlying category U B is a retract of a category obtained from the image of U A under U F by freely adjoining objects and then morphisms between objects.Proof.
This follows from [22, Lemma 4.1 and Corollary 4.12]. (cid:3)
Corollary 3.1.6. A -category A is cofibrant in if and only if its underlying category U A is free.Proof. This is given by [22, Theorem 4.8]. (cid:3)
Model structure for weakly horizontally invariant double categories.
Wenow present the weak equivalences, called double biequivalences , of the model structure onDblCat of [26], which were first introduced in [25, Definition 3.7]. These correspond to thedouble functors which are sent by both H and V to biequivalences (see [25, Proposition3.12]). Definition 3.2.1.
A double functor F : A → B is a double biequivalence if it is(i) horizontally bi-essentially surjective on objects, i.e., for every object B ∈ B , thereis an object A ∈ A together with a horizontal equivalence B ≃ −→ F A in B ,(ii) essentially full on horizontal morphisms, i.e., for every horizontal morphism g : F A → F C in B , there is a horizontal morphism f : A → C in A togetherwith a vertically invertible square in B F A F CF A F C , • • gF f ∼ = (iii) bi-essentially surjective on vertical morphisms, i.e., for every vertical morphism v : B B ′ in B , there is a vertical morphism u : A A ′ in A together with aweakly horizontally invertible square B F AB ′ F A ′ , • v • F u ≃≃≃ (iv) fully faithful on squares, i.e., for every square β in B of the form F A F CF A ′ F C ′ , • F u • F vF fF f ′ β there is a unique square α : ( u ff ′ v ) in A such that β = F α .We introduce two sets of double functors which correspond to sets of generating cofi-brations and generating trivial cofibrations in the model structure on DblCat of [26].
Notation 3.2.2.
We denote by I the set containing the following double functors:(i) the unique map I : ∅ → [0],(ii) the inclusion I : [0] ⊔ [0] → H [1],(iii) the inclusion I : [0] ⊔ [0] → V [1], DOUBLE ( ∞ , (iv) the inclusion I : δ S → S , where S is the free double category on a square, and δ S is its sub-double category containing the boundary of the square, i.e. it is free ontwo horizontal morphisms and two vertical morphisms sharing some boundaries,(v) the 2-functor I : S → S sending the two non-trivial squares of S to the non-trivial square of S , where S is the free double category on two parallel squares.We denote by J the set containing the following double functors:(i) the inclusion J : [0] → H E adj , where the 2-category E adj is the “free-living adjointequivalence”,(ii) the inclusion J : H [1] → H C inv , where the 2-category C inv is the “free-livinginvertible 2-cell”,(iii) the inclusion J : W − → W , where the double category W is the “free-living weaklyhorizontally invertible square with horizontal adjoint equivalence data”, and W − is its sub-double category as depicted below.0 W = 0 ′ ′ ≃ •• ≃≃ W − = 0 ′ ′ ≃ • ≃ Theorem 3.2.3.
There is a cofibrantly generated model structure on
DblCat in whichthe weak equivalences are the double biequivalences, and sets of generating cofibrationsand generating trivial cofibrations are given by I and J , respectively. In particular, thefibrant objects are precisely the weakly horizontally invariant double categories.Moreover, the model structure on DblCat is monoidal with respect to the Gray ten-sor product ⊗ G , and it is therefore enriched over the model structure for -categories ofTheorem 3.1.3 with respect to the -enrichment H [ − , − ] ps .Proof. The existence of the model structure is given in [26, Theorem 2.19]. The fibrantobjects are characterized in [26, Theorem 2.20], and the sets of generating (trivial) cofi-brations are described in [26, Definitions 2.10 and 2.12]. The monoidality and enrichmentare the content of [26, Theorem 4.4 and Remark 4.5]. (cid:3)
Remark . In particular, the model structure on DblCat being enriched over 2Catwith respect to H [ − , − ] ps implies that the pushout-product I (cid:3) ⊗ i (see Notation 2.1.16)of a cofibration I in DblCat and a cofibration i in 2Cat is a cofibration in DblCat, whichis trivial if I is a double biequivalence or i is a biequivalence.The following results state characterizations of cofibrations and cofibrant objects inDblCat. Proposition 3.2.5.
A double functor F : A → B is a cofibration in DblCat if and only if(i) it is injective on objects and faithful on horizontal and vertical morphisms,(ii) the horizontal underlying category U H B is a retract of a category obtained fromthe image of U H A under U H F by freely adjoining objects and then morphismsbetween objects, and(iii) the vertical underlying category U V B is a retract of a category obtained fromthe image of U V A under U V F by freely adjoining objects and then morphismsbetween objects.Proof. This is [26, Corollary 2.16]. (cid:3)
Corollary 3.2.6.
A double category A is cofibrant in DblCat if and only if its underlyinghorizontal and vertical categories U H A and U V A are free.Proof. This is [26, Corollary 2.17]. (cid:3)
The horizontal embedding functor H is not right Quillen with respect to Lack’s modelstructure on 2Cat and the model structure on DblCat of Theorem 3.2.3. However, itsbetter suited homotopical version H ≃ is such a right Quillen functor and it gives a homo-topically full embedding of 2-categories into double categories. In particular, the doublecategory H ≃ A associated to a 2-category A provides a fibrant replacement of the hori-zontal double category H A . Remark . The functor H : 2Cat → DblCat is not right Quillen. Indeed, its left adjoint L does not preserve cofibrations: it sends the cofibration I : [0] ⊔ [0] → V [1] to the 2-functor L ( I ) : [0] ⊔ [0] → [0] and this is not a cofibration in 2Cat since it is not injectiveon objects. See also [26, Remark 3.12].In [25], we first built another model structure on DblCat in which the weak equivalencesare also the double biequivalences, but with all double categories fibrant. This modelstructure is Quillen equivalent to the model structure of Theorem 3.2.3 and is such thatthe adjunction L ⊣ H is a Quillen pair. In particular, the double functor I : [0] ⊔ [0] → V [1]is not a cofibration in the model structure of [25]. Theorem 3.2.8.
The adjunction L ≃ H ≃ ⊥ is a Quillen pair between Lack’s model structure on and the model structure on DblCat for weakly horizontally invariant double categories. Moreover, the derived counitof this adjunction is level-wise a biequivalence in .Proof.
This is [26, Theorem 3.6]. (cid:3)
Proposition 3.2.9.
The inclusion H A → H ≃ A is a double biequivalence and exhibits H ≃ A as a fibrant replacement of H A in the model structure on DblCat for weakly hori-zontally invariant double categories.Proof.
This is [26, Proposition 3.13]. (cid:3) Model structures for ( ∞ , -categories and double ( ∞ , -categories The model for ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , n = 2 and i = 1. DOUBLE ( ∞ , Model structures for double ( ∞ , -categories. Let us denote by sSet the cat-egory of simplicial sets and by ∆ the simplex category. We endow the category sSetwith the Quillen model structure, constructed in [28]. Then we consider the Reedy orinjective model structure on sSet ∆ op × ∆ op , which coincide by results of Bergner and Rezk;see [3, Proposition 3.15 and Corollary 4.5]. This allows us to describe both the (trivial)cofibrations and the fibrant objects of this model structure.The objects of study here are bisimplicial spaces, i.e., trisimplicial sets, and we introducenotations for the representables in each of the three copies of ∆ op . Notation 4.1.1.
We denote by R [ m ], F [ k ], and ∆[ n ] the representable bisimplicial spacesin the first, second or third variable respectively. The first direction is called the horizontal direction, the second the vertical direction, and the last one the space direction. We alsodenote by ι Rm : δR [ m ] → R [ m ], ι Fk : δF [ k ] → F [ k ], and ι ∆ n : δ ∆[ n ] → ∆[ n ] their boundaryinclusions, and by ℓ ∆ n,t : Λ t [ n ] → ∆[ n ] the ( n, t )-horn inclusion in ∆[ n ]. Notation 4.1.2.
Given two maps f : X → Y and f ′ : X ′ → Y ′ in sSet ∆ op × ∆ op , we denoteby f (cid:3) × f ′ their pushout product f (cid:3) × f ′ : Y × X ′ G X × X ′ X × Y ′ → Y × Y ′ . Remark . A set of generating cofibrations for the Reedy/injective model structureon sSet ∆ op × ∆ op is given by the collection of maps(( ι Rm : δR [ m ] → R [ m ]) (cid:3) × ( ι Fk : δF [ k ] → F [ k ])) (cid:3) × ( ι ∆ n : δ ∆[ n ] → ∆[ n ])for m, k, n ≥
0, and a set of generating trivial cofibrations by the collection of maps(( ι Rm : δR [ m ] → R [ m ]) (cid:3) × ( ι Fk : δF [ k ] → F [ k ])) (cid:3) × ( ℓ ∆ n,t : Λ t [ n ] → ∆[ n ])for m, k ≥ n ≥
1, and 0 ≤ t ≤ n . In particular, the cofibrations are precisely themonomorphisms. Definition 4.1.4.
A bisimplicial space X : ∆ op × ∆ op → sSet is Reedy fibrant if themap X m,k ∼ = Map( R [ m ] × F [ k ] , X ) → Map( δR [ m ] × F [ k ] G δR [ m ] × δF [ k ] R [ m ] × δF [ k ] , X ) , induced by ι Rm (cid:3) × ι Fk , is a Kan fibration in sSet, where Map( − , − ) denotes the mappingsimplicial set in sSet ∆ op × ∆ op .We also introduce the following notation. Notation 4.1.5.
We denote by N R : Cat → Set (∆ op ) × the discrete nerve constant in thevertical and space directions. It is given by ( N R C ) m,k,n = Cat([ m ] , C ) at a category C . Example 4.1.6.
Let I = { x ∼ = y } ∈ Cat be the “free-living isomorphism”. Its discretenerve is given by ( N R I ) m,k,n = Cat([ m ] , I ). In particular, a functor [ m ] → I can bedescribed as a word of m letters in { x, y } . For example, when m = 0, we have that( N R I ) ,k,n = { x, y } ; and, when m = 1, ( N R I ) ,k,n = { xx, xy, yx, yy } where xx and yy are degenerate and represent the identities at x and y , and xy and yx represent the twoinverse morphisms between x and y . In particular, for X ∈ sSet ∆ op × ∆ op and k ≥ X − ,k ∈ sSet ∆ op is a Segal space, thenMap( N R I × F [ k ] , X ) ∼ = ( X ,k ) heq is the space of homotopy equivalences in X ,k , as described in [29, §5.7].We now present the ∞ -version of double categories of use in this paper. Definition 4.1.7. A horizontally complete double ( ∞ , -category is a bisimplicialspace X : ∆ op × ∆ op → sSet such that(i) X is Reedy/injective fibrant,(ii) X m, − : ∆ op → sSet is a Segal space, for every m ≥
0, i.e., the Segal mapsMap( F [ k ] , X m, − ) ∼ = X m,k ≃ −→ X m, × X m, . . . × X m, X m, ∼ = Map( F [1] ⊔ F [0] . . . ⊔ F [0] F [1] , X m, − )induced by the maps { i, i + 1 } : [1] → [ k ] of ∆ for 0 ≤ i ≤ k − m, k ≥ X − ,k : ∆ op → sSet is a complete Segal space, for every k ≥
0, i.e., the Segal mapsMap( R [ m ] , X − ,k ) ∼ = X m,k ≃ −→ X ,k × X ,k . . . × X ,k X ,k ∼ = Map( R [1] ⊔ R [0] . . . ⊔ R [0] R [1] , X − ,k )induced by the maps { j, j + 1 } : [1] → [ m ] of ∆ for 0 ≤ j ≤ m − m, k ≥
0, and the mapMap( N R I, X − ,k ) ∼ = ( X ,k ) heq ≃ −→ X ,k ∼ = Map( R [0] , X − ,k )induced by the inclusion x : [0] → I = { x ∼ = y } into the “free-living isomorphism”,is a weak equivalence in sSet, for all k ≥ F [ k ], R [ m ], and N R I arebisimplicial spaces (rather than simplicial spaces) which are constant in the horizontal orvertical direction. For a careful definition of a complete Segal space, we refer the readerto [29].We obtain a model structure on sSet ∆ op × ∆ op for horizontally complete double ( ∞ , Theorem 4.1.8.
There is a model structure on sSet ∆ op × ∆ op , denoted by DblCat h ∞ , ob-tained as a left Bousfield localization of the Reedy/injective model structure in which thefibrant objects are precisely the horizontally complete double ( ∞ , -categories.Proof. We localize the Reedy/injective model structure with respect to the cofibrations • id R [ m ] × g Fk : R [ m ] × G [ k ] → R [ m ] × F [ k ], for all m, k ≥
0, where g Fk is the inclusion g Fk : G [ k ] = F [1] ⊔ F [0] . . . ⊔ F [0] F [1] → F [ k ]induced by the maps { i, i + 1 } : [1] → [ k ] in ∆, for 0 ≤ i ≤ k − • q Rm × id F [ k ] : Q [ m ] × F [ k ] → R [ m ] × F [ k ], for all m, k ≥
0, where q Rm is the inclusion q Rm : Q [ m ] = R [1] ⊔ R [0] . . . ⊔ R [0] R [1] → R [ m ]induced by the maps { j, j + 1 } : [1] → [ m ] in ∆, for 0 ≤ j ≤ m − • e R × id F [ k ] : F [ k ] ∼ = R [0] × F [ k ] → N R I × F [ k ], for all k ≥
0, where e R is theinclusion e R : R [0] → N R I induced by the functor x : [0] → I = { x ∼ = y } , wherethe category I is the “free-living isomorphism” and N R I is its discrete nerveconstant in the vertical and space directions.The existence of this model structure is given by [17, Theorem 4.1.1]. Moreover, aReedy/injective fibrant bisimplicial set is local with respect to this collection of mapsif and only if it is a horizontally complete double ( ∞ , (cid:3) Remark . We could also have defined a notion of double ( ∞ , v ∞ the model structure for these vertically complete double ( ∞ , DOUBLE ( ∞ , obtained by localizing the Reedy/injective model structure on sSet ∆ op × ∆ op analogouslyto above. Then the functor t : ∆ op × ∆ op → ∆ op × ∆ op , ([ m ] , [ k ]) ([ k ] , [ m ])swapping the two copies of ∆ op induces a functor t ∗ : sSet ∆ op × ∆ op → sSet ∆ op × ∆ op , and weget a Quillen equivalence DblCat h ∞ DblCat v ∞ t ∗ t ∗ ⊥ between the two model structures for double ( ∞ , t ∗ can bethought of as a transpose functor .4.2. Model structure for -fold complete Segal spaces. Before defining 2-fold com-plete Segal spaces, we first introduce the following notation for the discrete nerve in thevertical direction.
Notation 4.2.1.
We denote by N F : Cat → Set (∆ op ) × the discrete nerve constant in thehorizontal and space directions. It is given by ( N F C ) m,k,n = Cat([ k ] , C ) at a category C . Definition 4.2.2.
A 2 -fold complete Segal space (or ( ∞ , -category ) is a bisimpli-cial space X : ∆ op × ∆ op → sSet such that(i) X is Reedy/injective fibrant,(ii) X m, − : ∆ op → sSet is a complete Segal space, for every m ≥
0, i.e., we have theSegal condition as in Definition 4.1.7 (ii), and the mapMap( N F I, X m, − ) ∼ = ( X m, ) heq ≃ −→ X m, ∼ = Map( F [0] , X m, − )induced by the inclusion x : [0] → I = { x ∼ = y } into the “free-living isomorphism”is a weak equivalence in sSet, for all m ≥ X − ,k : ∆ op → sSet is a complete Segal space, for every k ≥ X , − : ∆ op → sSet is essentially constant, for all k ≥
0, i.e., the mapMap( F [ k ] , X , − ) ∼ = X ,k ≃ −→ X , ∼ = Map( F [0] , X , − )induced by the map 0 : [0] → [ k ] in ∆ is a weak equivalence in sSet, for all k ≥ F [ k ] and N F I are bisimpli-cial spaces (rather than simplicial spaces) which are constant in the horizontal direction.We obtain a model structure for 2-fold complete Segal spaces as a left Bousfield local-ization of the model structure for horizontally complete double ( ∞ , Theorem 4.2.3.
There is a model structure on sSet ∆ op × ∆ op , denoted by , obtainedas a left Bousfield localization of the model structure DblCat h ∞ for horizontally completedouble categories in which the fibrant objects are precisely the -fold complete Segal spaces,i.e., the ( ∞ , -categories.Proof. We localize the model structure DblCat h ∞ with respect to the cofibrations • id R [ m ] × e F : R [ m ] ∼ = R [ m ] × F [0] → R [ m ] × N F I , for all m ≥
0, where e F isthe inclusion e F : F [0] → N F I induced by the functor x : [0] → I = { x ∼ = y } ,where the category I is the “free-living isomorphism” and N F I is its discretenerve constant in the horizontal and space directions, • c k : F [0] → F [ k ], for all k ≥
0, induced by the map 0 : [0] → [ k ] in ∆.The existence of this model structure is given by [17, Theorem 4.1.1]. Moreover, a hori-zontally complete double ( ∞ , (cid:3) The following result is obtained as a direct consequence of the fact that 2CSS is alocalization of DblCat h ∞ . Corollary 4.2.4.
The identity adjunction on sSet ∆ op × ∆ op is a Quillen pair h ∞ . idid ⊥ Moreover, the (derived) counit is level-wise a weak equivalence. In particular, this givesa homotopically full embedding of into
DblCat h ∞ . Nerve of double categories
This section gives a construction of a nerve functor from double categories to bisimpli-cial spaces. In Section 5.1, we define the nerve and its left adjoint, and in Section 5.2, weshow that they form a Quillen pair between the model structure on DblCat for weaklyhorizontally invariant double categories and the model structure DblCat h ∞ for horizontallycomplete double ( ∞ , ∞ , Definition of the nerve.
To define the nerve we make use of truncated versions ofthe n -orientals O ( n ), introduced by Street in [32]. More precisely: Definition 5.1.1.
For n ≥
0, let O ( n ) denote the 2-truncated n -oriental. It is the2-category described by the following data:(i) its set of objects is given by { , . . . , n } ,(ii) for 0 ≤ x, x ′ ≤ n , its hom-category O ( n )( x, x ′ ) is given by the poset O ( n )( x, x ′ ) = ( { I ⊆ [ x, x ′ ] | x, x ′ ∈ I } if x ′ ≤ x ∅ if x > x ′ where [ x, x ′ ] = { y ∈ { , . . . , n } | x ≤ y ≤ x ′ } .We define the 2-category O ∼ ( n ) obtained from O ( n ) by formally inverting every 2-cell, and we define the 2-category ^ O ( n ) obtained from O ∼ ( n ) by formally making everymorphism into an adjoint equivalence.In order to have a better sense of what these 2-categories look like, we describe thelower cases. Example 5.1.2.
For n = 0, the 2-categories O (0), O ∼ (0), and ^ O (0) are all given bythe terminal (2-)category [0].For n = 1, the 2-categories O (1) and O ∼ (1) are both given by the free (2-)category [1]on a morphism, while the 2-category ^ O (1) is the “free-living adjoint equivalence” E adj .For n = 2, the 2-categories O (2), O ∼ (2), and ^ O (2) are generated, respectively, by thefollowing data,0 1 2 0 1 2 ∼ = ≃ ≃≃∼ = DOUBLE ( ∞ , where ≃ −→ denotes the data of an adjoint equivalence.For n = 3, the 2-category O (3) is generated by the following data01 23 0= 1 23and the 2-category O ∼ (3) is given by the corresponding 2-category with all 2-cells in-vertible, while the 2-category ^ O (3) is given by the corresponding 2-category with allmorphisms being adjoint equivalences and all 2-cells being invertible.The nerve functor is then defined as the right adjoint of left Kan extension of the fol-lowing tricosimplicial double category along the Yoneda embedding. Recall the tensoringfunctor ⊗ : DblCat × → DblCat introduced in Definition 2.1.15.
Definition 5.1.3.
We define the tricosimplicial double category X : ∆ × ∆ × ∆ → DblCat([ m ] , [ k ] , [ n ]) X m,k,n := ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) , where the cosimplicial maps are induced by the ones of the cosimplicial objects∆ → DblCat ∆ → k ] V O ∼ ( k ) , [ m ] O ∼ ( m ) , and [ n ] ^ O ( n ) . Proposition 5.1.4.
The tricosimplicial double category X induces an adjunction ∆ × ∆ × ∆Set (∆ op ) × DblCat
XC N ⊤ where C is the left Kan extension of X along the Yoneda embedding, and we have that ( NA ) m,k,n ∼ = DblCat(( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) , A ) , for all A ∈ DblCat and all m, k, n ≥ ,Proof. This is a direct application of [8, Theorem 1.1.10]. (cid:3)
Remark . As expected from a nerve construction, the 0-simplices of the simplicialset ( NA ) , are given by the objects of A , the ones of ( NA ) , by the horizontal morphismsof A , the ones of ( NA ) , by the vertical morphisms of A , and the ones of ( NA ) , by thesquares of A . These can therefore be thought of as the spaces of objects, horizontalmorphisms, vertical morphisms, and squares . For a description of the 1- and 2-simplicesof these simplicial sets, we refer the reader to Appendix B.1. For m ≥ k ≥
2, thesimplicial sets ( NA ) m,k witness “compositions” in A of the data above. Remark . Since C is the left Kan extension of X along the Yoneda embedding, it isgiven on representables by C ( F [ k ] × R [ m ] × ∆[ n ]) = X m,k,n . In particular, we have that C ( F [ k ]) = V O ∼ ( k ) , C ( R [ m ]) = H O ∼ ( m ) and C (∆[ n ]) = H ^ O ( n ) . We also introduce a functor C , which takes values in 2-categories and coincides with C in the horizontal and space directions. Here ⊗ : 2Cat × → Notation 5.1.7.
We denote by X : ∆ × ∆ × ∆ → X m,k,n := O ∼ ( m ) ⊗ ^ O ( n ), and by C : Set (∆ op ) × → X along the Yoneda embedding. Remark . Note that X m, ,n = HX m, ,n . Therefore, if X ∈ Set (∆ op ) × is constant in thevertical direction, then C X = HC X . In particular, we have that C ( R [ m ]) = HC ( R [ m ])and C (∆[ n ]) = HC (∆[ n ]), where C ( R [ m ]) = O ∼ ( m ) and C (∆[ n ]) = ^ O ( n ).5.2. The nerve N is right Quillen. We now want to prove that the adjunction C ⊣ N is a Quillen pair between DblCat and DblCat h ∞ . To prove this result, we will make useof the following theorem. Theorem 5.2.1.
Let M and N be model categories and suppose that N M FU ⊥ is a Quillen pair. Let C be a set of cofibrations in M such that the left Bousfield localization L C M of M with respect to C exists. If F sends every morphism in C to a weak equivalencein N , then the adjunction N L C M FU ⊥ is also a Quillen pair.Proof. This is a direct consequence of [17, Theorem 3.3.20], since the localization of N with respect to maps in F C is N itself as maps in F C are already weak equivalencesin N . (cid:3) To apply this theorem, we first show that C ⊣ N is a Quillen pair between the modelstructure on DblCat and the Reedy/injective model structure on sSet ∆ op × ∆ op . Proposition 5.2.2.
The adjunction
DblCat sSet ∆ op × ∆ op CN ⊥ is a Quillen pair between the model structure on DblCat of Theorem 3.2.3 and theReedy/injective model structure on sSet ∆ op × ∆ op .Proof. It is enough to show that C sends generating cofibrations and generating trivialcofibrations in sSet ∆ op × ∆ op to cofibrations and trivial cofibrations in DblCat, respectively.Recall from Remark 4.1.3 that generating cofibrations and generating trivial cofibrationsare given by pushout-product of maps ( ι Fk (cid:3) × ι Rm ) (cid:3) × ι ∆ n and ( ι Fk (cid:3) × ι Rm ) (cid:3) × ℓ ∆ n,t , respec-tively. Note that the map ι Fk is constant in the horizontal and space directions, the map ι Rm is constant in the vertical and space directions, and the maps ι ∆ n and ℓ ∆ n,t are constantin the horizontal and vertical directions. Therefore, since the functor C preserves colimitsand by Remark 5.1.8, we have that C (( ι Fk (cid:3) × ι Rm ) (cid:3) × ι ∆ n ) ∼ = ( C ι Fk (cid:3) ⊗ G C ι Rm ) (cid:3) ⊗ G C ι ∆ n ∼ = ( C ι Fk (cid:3) ⊗ C ι Rm ) (cid:3) ⊗ C ι ∆ n , and similarly for ℓ ∆ n,t in place of ι ∆ n . Since the model structure DblCat is enriched over2Cat, pushout-products of cofibrations with respect to ⊗ are cofibrations, which aretrivial if one of the maps involved is a weak equivalence, by Remark 3.2.4. Therefore, it DOUBLE ( ∞ , is enough to show that C ι Fk is a cofibration in DblCat, for all k ≥
0, that C ι Rm and C ι ∆ n are cofibrations in 2Cat, for all m, n ≥
0, and that C ℓ ∆ n,t is a trivial cofibration in 2Cat,for all n ≥
1, 0 ≤ t ≤ n . These statements are verified in Lemmas 5.2.5 to 5.2.7. (cid:3) To prove that the boundary and horn inclusions mentioned above are sent to cofibra-tions in 2Cat and DblCat, we introduce the following definitions of the boundary of O ( n )and the ( n, t )-horn of O ( n ), which will be used to describe the images under C of theboundary and horn inclusions. Definition 5.2.3.
For n ≥
0, we define the boundary -category δO ( n ) as the co-equalizer in 2Cat G ≤ i 1) and d s : O ( n − → O ( n ) denote the face maps for r = i, j and s = i, j − 1. In particular, there is an inclusion δO ( n ) → O ( n ) induced bythe face maps d i : O ( n − → O ( n ) for 0 ≤ i ≤ n . More explicitly, these 2-categoriesare given by the following: • for n = 0, δO (0) = ∅ with δO (0) = ∅ → O (0) = [0] given by the unique map, • for n = 1, δO (1) = [0] ⊔ [0] with δO (1) = [0] ⊔ [0] → O (1) = [1] given byincluding the two copies of [0] as the two endpoints of the morphism in [1], • for n = 2, δO (2) is the sub-2-category of O (2) where the 2-cell is missing andthe inclusion δO (2) → O (2) is given by the following.0 1 2 0 −→ • for n = 3, δO (3) is the sub-2-category of O (3) where only the equality betweenthe two pasting diagrams in O (3) – as depicted in Example 5.1.2 – is missing, • for n ≥ δO ( n ) = O ( n ).Similarly, we define the boundary 2-categories δO ∼ ( n ) and δ ^ O ( n ). Definition 5.2.4. For n ≥ ≤ t ≤ n , we define the ( n, t ) -horn -category Λ t O ( n ) as the coequalizer in 2Cat G ≤ i 1) and d s : O ( n − → O ( n ) denote the face maps for r = i, j and s = i, j − 1. In particular, there is an inclusion Λ t O ( n ) → O ( n ) inducedby the face maps d i : O ( n − → O ( n ) for 0 ≤ i ≤ n , i = t . More explicitly, these2-categories are given by the following: • for n = 1, Λ t O (1) = [0] with Λ t O (1) = [0] → O (1) = [1] given by the inclusionof [0] at the source of the morphism in [1] if t = 1 and at the target if t = 0, • for n = 2, Λ O (2), Λ O (2), and Λ O (2) are generated, respectively, by thefollowing data0 1 2 0 1 2 0 1 2with the obvious inclusions into O (2), • for n = 3 and 0 ≤ t ≤ 3, Λ t O (3) is the sub-2-category where the equalitybetween the two pasting diagrams in O (3) and the 2-cell opposite to the object t are missing. For example, when t = 0, the inclusion Λ O (3) → O (3) is given bythe following.01 23 01 23 0 −→ • for n ≥ ≤ t ≤ n , Λ t O ( n ) = O ( n ).Similarly, we define the ( n, t )-horn 2-categories Λ t O ∼ ( n ) and Λ t ^ O ( n ).We are now ready to prove the promised lemmas which complete the proof of Propo-sition 5.2.2. Lemma 5.2.5. For all k ≥ , the double functor C ( ι Fk ) : C ( δF [ k ]) → C ( F [ k ]) is a cofi-bration in DblCat .Proof. We have that δF [ k ] is defined as the coequalizer in sSet ∆ op × ∆ op G ≤ i 0. Therefore, the double functors C ( ι Fk ) are given by • for k = 0, the generating cofibration I : ∅ → [0], • for k = 1, the generating cofibration I : [0] ⊔ [0] → V [1], • for k = 2, the inclusion 01202 ••• −→ ••• ∼ = which is a cofibration by Proposition 3.2.5 since it is the identity on underlyinghorizontal and vertical categories, • for k = 3, the inclusion V δO ∼ (3) → V O ∼ (3), which is a cofibration by Proposi-tion 3.2.5 since it is the identity on underlying horizontal and vertical categories, • for k ≥ 4, the identity.This shows that the double functor C ( ι Fk ) is a cofibration in DblCat, for all k ≥ (cid:3) Lemma 5.2.6. For all m, n ≥ , the -functors C ( ι Rm ) : C ( δR [ m ]) → C ( R [ m ]) and C ( ι ∆ n ) : C ( δ ∆[ n ]) → C (∆[ n ]) are cofibrations in .Proof. We first prove the statement for C ( ι Rm ). As in the proof of Lemma 5.2.5 and byRemark 5.1.8, we find that C ( δR [ m ]) = δO ∼ ( m ) and C ( R [ m ]) = O ∼ ( m ) , for all m ≥ 0. Therefore, the 2-functors C ( ι Rm ) are given by • for m = 0, the generating cofibration i : ∅ → [0], DOUBLE ( ∞ , • for m = 1, the generating cofibration i : [0] ⊔ [0] → [1], • for m = 2, the inclusion δO ∼ (2) → O ∼ (2), which is a cofibration by Proposi-tion 3.1.5 since it is the identity on underlying categories, • for m = 3, the inclusion δO ∼ (3) → O ∼ (3), which is a cofibration by Proposi-tion 3.1.5 since it is the identity on underlying categories, • for m ≥ 4, the identity.Therefore, the 2-functor C ( ι Rm ) is a cofibration in 2Cat, for all m ≥ C ( ι ∆ n ). As above, we find that C ( δ ∆[ n ]) = δ ^ O ( n ) and C (∆[ n ]) = ^ O ( n ) , for all n ≥ 0. Therefore the 2-functors C ( ι ∆ n ) : δ ^ O ( n ) → ^ O ( n ) can be described asthe 2-functors C ( ι Rm ) above, but where all the morphisms of the 2-categories in play areadjoint equivalences. In particular, the 2-functor C ( ι ∆ n ) is also a cofibration in 2Cat, forall n ≥ (cid:3) Lemma 5.2.7. For all n ≥ and ≤ t ≤ n , the -functor C ( ℓ ∆ n,t ) : C (Λ t [ n ]) → C (∆[ n ]) is a trivial cofibration in .Proof. We have that Λ t [ n ] is defined as the coequalizer in sSet ∆ op × ∆ op G ≤ i 1, the generating trivial cofibration j : [0] → ^ O (1) = E adj ,including [0] as one of the two end points, • for n = 2 and 0 ≤ t ≤ 2, the inclusion Λ t ^ O (2) → ^ O (2), which is a cofibrationby Proposition 3.1.5 since it is given by adding two morphisms x → y and y → x freely between objects x < y ∈ { , , } \ { t } on underlying categories. Moreover,it is a biequivalence, since it is bijective on objects, essentially full on morphisms,and fully faithful on 2-cells, where essential fullness on morphisms can be shownusing the fact that all the morphisms are adjoint equivalences. • for n = 3 and 0 ≤ t ≤ 3, the inclusion Λ t ^ O (3) → ^ O (3), which is a cofibrationby Proposition 3.1.5 since it is the identity on underlying categories. Moreover,it is a biequivalence, since it is bijective on objects and morphisms, and it is fullyfaithful on 2-cells, where fully faithfulness follows from the fact that there is aunique invertible 2-cell filling the triangle of the missing invertible 2-cell and it isgiven by the obvious composite of the three other invertible 2-cells. • for n ≥ ≤ t ≤ n , the identity.Therefore, the 2-functor C ( ℓ ∆ n,t ) is a trivial cofibration in 2Cat, for all n ≥ ≤ t ≤ n . (cid:3) We now state and prove the theorem. Theorem 5.2.8. The adjunction DblCat DblCat h ∞ CN ⊥ is a Quillen pair between the model structure on DblCat for weakly horizontally invariantdouble categories and the model structure on sSet ∆ op × ∆ op for horizontally complete double ( ∞ , -categories.Proof. By Theorem 5.2.1 and Proposition 5.2.2, it is enough to show that the cofibra-tions g Fk × id R [ m ] , id F [ k ] × q Rm , and id F [ k ] × e R , with respect to which we localize theReedy/injective model structure on sSet ∆ op × ∆ op in order to obtain the model structureDblCat h ∞ of Theorem 4.1.8, are sent by C to weak equivalences in DblCat. By definitionof C and by Remark 5.1.8, we have that C ( g Fk × id R [ m ] ) ∼ = C ( g Fk ) ⊗ id C R [ m ] = C ( g Fk ) (cid:3) ⊗ ( ∅ → C R [ m ])and similarly, that C (id F [ k ] × q Rm ) ∼ = ( ∅ → C F [ k ]) (cid:3) ⊗ C ( q Rm ) , C (id F [ k ] × e R ) ∼ = ( ∅ → C F [ k ]) (cid:3) ⊗ C ( e R ) . Since C is left Quillen from the Reedy/injective model structure on sSet ∆ op × ∆ op in whichevery object is cofibrant, the unique maps ∅ → C R [ m ] and ∅ → C F [ k ] are cofibrationsin 2Cat and DblCat, respectively. Moreover, the maps C ( g Fk ), C ( q Rm ) and C ( e R ) arecofibrations in DblCat and 2Cat, since they are images of monomorphisms in sSet ∆ op × ∆ op .As the model structure on DblCat is 2Cat-enriched, it is enough to show that C ( g Fk ) isa double biequivalence and that C ( q Rm ), and C ( e R ) are biequivalences by Remark 3.2.4.These statements are the content of Lemmas 5.2.9 and 5.2.10, respectively. (cid:3) The following two lemmas complete the proof of Theorem 5.2.8. Lemma 5.2.9. For all k ≥ , the double functor C ( g Fk ) : C ( G [ k ]) → C ( F [ k ]) is a doublebiequivalence in DblCat .Proof. Since C preserve colimits and [ k ] = [1] ⊔ [0] . . . ⊔ [0] [1], we have that C ( G [ k ]) = V [ k ] and C ( F [ k ]) = V O ∼ ( k ) , for all k ≥ 0. First note that, when k = 0 , 1, the double functor C ( g Fk ) is an identity. For k ≥ 2, let us give an example. When k = 2, the double functor C ( g F ) is given by theinclusion 012 •• 012 .0 −→ ••• ∼ = Having this example in mind, we can see that, for all k ≥ C ( g Fk ) : V [ k ] → V O ∼ ( k )is the identity on objects and horizontal morphisms, and it is fully faithful on squares,since all squares in V [ k ] are trivial. The double functor C ( g Fk ) is also injective on verticalmorphisms. Moreover, since every vertical morphism i j in V O ∼ ( k ) is related bya horizontally invertible square to the composite i i + 1 . . . j , then C ( g Fk ) isessentially full on vertical morphisms. This shows that the double functor C ( g Fk ) is adouble biequivalence, for all k ≥ (cid:3) DOUBLE ( ∞ , Lemma 5.2.10. For all m ≥ , the -functors C ( q Rm ) : C ( Q [ m ]) → C ( R [ m ]) and C ( e R ) : C ( R [0]) → C ( N R I ) are biequivalences in .Proof. We first show the result for C ( q Rm ). As in the proof of Lemma 5.2.9 and by Re-mark 5.1.8, we have that C ( Q [ m ]) = [ m ] and C ( R [ m ]) = O ∼ ( m ) , for all m ≥ 0. First note that, when m = 0 , 1, the 2-functor C ( q Rm ) is an identity. For m ≥ 2, let us give an example. When m = 3, the 2-functor C ( q R ) is given by the inclusion01 23 0 −→ ∼ = ∼ = 0= 1 23 . ∼ = ∼ = Having this example in mind, we can see that, for all m ≥ C ( q Rm ) : [ m ] → O ∼ ( m ) is theidentity on objects, and it is fully faithful on 2-cells, since all 2-cells in [ m ] are trivial. The2-functor C ( q Rm ) is also injective on morphisms. Moreover, since every morphism i → j in O ∼ ( m ) is related by an invertible 2-cell to the composite i → i + 1 → . . . → j , then C ( q Rm )is essentially full on morphisms. This shows that the 2-functor C ( q Rm ) is a biequivalence,for all m ≥ C ( e R ) is a biequivalence. We have that C ( R [0]) = [0], and wecompute C ( N R I ). Recall from Example 4.1.6 that m -simplices of the bisimplicial space N R I constant in the vertical and space directions are given by words of m letters in { x, y } .Since C ( N R I ) is obtained by gluing a copy of O ∼ ( m ) for each m -simplex of N R I , we havethat C ( N R I ) has • two objects 0 and 1, given by the 0-simplices x and y , • two morphisms f : 0 → g : 1 → 0, given by the 1-simplices xy and yx , • two invertible 2-cells η : id x ∼ = gf and ǫ : id y ∼ = f g , given by the 2-simplices xyx and yxy ,such that η and ǫ satisfy the triangle identities, expressed by the 3-simplices yxyx and xyxy . Higher simplices of N R I do not add any relations. Therefore, the 2-category C ( N R I ) = E adj is the “free-living adjoint equivalence”, and C ( e R ) = j : [0] → E adj is agenerating trivial cofibration in 2Cat. (cid:3) The nerve N is homotopically fully faithful. We now show that the nervefunctor is homotopically fully faithful. For this, we show that the derived counit of theadjunction C ⊣ N is level-wise a double biequivalence. Since all objects are cofibrant inDblCat h ∞ , the derived counit coincides with the counit. Theorem 5.3.1. The components ǫ A : CNA → A of the (derived) counit are trivial fibra-tions in DblCat , for all double categories A . In particular, these are double biequivalencesand therefore the nerve functor N : DblCat → DblCat h ∞ is homotopically fully faithful.Proof. Let A be a double category. We first compute the double category CNA . By aformula for left Kan extensions, we have that CNA = colim( Y ↓ NA −→ ∆ × ∆ × ∆ X −→ DblCat) , where Y : ∆ × ∆ × ∆ → Set (∆ op ) × denotes the Yoneda embedding and Y ↓ NA is theslice category over NA . An object in Y ↓ NA is a map R [ m ] × F [ k ] × ∆[ n ] → NA , orequivalently, by the adjunction C ⊣ N , a double functor ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) → A . Therefore, for each double functor ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) → A , we glue a copy of X m,k,n = ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) in CNA .The double category CNA is cofibrant, since every object in DblCat h ∞ is cofibrant and C is left Quillen. Therefore its underlying horizontal and vertical categories are free byCorollary 3.2.6 and it is enough to describe the generators. First note that CNA has thesame objects as A . The horizontal morphisms in CNA are given by • a horizontal morphism f : A → B , for each horizontal morphism f of A , • a horizontal morphism e f ( f,g,η,ǫ ) : A → B together with a horizontal morphism e g ( f,g,η,ǫ ) : B → A , for each horizontal adjoint equivalence ( f, g, η, ǫ ) in A .where id A , e f (id A , id A , id id A , id id A ) , and e g (id A , id A , id id A , id id A ) are identified with the identity id A at the object A of CNA . The underlying horizontal category of CNA is the free categorygenerated by these horizontal morphisms. The vertical morphisms in CNA are given bya vertical morphism u : A A ′ , for each vertical morphism u of A , where e A is identifywith the identity e A at the object A of CNA . The underlying vertical category is thefree category generated by these vertical morphisms. It remains to identify the squaresof CNA . They are given by: • vertically invertible squares e η ( f,g,η,ǫ ) : ( e A id A e g e f e A ) and e ǫ ( f,g,η,ǫ ) : ( e B e f e g id B e B ) sat-isfying the triangle identities, for each horizontal adjoint equivalence ( f, g, η, ǫ )in A , • a square α : ( u fg v ), for each square α in A , • a square e α : ( u e f e g v ), for each square α in A whose horizontal boundaries arehorizontal adjoint equivalences ( f, f ′ , η, ǫ ) and ( g, g ′ , η ′ , ǫ ′ ), • a vertically invertible square θ f,k,g,h : ( e A e g fk e h e C ), for each vertically invertiblesquare θ in A as depicted below, AA B ′ CB C • • f g ≃≃ h kθ ∼ = where g and h are horizontal adjoint equivalences ( g, g ′ , η, ǫ ) and ( h, h ′ , η ′ , ǫ ′ ), • a vertically invertible square ϕ f,g,h : ( e A hgf e C ), for each vertically invertible square ϕ in A as depicted below, AA B CC • • hf gϕ ∼ = • a vertically invertible square e ϕ f,g,h : ( e A e h e g e f e C ), for each vertically invertiblesquare ϕ in A as above, but where the morphisms f , g , and h are all horizontaladjoint equivalences, • a horizontally invertible square ψ u,v,w : ( w id A id A ′′ v u ), for each horizontally invertiblesquare ψ in A as depicted below. DOUBLE ( ∞ , AA ′ A ′′ AA ′′ • u • v • w ∼ = ψ Furthermore, these squares are submitted to relations represented by double functors( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) → A , where k + m + n ≥ 3. In particular, these relationshold for the squares that represent them in A .The double functor ǫ A : CNA → A is given by the identity on objects and by sendingeach horizontal morphism, vertical morphism, and square to the horizontal morphism,vertical morphism, and square representing it. This defines a double functor since theunderlying horizontal and vertical categories are free, and the relations on squares in CNA are satisfied by the squares representing them in A . Moreover, it is straightforward to seethat this double functor is surjective on objects, full on horizontal, and full on verticalmorphisms. Fully faithfulness on squares follows from the fact that, given a boundary in CNA , for each square in A in the representing boundary, we added a unique square, andthe fact that the relations satisfied for squares in A are also satisfied in CNA . (cid:3) Remark . Since DblCat h ∞ is obtained as a localization of the Reedy/injective modelstructure on sSet ∆ op × ∆ op , all objects are cofibrant in DblCat h ∞ , and hence the functor C : DblCat h ∞ → DblCat preserves weak equivalences by Ken Brown’s Lemma (see [18,Lemma 1.1.12]). Therefore, since the components ǫ A : CNA → A of the counit are doublebiequivalences by Theorem 5.3.1, for all A ∈ DblCat, the nerve N : DblCat → DblCat h ∞ reflects weak equivalences by 2-out-of-3.5.4. Level of fibrancy of nerves of double categories. The nerve of any doublecategory is almost fibrant in the model structure DblCat h ∞ of Theorem 4.1.8. Indeed,aside from the vertical Reedy/injective fibrancy, the nerve of a double category satisfiesthe conditions of a horizontally complete double ( ∞ , A is satisfied if and only ifthe double category A is weakly horizontally invariant.Let us first summarize the properties of the nerve of a general double category in thefollowing theorem. Theorem 5.4.1. The nerve of a double category A is such that(i) ( NA ) − ,k : ∆ op → sSet is Reedy/injective fibrant, for all k ≥ ,(ii) ( NA ) m, − : ∆ op → sSet satisfies the Segal condition, for all m ≥ ,(iii) ( NA ) − ,k : ∆ op → sSet is a complete Segal space, for all k ≥ . To show this theorem we will need several technical results. The first piece is a Quillenpair between 2Cat and sSet whose left adjoint is given by the restriction of the functor C : sSet ∆ op × ∆ op → Definition 5.4.2. We define the cosimplicial 2-category X : ∆ → n ] ^ O ( n ) . Proposition 5.4.3. The cosimplicial -category X induces an adjunction ∆Set ∆ op X C N ⊤ where C is the left Kan extension of X along the Yoneda embedding, and we have that ( N A ) n ∼ = 2Cat( ^ O ( n ) , A ) , for all A ∈ and all n ≥ .Proof. This is a direct application of [8, Theorem 1.1.10]. (cid:3) This adjunction gives the desired Quillen pair between the model structures for 2-categories and for Kan complexes. Proposition 5.4.4. The adjunction C N ⊥ is a Quillen pair between Lack’s model structure on and the Quillen model structureon sSet .Proof. It is enough to show that C sends generating cofibrations and generating trivialcofibrations in sSet to cofibrations and trivial cofibrations in 2Cat, respectively. Recallthat generating cofibrations and generating trivial cofibrations in sSet are given by maps ι ∆ n : δ ∆[ n ] → ∆[ n ], for n ≥ 0, and ℓ ∆ n,t : Λ t [ n ] → ∆[ n ], for n ≥ ≤ t ≤ n ,respectively. Note that we have C ( ι ∆ n ) = C ( ι ∆ n ) and C ( ℓ ∆ n,t ) = C ( ℓ ∆ n,t ). Therefore, byLemmas 5.2.6 and 5.2.7, we see that these are cofibrations and trivial cofibrations in 2Cat,respectively. This shows the result. (cid:3) We will reformulate conditions (i-iii) of Theorem 5.4.1, which are for now given in termsof weak equivalences between mapping spaces, using the right Quillen functor N of theabove proposition. This can be done by applying the following lemma. Lemma 5.4.5. Let X ∈ sSet ∆ op × ∆ op be a bisimplicial space which is constant in the spacedirection. Then, for every double category A , we have an isomorphism of simplicial sets Map( X, NA ) ∼ = N ( H [ C ( X ) , A ] ps ) natural in X and A .Proof. For all n ≥ 0, we have natural isomorphisms of setsMap( X, NA ) n ∼ = sSet ∆ op × ∆ op ( X × ∆[ n ] , NA ) ∼ = DblCat( C ( X × ∆[ n ]) , A ) ∼ = DblCat( C ( X ) ⊗ ^ O ( n ) , A ) ∼ = 2Cat( ^ O ( n ) , H [ C ( X ) , A ] ps ) ∼ = N ( H [ C ( X ) , A ] ps ) n , where the first isomorphism holds by definition of the mapping space, the second by theadjunction C ⊣ N , the third by definition of C and the fact that X is constant in thespace direction, the fourth by the universal property of ⊗ (see Definition 2.1.15), andthe last isomorphism by definition of N . These isomorphisms of sets assemble into an DOUBLE ( ∞ , isomorphism Map( X, NA ) ∼ = N ( H [ C ( X ) , A ] ps ) of simplicial sets, which is natural in X and A . (cid:3) We now prove Theorem 5.4.1 assuming Lemmas 5.4.6 and 5.4.7 below. Proof of Theorem 5.4.1. Let A be a double category. By Lemmas 5.4.6 and 5.4.7, the2-functor H [ C (id F [ k ] × ι Rm ) , A ] ps is a fibration in 2Cat, and the 2-functors H [ C (id F [ k ] × q Rm ) , A ] ps , H [ C (id F [ k ] × e R ) , A ] ps and H [ C ( ι Fk × id R [ m ] ) , A ] ps are trivial fibrations in 2Cat, for all m, k ≥ 0. As N : 2Cat → sSet is right Quillenby Proposition 5.4.4, these are sent by N to fibrations and trivial fibrations in sSet,respectively. As the map id F [ k ] × ι Rm is constant in the space direction, by Lemma 5.4.5,we have that Map(id F [ k ] × ι Rm , NA ) ∼ = N ( H [ C (id F [ k ] × ι Rm ) , A ] ps ) . By the above arguments, this is a fibration in sSet, for all m, k ≥ 0, which shows (i) sayingthat ( NA ) − ,k is Reedy/injective fibrant. Similarly, we have thatMap(id F [ k ] × q Rm , NA ) ∼ = N ( H [ C (id F [ k ] × q Rm ) , A ] ps )Map(id F [ k ] × e R , NA ) ∼ = N ( H [ C (id F [ k ] × e R ) , A ] ps )Map( ι Fk × id R [ m ] , NA ) ∼ = N ( H [ C ( ι Fk × id R [ m ] ) , A ] ps )and these are trivial fibrations in sSet by the above arguments, for all m, k ≥ 0. The factthat Map(id F [ k ] × q Rm , NA ) and Map(id F [ k ] × e R , NA ) are in particular weak equivalencesin sSet shows that (iii) holds, i.e., we have the Segal and completeness conditions for( NA ) − ,k , and the fact that Map( ι Fk × id R [ m ] , NA ) is in particular a weak equivalence insSet gives (ii), i.e., the Segal condition for ( NA ) m, − . (cid:3) The following two lemmas complete the proof of Theorem 5.4.1. Lemma 5.4.6. Let A be a double category. The -functor H [ C (id F [ k ] × ι Rm ) , A ] ps is afibration in , and the -functors H [ C (id F [ k ] × q Rm ) , A ] ps and H [ C (id F [ k ] × e R ) , A ] ps aretrivial fibrations in , for all m, k ≥ .Proof. By promoting the bijections in Definition 2.1.15 of the tensor ⊗ , we get isomor-phisms of 2-categories as in the following commutative square. H [ V O ∼ ( k ) ⊗ O ∼ ( m ) , A ] ps H [ V O ∼ ( k ) ⊗ δO ∼ ( m ) , A ] ps [ O ∼ ( m ) , H [ V O ∼ ( k ) , A ] ps ] , ps [ δO ∼ ( m ) , H [ V O ∼ ( k ) , A ] ps ] , ps H [ C (id F [ k ] × ι Rm ) , A ] ps [ C ( ι Rm ) , H [ V O ∼ ( k ) , A ] ps ] , ps ∼ = ∼ = As every 2-category is fibrant and C ( ι Rm ) is a cofibration in 2Cat by Lemma 5.2.6, the 2-functor [ C ( ι Rm ) , H [ V O ∼ ( k ) , A ] ps ] , ps is a fibration in 2Cat by monoidality of Lack’s modelstructure. Hence H [ C (id F [ k ] × ι Rm ) , A ] ps is also a fibration in 2Cat.Similarly, we have isomorphisms H [ C (id F [ k ] × q Rm ) , A ] ps ∼ = [ C ( q Rm ) , H [ V O ∼ ( k ) , A ] ps ] , ps and H [ C (id F [ k ] × e R ) , A ] ps ∼ = [ C ( e R ) , H [ V O ∼ ( k ) , A ] ps ] , ps . By Lemma 5.2.10 and since C preserves cofibrations, the 2-functors C ( q Rm ) and C ( e R ) are trivial cofibrations in 2Cat.Therefore, by monoidality of Lack’s model structure, the 2-functors[ C ( q Rm ) , H [ V O ∼ ( k ) , A ] ps ] , ps and [ C ( e R ) , H [ V O ∼ ( k ) , A ] ps ] , ps are trivial fibrations in 2Cat and hence so are H [ C (id F [ k ] × q Rm ) , A ] ps and H [ C (id F [ k ] × e R ) , A ] ps . (cid:3) The last piece for the proof of Theorem 5.4.1 makes use of the data in the 2-category H [ − , − ] ps of double functors, horizontal pseudo-natural transformations, and modifica-tions, whose definitions can be found in Appendix A.3. Lemma 5.4.7. Let A be a double category. The -functor H [ C ( ι Fk × id R [ m ] ) , A ] ps is atrivial fibration in , for all m, k ≥ .Proof. By promoting the bijections in Definition 2.1.14 of the Gray tensor ⊗ G , we getisomorphisms of 2-categories as in the following commutative square. H [ V O ∼ ( k ) ⊗ O ∼ ( m ) , A ] ps H [ V [ k ] ⊗ O ∼ ( m ) , A ] ps H [ V O ∼ ( k ) , [ H O ∼ ( m ) , A ] ps ] ps H [ V [ k ] , [ H O ∼ ( m ) , A ] ps ] ps H [ C ( ι Fk × id R [ m ] ) , A ] ps H [ C ( ι Fk ) , [ H O ∼ ( m ) , A ] ps ] ps ∼ = ∼ = To see that the 2-functor H [ C ( ι Fk × id R [ m ] ) , A ] ps is a trivial fibration in 2Cat, it is enoughto show that the 2-functor H [ C ( ι Fk ) , B ] ps : H [ V O ∼ ( k ) , B ] ps → H [ V [ k ] , B ] ps is a trivial fibration in 2Cat, for any B ∈ DblCat. Then, by applying this result to B = [ H O ∼ ( m ) , A ] ps , we get that H [ C ( ι Fk ) , [ H O ∼ ( m ) , A ] ps ] ps is a trivial fibration in 2Cat.By the commutative square above, we get that H [ C ( ι Fk × id R [ m ] ) , A ] ps is also a trivialfibration in 2Cat.We first describe the double functor C ( ι Fk ) : V [ k ] → V O ∼ ( k ) on objects and verticalmorphisms. Since the horizontal morphisms and squares of V [ k ] are all trivial, this de-scribes the image of C ( ι Fk ) completely. Let us denote by u i : i i + 1, for 0 ≤ i < k ,the generating vertical morphisms of V [ k ]. Then the double functor C ( ι Fk ) is the identityon objects and sends a generating vertical morphism u i : i i + 1 of V [ k ] to the verticalmorphism i i + 1 of V O ∼ ( k ) represented by { i, i + 1 } .Now let B be a double category. We show that the 2-functor H [ C ( ι Fk ) , B ] ps is a trivialfibration in 2Cat, by verifying that it is surjective on objects, full on morphisms, and fullyfaithful on 2-cells.Given a double functor F : V [ k ] → B , consider the composite V O ∼ ( k ) V π −→ V [ k ] F −→ B , where π : O ∼ ( k ) → [ k ] is the identity on objects and acts on hom-categories as theunique functor O ∼ ( k )( i, j ) → [ k ]( i, j ) = [0]. The composite above is a double functorin H [ V O ∼ ( k ) , B ] such that F ( V π ) C ( ι Fk ) = F , which proves surjectivity on objects.Let F, G : V O ∼ ( k ) → B be double functors, and ϕ : F C ( ι Fk ) ⇒ G C ( ι Fk ) be a horizontalpseudo-natural transformation in H [ V [ k ] , B ] ps . We want to define a horizontal pseudo-natural transformation ϕ : F ⇒ G in H [ V O ∼ ( k ) , B ] ps such that ϕ C ( ι Fk ) = ϕ . It is enoughto define ϕ on the generating vertical morphisms of V O ∼ ( k ) which are represented by { i, j } for i < j . When j = i + 1, we set ϕ { i,i +1 } := ϕ u i . For j > i + 1, let θ denote theunique horizontally invertible square in V O ∼ ( k ) from the vertical morphism respresentedby { i, j } to the vertical composite of morphisms represented by [ i, j ] = { k | i ≤ k ≤ j } .Then there is a unique way of defining ϕ { i,j } so that ϕ is natural; namely as follows. DOUBLE ( ∞ , F iF j GiGj • F { i,j } • G { i,j } ϕ { i,j } F i F iF ( i + 1)... F jF j • F { i,j } = • F { i,i +1 } •• F θ ∼ = Gi GiG ( i + 1)... Gj Gj • G { i,j } • G { i,i +1 } •• ( Gθ ) − ∼ = ϕ u i ϕ u i +1 ϕ u j − ...This defines a horizontal pseudo-natural transformation ϕ : F ⇒ G which maps to ϕ via H [ C ( ι Fk ) , B ] ps , and hence shows fullness on morphisms.Let ϕ, ψ : F ⇒ G be horizontal pseudo-natural transformations in H [ V O ∼ ( k ) , B ] ps , andlet µ : ϕ := ϕ C ( ι Fk ) → ψ := ψ C ( ι Fk ) be a modification in H [ V [ k ] , B ] ps . The modification µ comprises the data of squares µ i : ( e F i ϕ i ψ i e Gi ), for 0 ≤ i ≤ k , natural with respect tothe square components of ϕ and ψ . By the relations between the square components of ϕ and ϕ , and the ones of ψ and ψ as indicated in the pasting equality above, one canshow that the squares µ i of µ are also natural with respect to the square components of ϕ and ψ . Therefore µ also defines a modification µ : ϕ → ψ in H [ V O ∼ ( k ) , B ] ps . As it isthe unique such modification in H [ V O ∼ ( k ) , B ] ps that maps to µ via H [ C ( ι Fk ) , B ] ps , thisshows fully faithfulness on 2-cells. (cid:3) Finally, we show that the nerve of a double category satisfies the missing conditionof a horizontally complete double ( ∞ , Theorem 5.4.8. The nerve of a double category A is such that ( NA ) m, − : ∆ op → sSet is Reedy/injective fibrant, for all m ≥ , if and only if the double category A is weaklyhorizontally invariant.Proof. Let A be a double category. Suppose that A is weakly horizontally invariant,then NA is a horizontally complete double ( ∞ , N : DblCat → DblCat h ∞ is right Quillen. In particular, this says that ( NA ) m, − : ∆ op → sSet is Reedy/injectivefibrant, for all m ≥ NA ) m, − : ∆ op → sSet is Reedy/injective fibrant, for all m ≥ 0. Then ( NA ) , − is Reedy/injective fibrant and therefore the map( ι F ) ∗ : ( NA ) , ∼ = Map( F [1] , NA ) → Map( δF [1] , NA ) ∼ = ( NA ) , × ( NA ) , . is a fibration in sSet, by Definition 4.1.4. In particular, it has the right lifting propertywith respect to ℓ ∆1 , : ∆[0] → ∆[1], i.e., there is a lift in every commutative diagram asbelow. ∆[0] ( NA ) , ∆[1] ( NA ) , × ( NA ) , ℓ ∆1 , ( ι F ) ∗ v ( f, f ′ ) By Descriptions B.1.2 and B.1.4, the upper map v is the data of a vertical morphism v : B B ′ in A , while the bottom map ( f, f ′ ) is the data of a pair of horizontal adjointequivalences ( f : A ≃ −→ B, f ′ : A ′ ≃ −→ B ′ ) in A . Therefore, the existence of a lift in eachdiagram as above corresponds to the existence of a weakly horizontally invertible squarein A of the form A BA ′ B ′ • u • v ≃ ff ′ ≃≃ for each such data ( v, f, f ′ ). In other words, this says that A is weakly horizontallyinvariant. (cid:3) Remark . In particular, since a horizontal double category is not generally weaklyhorizontally invariant (see [26, Remark 3.12]), the nerve NH A of a 2-category A is notgenerally fibrant. Since every 2-category is fibrant in Lack’s model structure on 2Cat,this shows that the composite NH is not right Quillen from 2Cat to DblCat h ∞ . Therefore,we will need to define the nerve for 2-categories differently in the next section.6. Nerve of -categories As 2-categories are embedded in double categories, we hope that the nerve functor N : DblCat → DblCat h ∞ restricts to a nerve functor 2Cat → H A associated to a 2-category A is not generally fibrant, as explainedin Remark 5.4.9, we need to define the nerve of a 2-category as the nerve of its fibrantreplacement H ≃ A in DblCat; see Proposition 3.2.9. In Section 6.1, we show that thecomposite of the Quillen pairs L ≃ ⊣ H ≃ and C ⊣ N restrict to a Quillen pair between2Cat and 2CSS. The (derived) counit of the composite of these adjunctions is also level-wise a biequivalence, and we get a homotopically full embedding of 2Cat into 2CSS.As all objects are fibrant in 2Cat, the nerve NH ≃ preserves weak equivalences, and wecan further show in Section 6.2 that Lack’s model on 2Cat is right-induced from 2CSSalong NH ≃ . In particular, as the weak equivalences and fibrations are determined throughtheir images under NH ≃ , this says that the homotopy theory of 2-categories is created bythat of 2-fold complete Segal spaces. In Section 6.3, we compare the nerve of the doublecategories H A and H ≃ A , by showing that the nerve of the latter is a fibrant replacementof the nerve of the former in 2CSS, and hence also in DblCat h ∞ .6.1. The nerve NH ≃ is right Quillen and homotopically fully faithful. We con-sider the composite of the Quillen pairs2Cat DblCat DblCat h ∞ L ≃ H ≃ ⊥ CN ⊥ and show that this gives a Quillen pair between 2Cat and the localization 2CSS ofDblCat h ∞ . Theorem 6.1.1. The adjunction L ≃ CNH ≃ ⊥ DOUBLE ( ∞ , is a Quillen pair between Lack’s model structure on and the model structure on sSet ∆ op × ∆ op for -fold complete Segal spaces, i.e., ( ∞ , -categories.Remark . Note that the functor L ≃ : DblCat → L ≃ ( V [1] ⊗ [1]) is as below-left, while the 2-category L ≃ ( V [1]) ⊗ [1]is as below-right. 0 10 ′ ′ ≃ ≃ ′ ′ ≃ ≃∼ = The fact that the left-hand 2-cell is not invertible in a square coming from a pair of avertical morphism and a horizontal morphism is the only difference between L ≃ ( − ⊗ − )and L ≃ ( − ) ⊗ L ≃ ( − ). Proof. First note that the adjunction L ≃ C ⊣ NH ≃ is a Quillen pair between 2Cat andDblCat h ∞ , since it is a composite of two Quillen pairs. By Theorem 5.2.1, it is enoughto show that the functor L ≃ C sends the cofibrations e F × id R [ m ] and c k , with respect towhich we localize DblCat h ∞ to obtain 2CSS, to weak equivalences in 2Cat.We first show that L ≃ C ( e F × id R [ m ] ) is a biequivalence. By a similar computation tothe one of C ( N R I ) in the proof of Lemma 5.2.10, we obtain that L ≃ C ( N F I × R [ m ]) ∼ = L ≃ ( V ^ O ( k ) ⊗ O ∼ ( m )) . Then the squares in the tensor V ^ O ( k ) ⊗ O ∼ ( m ) induced from vertical morphisms in V ^ O ( k ) and morphisms in O ∼ ( m ) must be weakly vertically invertible, since all verticalmorphisms in V ^ O ( k ) are vertical equivalences, and these correspond to invertible 2-cellsin L ≃ ( V ^ O ( k ) ⊗ O ∼ ( m )), by a dual version of Lemma A.2.4. By Remark 6.1.2, we deducethat L ≃ preserves this tensor: L ≃ ( V ^ O ( k ) ⊗ O ∼ ( m )) ∼ = ^ O ( k ) ⊗ O ∼ ( m ) ∼ = L ≃ C ( N F I ) ⊗ L ≃ C ( R [ m ]) . Therefore, L ≃ C ( e F × id R [ m ] ) ∼ = L ≃ C ( e F ) (cid:3) ⊗ ( ∅ → L ≃ C R [ m ]). Both maps in thispushout-product are cofibrations in 2Cat since L ≃ C is left Quillen from DblCat h ∞ , andtherefore, by Remark 3.1.4, it is enough to show that L ≃ C ( e F ) is a biequivalence. Butthis is clear since the 2-functor L ≃ C ( e F ) : L ≃ C ( F [0]) → L ≃ C ( N F I ) can be identified withthe generating trivial cofibration j : [0] → E adj in 2Cat.We now show that the 2-functor L ≃ C ( c k ) : L ≃ C ( F [0]) → L ≃ C ( F [ k ]) is a biequivalence.It is given by the inclusion [0] → ^ O ( k ). First note that for k = 0, this is the identity.For k ≥ 1, it is a biequivalence since it is • bi-essentially surjective on objects as every object in ^ O ( k ) is related by an adjointequivalence to the object 0, • essentially full on morphisms since every composite of equivalences 0 → ^ O ( k )is related by an invertible 2-cell to id , which is given by a pasting of units andcounits of the adjoint equivalences, • fully faithful on 2-cells since the only 2-cell id ⇒ id in ^ O ( k ) is the identity.This proves the theorem. (cid:3) As in the double categorical case, the nerve NH ≃ is homotopically fully faithful. Again,since all objects in 2CSS are cofibrant, the derived counit of the adjunction L ≃ C ⊣ NH ≃ coincide with the counit, and we show that it is level-wise a biequivalence. Theorem 6.1.3. The components ǫ A : L ≃ CNH ≃ A → A of the (derived) counit arebiequivalences, for all -categories A . In particular, the nerve NH ≃ : 2Cat → ishomotopically fully faithful.Proof. This follows from the fact that the (derived) counits of the adjunctions C ⊣ N and L ≃ ⊣ H ≃ are weak equivalences, by Theorems 3.2.8 and 5.3.1, respectively. (cid:3) Remark . Let us denote by D : Cat → C tothe 2-category D C with the same objects and morphisms as C and only trivial 2-cells.The functor D has a left adjoint P : 2Cat → Cat given by base change along the functor π : Cat → Set sending a category to its set of connected components. By [22, Theo-rem 8.2], these functors form a Quillen pair between the canonical model structure onCat and Lack’s model structure on 2Cat, and its derived counit is level-wise an equivalenceof categories.Composing with the Quillen pair of Theorem 6.1.1, we obtain a Quillen pairCat 2Cat 2CSS PD ⊥ L ≃ CNH ≃ ⊥ between the canonical model structure on Cat and the model structure on sSet ∆ op × ∆ op for 2-fold complete Segal spaces, i.e., ( ∞ , is right-induced from along NH ≃ . We now show that Lack’s modelstructure on 2Cat is right-induced from 2CSS along the nerve NH ≃ . In particular, thissays that the homotopy theory of 2-categories is determined by the homotopy theory of2-fold complete Segal spaces through its image under NH ≃ . Theorem 6.2.1. Lack’s model structure on is right-induced along the adjunction L ≃ C , NH ≃ ⊥ where denotes the model structure on sSet ∆ op × ∆ op for -fold complete Segal spaces.Proof. It is enough to show that a 2-functor F is a weak equivalence (resp. fibration)in 2Cat if and only if NH ≃ F is a weak equivalence (resp. fibration) in 2CSS, as modelstructures are uniquely determined by their classes of weak equivalences and fibrations.Since the functor NH ≃ is right Quillen, it preserves fibrations. Moreover, since allobjects are fibrant in 2Cat, the functor NH ≃ also preserves weak equivalences by KenBrown’s Lemma (see [18, Lemma 1.1.12]). This shows that, if F is a weak equivalence(resp. fibration) in 2Cat, then NH ≃ F is a weak equivalence (resp. fibration) in 2CSS.Now let F : A → B be a 2-functor such that NH ≃ F : NH ≃ A → NH ≃ B is a weakequivalence in 2CSS. Since all objects are cofibrant in 2CSS, by Ken Brown’s Lemma (see[18, Lemma 1.1.12]), the left Quillen functor L ≃ C preserves weak equivalences. Therefore,the 2-functor L ≃ CNH ≃ F is a biequivalence. We then have a commutative square L ≃ CNH ≃ A L ≃ CNH ≃ BA B L ≃ CNH ≃ F ≃ ǫ A ≃ ǫ B ≃ F DOUBLE ( ∞ , where the vertical 2-functors are biequivalences by Theorem 6.1.3. By 2-out-of-3, we getthat F is also a biequivalence.Finally, let F : A → B be a 2-functor such that NH ≃ F : NH ≃ A → NH ≃ B is a fibrationin 2CSS. We show that F has the right lifting property with respect to the generatingtrivial cofibrations j : [0] → E adj and j : [1] → C inv in 2Cat as described in Notation 3.1.2,where E adj denotes the “free-living adjoint equivalence” and C inv denotes the “free-livinginvertible 2-cell”. First note that, if NH ≃ F is a fibration, then ( NH ≃ F ) m,k is a fibrationin sSet for all m, k ≥ 0, since fibrations between fibrant objects in 2CSS are in particularlevel-wise fibrations.By taking m = k = 0, as ( NH ≃ F ) , is a fibration in sSet, there is a lift in everycommutative diagram as below left.∆[0] ( NH ≃ A ) , ∆[1] ( NH ≃ B ) , ℓ ∆1 , ( NH ≃ F ) , [0] A E adj B j F By Description B.2.1, a 0-simplex in ( NH ≃ A ) , is an object of A , and a 1-simplex in( NH ≃ A ) , is an adjoint equivalence in A . Therefore, the existence of a lift in eachdiagram as above left corresponds to the existence of a lift in each diagram as aboveright. This shows that F has the right lifting property with respect to j .Now take m = 1 and k = 0. As ( NH ≃ A ) , is a fibration in sSet, there is a lift in everycommutative diagram as below left.∆[0] ( NH ≃ A ) , ∆[1] ( NH ≃ B ) , ℓ ∆1 , ( NH ≃ F ) , [1] A [1] ⊗ E adj B j ′ F By Description B.2.2, a 0-simplex in ( NH ≃ A ) , is a morphism of A , and a 1-simplex in( NH ≃ A ) , is an invertible 2-cell in A , as depicted in Description B.2.3 (1). Therefore,the existence of a lift in each diagram as above left corresponds to the existence of a liftin each diagram as above right. Now the generating trivial cofibration j : [1] → C inv is aretract of j ′ , [1] [1] [1] C inv [1] ⊗ E adj C inv j j ′ j i r where i sends the invertible 2-cell of C inv to the invertible 2-cell of [1] ⊗ E adj , and r sendsthe adjoint equivalences of E adj to identities, and the invertible 2-cell of [1] ⊗ E adj tothe invertible 2-cell of C inv . Therefore, since F has the right lifting property with respectto j ′ , then F also has the right lifting property with respect to j . This shows that F isa fibration in 2Cat and concludes the proof. (cid:3) Comparison between the nerves NH and NH ≃ . We now want to compare thenerves NH A and NH ≃ A of a 2-category A . For this, we will construct a homotopy equivalence between the spaces ( NH A ) m,k and ( NH ≃ A ) m,k . Their sets of n -simplices aregiven by ( NH A ) m,k,n = DblCat( X m,k,n , H A ) ∼ = 2Cat( L X m,k,n , A )and ( NH ≃ A ) m,k,n = DblCat( X m,k,n , H ≃ A ) ∼ = 2Cat( L ≃ X m,k,n , A ) . Let us first describe the 2-categories L ≃ X m,k,n and L X m,k,n . Description 6.3.1. The 2-category L X m,k,n is obtained from the double category X m,k,n = ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n )by identifying the objects ( x, y, z ) ∼ ( x, y ′ , z ), for all 0 ≤ x ≤ m , 0 ≤ y, y ′ ≤ k , and0 ≤ z ≤ n , and by identifying the vertical morphisms ( x, g, z ) : ( x, y, z ) ( x, y ′ , z ), where g ∈ O ∼ ( k )( y, y ′ ), with the identity at ( x, y, z ) ∼ ( x, y ′ , z ). We denote by [ x, z ] the equiv-alence class { ( x, y, z ) | ≤ y ≤ k } . Then, the 2-category L X m,k,n has • objects [ x, z ] for all 0 ≤ x ≤ m and 0 ≤ z ≤ n , • morphisms freely generated by – a morphism ( f, y, z ) : [ x, z ] → [ x ′ , z ] where f ∈ O ∼ ( m )( x, x ′ ) is representedby the set { x, x ′ } , for all 0 ≤ x, x ′ ≤ m , 0 ≤ y ≤ k , and 0 ≤ z ≤ n , – a morphism ( x, y, h ) : [ x, z ] → [ x, z ′ ] where h ∈ ^ O ( n )( z, z ′ ) is represented bythe set { z, z ′ } , for all 0 ≤ x ≤ m , 0 ≤ y ≤ k , and 0 ≤ z, z ′ ≤ n , • a 2-cell α : f ⇒ f ′ for each square α : ( u ff ′ v ) in X m,k,n . Description 6.3.2. The 2-category L ≃ X m,k,n has • the same objects as the double category X m,k,n = ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ),i.e., triples ( x, y, z ) with 0 ≤ x ≤ m , 0 ≤ y ≤ k , 0 ≤ z ≤ n , • morphisms generated by – a morphism ( f, y, z ) : ( x, y, z ) → ( x ′ , y, z ) where f ∈ O ∼ ( m )( x, x ′ ) is repre-sented by the set { x, x ′ } , for all 0 ≤ x, x ′ ≤ m , 0 ≤ y ≤ k , and 0 ≤ z ≤ n , – a morphism ( x, y, h ) : ( x, y, z ) → ( x, y, z ′ ) where h ∈ ^ O ( n )( z, z ′ ) is repre-sented by the set { z, z ′ } , for all 0 ≤ x ≤ m , 0 ≤ y ≤ k , and 0 ≤ z, z ′ ≤ n , – an adjoint equivalence ( x, g, z ) : ( x, y, z ) ≃ −→ ( x, y ′ , z ) where g ∈ O ∼ ( k )( y, y ′ )is represented by the set { y, y ′ } , for all 0 ≤ x ≤ m , 0 ≤ y, y ′ ≤ k , and0 ≤ z ≤ n , • a 2-cell α : vf ⇒ f ′ u for each square α : ( u ff ′ v ) in X m,k,n . Example 6.3.3. We compute these 2-categories in the case where m = 1, k = 1, and n = 0. Let us denote by u : 0 ′ ′ the vertical morphism in V [1] and by f : 0 → L ( V [1] ⊗ [1]) is the free 2-category on a 2-cell as depictedbelow left, while L ≃ ( V [1] ⊗ [1]) is the 2-category as depicted below right. We omit the z -component since it is always 0.[0] [1] ( f, ′ )( f, ′ )( f, u ) (0 , ′ ) (1 , ′ )(0 , ′ ) (1 , ′ ) ( f, ′ )( f, ′ )(0 , u ) ≃ ≃ (1 , u )( f, u ) Remark . Using these descriptions, we can see that the 0-simplices of the simpli-cial sets ( NH A ) , and ( NH ≃ A ) , are the objects of A , and the ones of ( NH A ) , and( NH ≃ A ) , the morphisms of A . The 0-simplices in ( NH A ) , are the 2-cells of A as in the DOUBLE ( ∞ , above left diagram of Example 6.3.3, while the ones of ( NH ≃ A ) , are the 2-cells of A asin the above right diagram of Example 6.3.3. In particular, the 0-simplices in ( NH A ) , are just objects of A , while the ones of ( NH ≃ A ) , are adjoint equivalences in A . Wedescribe these simplicial sets in greater details in Appendices B.2 and B.3.There is a comparison map π m,k,n : L ≃ X m,k,n → L X m,k,n which sends an object ( x, y, z )to the object [ x, z ], morphisms ( f, y, z ) and ( x, y, h ) to the morphisms ( f, y, z ) and ( x, y, h ),the adjoint equivalences ( x, g, z ) to the identity at [ x, z ], and a 2-cell α : vf ⇒ f ′ u to thecorresponding 2-cell α : f ⇒ f ′ . Note that this is a 2-functor since the adjoint equivalencesare sent to identities. Moreover, this 2-functor is clearly surjective on objects, full onmorphisms, and fully faithful on 2-cells. By constructing an inverse 2-functor up topseudo-natural equivalence to this comparison map π m,k,n , we obtain the following result. Theorem 6.3.5. Let A be a -category. The map π ∗ : NH A → NH ≃ A induced by thecomparison maps π m,k,n : L ≃ X m,k,n → L X m,k,n is level-wise a homotopy equivalence in sSet ∆ op × ∆ op . In particular, this exhibits NH ≃ A as a fibrant replacement of NH A in (or in DblCat h ∞ ).Proof. We first construct an inverse 2-functor up to pseudo-natural equivalence ι m,k,n : L X m,k,n → L ≃ X m,k,n to the 2-functor π m,k,n such that the composite π m,k,n ι m,k,n is the identity at L X m,k,n . Itsends an object [ x, z ] to the object ( x, , z ), a morphism ( f, y, z ) : [ x, z ] → [ x ′ , z ] to thecomposite ( x, , z ) ( x,g,z ) −−−−→ ≃ ( x, y, z ) ( f,y,z ) −−−−→ ( x ′ , y, z ) ( x ′ ,g ′ ,z ) −−−−−→ ≃ ( x ′ , , z ) , and a morphism ( x, y, h ) : [ x, z ] → [ x, z ′ ] to the composite( x, , z ) ( x,g,z ) −−−−→ ≃ ( x, y, z ) ( x,y,h ) −−−−→ ( x, y, z ′ ) ( x,g ′ ,z ′ ) −−−−−→ ≃ ( x, , z ′ )where g ∈ ^ O ( k )(0 , y ) is represented by the set { , y } and g ′ ∈ ^ O ( k )( y, 0) is its weak in-verse. The assignment on 2-cells is uniquely determined by these assignments on objectsand morphisms, since the 2-functor π m,k,n is fully faithful on 2-cells and we impose that π m,k,n ι m,k,n = id L X m,k,n . In particular, since the morphisms in the 2-category L X m,k,n are freely generated by the morphisms ( f, y, z ) and ( x, y, h ), this defines a 2-functor ι m,k,n : L X m,k,n → L ≃ X m,k,n .We now construct a pseudo-natural adjoint equivalence ǫ m,k,n : ι m,k,n π m,k,n ⇒ id L ≃ X m,k,n .At an object ( x, y, z ) ∈ L ≃ X m,k,n , we define ǫ ( x,y,z ) to be the morphism ǫ ( x,y,z ) := ( x, g, z ) : ( x, , z ) ≃ −→ ( x, y, z )where g ∈ ^ O ( k )(0 , y ) is represented by the set { , y } . Note that the morphism ǫ ( x,y,z ) asdefined above is an adjoint equivalence. Given a morphism ( f, y, z ) : ( x, y, z ) → ( x ′ , y, z ),we define ǫ ( f,y,z ) to be the following invertible 2-cell ( x, , z ) ( x, y, z )( x, y, z )( x ′ , y, z )( x ′ , , z ) ( x ′ , y, z ) ǫ ( x,y,z ) = ( x, g, z ) ≃ ( x, g, z ) ≃ ( f, y, z )( x ′ , g ′ , z ) ≃ ǫ ( x ′ ,y,z ) = ( x ′ , g, z ) ≃ ( f, y, z ) ∼ = = induced by the counit gg ′ ∼ = id y of the adjoint equivalence ( g, g ′ ). We define ǫ ( x,y,h ) for amorphism ( x, y, h ) : ( x, y, z ) → ( x, y, z ′ ) similarly. This defines a pseudo-natural adjointequivalence ǫ m,k,n : ι m,k,n π m,k,n ⇒ id L ≃ X m,k,n , which can be represented by a 2-functor O ∼ (1) → [ L ≃ X m,k,n , L ≃ X m,k,n ] ,ps since it corresponds to an adjoint equivalence in thehom 2-category. By definition of the Gray tensor product ⊗ (see Definition 2.1.5), thispseudo-natural adjoint equivalence can equivalently be seen as a 2-functor L ≃ X m,k,n ⊗ ^ O (1) L ≃ X m,k,n L ≃ X m,k,n L ≃ X m,k,n id ⊗ d id ⊗ d ǫ m,k,n ι m,k,n ◦ π m,k,n We claim that these 2-functors ǫ m,k,n induce a homotopy ǫ ∗ m,k as in( NH ≃ A ) m,k × ∆[1]( NH ≃ A ) m,k ( NH ≃ A ) m,k ( NH ≃ A ) m,k id × d id × d ǫ ∗ m,k π ∗ m,k ◦ ι ∗ m,k where the n th component of ǫ ∗ m,k is obtained by applying the functor 2Cat( − , A ) to ǫ m,k,n ,for all n ≥ F ∈ ( NH ≃ A ) m,k,n , we want to describe the corresponding (∆[ n ] × ∆[1])-prismof the homotopy, which coincide with F ι m,k,n π m,k,n at 0 ∈ ∆[1] and with F at 1 ∈ ∆[1].Note that a (∆[ n ] × ∆[1])-prism in ( NH ≃ A ) m,k corresponds to a 2-functor L ≃ (( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ( ^ O ( n ) ⊗ ^ O (1))) −→ A . The squares induced by vertical morphisms in V O ∼ ( k ) and morphisms in ^ O (1) mustbe weakly horizontally invertible in ( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ( ^ O ( n ) ⊗ ^ O (1)), since themorphisms in ^ O (1) are adjoint equivalences. It follows from Lemma A.2.4 that the DOUBLE ( ∞ , corresponding 2-cells in L ≃ (( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ( ^ O ( n ) ⊗ ^ O (1))) are invertible andtherefore, by Remark 6.1.2, we get that L ≃ (( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ( ^ O ( n ) ⊗ ^ O (1))) ∼ = L ≃ (( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n )) ⊗ ^ O (1)= L ≃ X m,k,n ⊗ ^ O (1) . This says that a (∆[ n ] × ∆[1])-simplex in ( NH ≃ A ) m,k corresponds to a 2-functor L ≃ X m,k,n ⊗ ^ O (1) → A . We can therefore define the component of the homotopy at F ∈ ( NH ≃ A ) m,k,n to be F ǫ m,k,n . This shows the claim.Since ι ∗ m,k ◦ π ∗ m,k = id ( NH A ) m,k and by the above homotopy, we see that ι ∗ m,k and π ∗ m,k give a homotopy equivalence between ( NH A ) m,k and ( NH ≃ A ) m,k , for all m, k ≥ 0. Theseassemble into maps ι ∗ and π ∗ of sSet ∆ op × ∆ op which give a level-wise weak equivalencebetween NH A and NH ≃ A . This is in particular a weak equivalence in 2CSS and inDblCat h ∞ . Since NH ≃ A is fibrant in 2CSS and in DblCat h ∞ , we conclude that it gives afibrant replacement of NH A . (cid:3) Remark . Recall from Remark 6.1.4 the Quillen pair P ⊣ D between Cat and 2Catand let C be a category. We compute the nerve of H D C ( NH D C ) m,k,n = 2Cat( L X m,k,n , D C ) ∼ = Cat( P L X m,k,n , C ) . By applying the functor P to the 2-category L X m,k,n as given in Description 6.3.1, wecan see that P L X m,k,n ∼ = [ m ] × I [ n ], where I [ n ] is the category with object set { , . . . , n } and a unique isomorphism between any two objects. Therefore,( NH D C ) m,k,n ∼ = Cat([ m ] × I [ n ] , C ) = N Rezk ( C ) m,n is given by the Rezk nerve (defined in [29, §3.5]) constant in the vertical direction. Simi-larly, we compute the nerve of H ≃ D C and find that( NH ≃ D C ) m,k,n = 2Cat( L ≃ X m,k,n , D C ) ∼ = Cat( P L ≃ X m,k,n , C ) ∼ = Cat(( I [ k ] × [ m ]) × I [ n ] , C ) . By Theorem 6.3.5, we get a level-wise homotopy equivalence NH D C → NH ≃ D C whichexhibits NH ≃ D C as a fibrant replacement of the Rezk nerve of C in 2CSS (or DblCat h ∞ ). Appendix A. Weakly horizontally invertible squares In this first appendix, we give some technical results about weakly horizontally invert-ible squares, which will be of use to describe the nerves in low dimensions in Appendix B.These results also find their utility in the papers [25, 26] by the author, Sarazola, andVerdugo. Some of the lemmas presented here (Lemmas A.1.1, A.2.1 and A.3.3) were alsoproven independently in another context by proven by Grandis and Paré in [14] – theirterminology for weakly horizontally invertible squares is that of equivalence cells . In Ap-pendix A.1, we first prove that the weak inverse of a weakly horizontally invertible squareis unique when one first fixes horizontal adjoint equivalence data. In Appendix A.2, weconsider weakly horizontally invertible squares of special forms and characterize them. Fi-nally, in Appendix A.3, we give a definition of horizontal pseudo-natural transformationsand modifications, which correspond to the morphisms and 2-cells in the hom 2-categories H [ − , − ] ps of the 2Cat-enrichment of DblCat given in Definition 2.1.15. We then charac-terize the equivalences in these hom 2-categories. A.1. Unique inverse lemma. We first show the existence and uniqueness of a weakinverse for a weakly horizontally invertible square with respect to fixed horizontal adjointequivalence data. Lemma A.1.1. Let α : ( u ff ′ v ) be a weakly horizontally invertible square in a doublecategory A . Suppose ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ) are horizontal adjoint equivalences. Thenthere is a unique square β : ( v gg ′ u ) in A which is the weak inverse of α with respect tothe horizontal adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ) .Proof. Since α is weakly horizontally invertible, by definition, there is a weak inverse γ of α with respect to horizontal adjoint equivalence data ( f, h, µ, δ ) and ( f ′ , h ′ , µ ′ , δ ′ ). Wedefine β to be given by the following pasting. B A A g B A B A • • • g f he g µ ∼ = B B A h • • • e h ǫ ∼ = ABB ′ A ′ g • v g ′ • uβ = B ′ B ′ A ′ h ′ • v • v • u id v γ B ′ A ′ B ′ A ′ • • • g ′ f ′ h ′ e h ′ ( ǫ ′ ) − ∼ = B ′ A ′ A ′ g ′ • • • e g ′ ( µ ′ ) − ∼ = We check that β is a weak inverse of α with respect to the horizontal adjoint equivalencedata ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ). We have that A AA B AA ′ B ′ A ′ A ′ A ′ • •• • f gf ′ g ′ • u • v • uη ∼ = α β ( η ′ ) − ∼ = A = B B AA ′ B ′ B ′ A ′ f hf ′ h ′ • u • v • v • uα γ id v A ′ B ′ A ′ B ′ A ′ f ′ g ′ f ′ h ′ • • • • ( ǫ ′ ) − ∼ = e f ′ e h ′ A ′ A ′ A ′ • • • ( η ′ ) − ∼ = ( µ ′ ) − ∼ = A B A B A f g f h • • • • ǫ ∼ = e f e h A A A • • • η ∼ = µ ∼ = DOUBLE ( ∞ , A AA B AA ′ B ′ A ′ A ′ A ′ • •• • f hf ′ h ′ • u • v • u = µ ∼ = α γ ( µ ′ ) − ∼ = A AA ′ A ′ • u • u id u = where the first equality holds by definition of β , the second by the triangle identities for( η, ǫ ) and ( η ′ , ǫ ′ ), and the last by definition of γ being a weak inverse of α with respectto the horizontal adjoint equivalence data ( f, h, µ, δ ) and ( f ′ , h ′ , µ ′ , δ ′ ). The other pastingequality for α , β , ǫ − , and ǫ ′ also holds by definition of γ being a weak inverse of α , andby the triangle identities for ( µ, δ ) and ( µ ′ , δ ′ ). This shows that β is a weak inverse of α with respect to the horizontal adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ).Now suppose that β ′ : ( v gg ′ u ) is another weak inverse of α with respect to the horizontaladjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ). Then we have that ABB ′ A ′ g • v g ′ • uβ ′ B = B ′ B AB ′ A ′ g • v • v g ′ • uβ ′ id v B B AB = A B AB ′ A ′ B ′ A ′ B ′ B ′ A ′ • • •• • • ggg ′ g ′ g fg ′ f ′ • v • u • v • vǫ − ∼ = β α β ′ ǫ ′ ∼ = e g e g ′ B A AB ′ A ′ A ′ g • v g ′ • u • u = β id u B A B A g f g • • • e g η − ∼ = B B A • • • gǫ − ∼ = e g B ′ A ′ B ′ A ′ g ′ f ′ g ′ • • • e g ′ η ′ ∼ = B ′ B ′ A ′ g ′ e g ′ • • • ǫ ′ ∼ = B ′ B A = A ′ g • v g ′ • uβ where the second equality holds by definition of β being a weak inverse of α with respect tothe horizontal adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ), the third by definitionof β ′ being a weak inverse of α with respect to the horizontal adjoint equivalence data( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ), and the last by the triangle identities for ( η, ǫ ) and ( η ′ , ǫ ′ ).This shows that β ′ = β and therefore such a weak inverse is unique. (cid:3) A.2. Weakly horizontally invertible square in H A , H ≃ A , and L ≃ A . We first showthat weakly horizontally invertible square with trivial vertical boundaries correspond tovertically invertible squares between horizontal equivalences. Lemma A.2.1. Let α be a square in a double category A of the form A BA B • • ff ′ α where f and f ′ are horizontal equivalences in A . Then the square α is weakly horizontallyinvertible if and only if it is vertically invertible.Proof. Suppose first that α is weakly horizontally invertible. Let β be its weak inversewith respect to horizontal adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ). We define γ to be given by the following pasting. A BA B • • f ′ fγ A = B A f g B A B g ′ f ′ A A B f ′ BA B f • • •• • •• • η ∼ = ǫ ′ ∼ = e f e f ′ β We show that γ is a vertical inverse of α . We have that A BA BA B • •• • f ′ ffγα A = B A B f g f B A B g ′ f ′ A A B f BA B f • • ••• • •• • η ∼ = ǫ ′ ∼ = e f e f αβ DOUBLE ( ∞ , A A BA = B A BA B B ff g ff • • •• • • η ∼ = ǫ ∼ = e f e f A BA B • = • ffe f where the first equality holds by definition of γ , the second by definition of β beinga weak inverse of α with respect to horizontal adjoint equivalence data ( f, g, η, ǫ ) and( f ′ , g ′ , η ′ , ǫ ′ ), and the last by the triangle identities for ( η, ǫ ). Similarly, one can show thatthe other vertical composite gives the vertical identity e f ′ by using the definition of β being a weak inverse of α , and the triangle identities for ( η ′ , ǫ ′ ). This shows that α isvertically invertible with α − = γ .Suppose now that α is vertically invertible. Let ( f, g, η, ǫ ) be an adjoint equivalencedata and define η ′ and ǫ ′ to be the following pasting. A AA B A f ′ g • • η ′ ∼ = A AA = B A f g • • η ∼ = A B A f ′ g • • • α ∼ = e g B A BB B g f ′ • • ǫ ′ ∼ = B = A BB B g f • • ǫ ∼ = B A B g f ′ • • • α − ∼ = e g Then ( f ′ , g, η ′ , ǫ ′ ) is a horizontal adjoint equivalence, and e g is a weak inverse of α withrespect to the horizontal adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g, η ′ , ǫ ′ ). This showsthat α is weakly horizontally invertible. (cid:3) Remark A.2.2 . Given a 2-category A , Lemma A.2.1 shows that a square in H A is weaklyhorizontally invertible if and only if its associated 2-cell is invertible.We now use the result above to characterize the weakly horizontally invertible squaresin H ≃ A as invertible 2-cells. Lemma A.2.3. Let A be a -category and let α : ( u ff ′ v ) be a square in H ≃ A , where f and f ′ are equivalences in A . Then α is weakly horizontally invertible if and only if itsassociated -cell α : vf ⇒ f ′ u is invertible.Proof. Consider a square α in H ≃ A of the form A BA ′ B ′ , ff ′ u = ( u, u ′ , η u , ǫ u ) ≃ v = ( v, v ′ , η v , ǫ v ) ≃ α where f and f ′ are horizontal equivalences. Note that the 2-cell α : vf ⇒ f ′ u also givesrise to a square in H ≃ A A B B ′ A A ′ B ′ , f f ′ u vα where the composites vf and f ′ u are horizontal equivalences. We show that α is weaklyhorizontally invertible if and only if its associated square α is weakly horizontally invert-ible. We can then conclude by applying Lemma A.2.1.Let us fix adjoint equivalence data ( f, g, η, ǫ ) and ( f ′ , g ′ , η ′ , ǫ ′ ). Suppose first that thefollowing square in H ≃ A B AB ′ A ′ gg ′ v ≃ u ≃ β is a weak inverse of α with respect to the adjoint equivalence data ( f, g, η, ǫ ), ( f ′ , g ′ , η ′ , ǫ ′ ).Then its mate B ′ B AB ′ A ′ A v ′ u ′ g ′ gβ ∗ B ′ = B AB ′ A ′ A v ′ u ′ g ′ gv uβǫ v η u is a weak inverse for the square α with respect to the composite of adjoint equivalencedata ( vf, gv ′ , ( g ∗ η v ∗ f ) η, ǫ v ( v ∗ ǫ ∗ v ′ ) and ( f ′ u, u ′ g ′ , ( u ′ ∗ η ′ ∗ u ) η u , ǫ ′ ( f ′ ∗ ǫ u ∗ g ′ )), where ∗ denotes the whisker of a morphism and a 2-cell. This follows from the triangle identitiesfor ( η u , ǫ u ) and ( η v , ǫ v ) and the definition of β being a weak inverse of α with respect tothe adjoint equivalence data ( f, g, η, ǫ ), ( f ′ , g ′ , η ′ , ǫ ′ ).Conversely, suppose that the following square in H ≃ A B ′ B AB ′ A ′ A v ′ u ′ g ′ gβ is a weak inverse of α with respect to the composite of adjoint equivalence data ( vf, gv ′ , ( g ∗ η v ∗ f ) η, ǫ v ( v ∗ ǫ ∗ v ′ ) and ( f ′ u, u ′ g ′ , ( u ′ ∗ η ′ ∗ u ) η u , ǫ ′ ( f ′ ∗ ǫ u ∗ g ′ )). Then its mate DOUBLE ( ∞ , B AB ′ A ′ gg ′ v ≃ u ≃ β ! B ′ = BA ′ AB A ′ v ′ v uu ′ g ′ gβη v ǫ u is a weak inverse of α with respect to the adjoint equivalence data ( f, g, η, ǫ ), ( f ′ , g ′ , η ′ , ǫ ′ ). (cid:3) In particular, we can see that a 2-cell in L ≃ A corresponding to a weakly horizontallyinvertible square in a double category A is invertible. Lemma A.2.4. Consider the left adjoint L ≃ : DblCat → of the functor H ≃ and let A be a double category.(i) If f : A → B is a horizontal equivalence in A , then the corresponding morphism f : A → B in L ≃ A is an equivalence.(ii) If α : ( u ff ′ v ) is a weakly horizontally invertible square in A , then the corresponding -cell A BA ′ B ′ ff ′ u ≃ v ≃ α in L ≃ A is invertible.Proof. Given a horizontal equivalence ( f, g, η, ǫ ) in A , then there are corresponding mor-phisms f and g and corresponding invertible 2-cells η : id ∼ = gf and ǫ : f g ∼ = id in L ≃ A ,i.e., this is the data of an equivalence in L ≃ A . This proves (i).Now, given a weakly horizontally invertible square α : ( u ff ′ v ), then the correspondingmorphisms f and f ′ are equivalences in L ≃ A by (i). The relations expressing the factthat α is a weakly horizontally invertible square in A translate to relations in H ≃ L ≃ A implying that the corresponding square A BA ′ B ′ ff ′ u ≃ v ≃ α is weakly horizontally invertible in H ≃ L ≃ A . By Lemma A.2.3, we obtain that the associ-ated 2-cell α : vf ⇒ f ′ u is invertible. (cid:3) A.3. Horizontal pseudo-natural equivalences. We now give complete definitions ofthe morphisms and 2-cells of the hom 2-category H [ I , A ] ps of double functors defined inDefinition 2.1.15. Definition A.3.1. Let F, G : I → A be double functors. A horizontal pseudo-naturaltransformation ϕ : F ⇒ G consists of(i) a horizontal morphism ϕ i : F i → Gi in A , for each object i ∈ I ,(ii) a square ϕ u : ( F u ϕ i ϕ i ′ Gu ) in A , for each vertical morphism u : i i ′ in I ,(iii) a vertically invertible square in A F i Gi GjF i F j Gj , ϕ i GfF f ϕ j • • ϕ f ∼ = for each horizontal morphism f : i → j in I ,such that the following conditions hold:(1) for every object i ∈ I , ϕ e i = e ϕ i : ( e F i ϕ i ϕ i e Gi ),(2) for every pair of composable vertical morphisms u : i i ′ and v : i ′ i ′′ in I , F i GiF i ′ Gi ′ F i ′′ Gi ′′ ϕ i ϕ i ′ ϕ i ′′ • F u • Gu • F v • Gvϕ u ϕ v F i GiF i ′′ Gi ′′ , ϕ i ϕ i ′′ • F ( vu )= • G ( vu ) ϕ vu (3) for every object i ∈ I , ϕ id i = e ϕ i : ( e F i ϕ i ϕ i e Gi ),(4) for every pair of composable horizontal morphisms f : i → j and g : j → k in I , F i Gi GjF i F j Gj GkGk F f ϕ j GgGgϕ i Gf • • • ϕ f ∼ = e Gg F i F j F k Gk F f F g ϕ k • • • e Ff ϕ g ∼ = F i Gi GkF i F k Gk , ϕ i G ( gf ) F ( gf ) ϕ k • = • ϕ gf ∼ = (5) for every α : ( u ff ′ v ) in I , F i Gi GjF i F j Gj ϕ i Gf • • F f ϕ j ϕ f ∼ = F i ′ F j ′ Gj ′ • F u • F v • GvF f ′ ϕ j ′ F α ϕ v F i = Gi GjF i ′ Gi ′ Gj ′ ϕ i Gf • F u • Gu • Gvϕ i ′ Gf ′ ϕ u Gα F i ′ F j ′ Gj ′ . F f ′ ϕ j ′ • • ϕ f ′ ∼ = Definition A.3.2. Let ϕ, ψ : F ⇒ G be horizontal pseudo-natural transformations be-tween double functors F, G : I → A . A modification µ : ϕ → ψ consists of a square µ i : ( e F i ϕ i ψ i e Gi ) in A , for each object i ∈ I , such that(1) for every horizontal morphisms f : i → j in I , DOUBLE ( ∞ , F i Gi GjF i F j Gj ϕ i Gf • • F f ϕ j ϕ f ∼ = F i F j Gj • • • F f ψ j e Ff µ j F i = Gi GjF i Gi Gj ϕ i Gf • • • ψ i Gfµ i e Gf F i F j Gj , F f ψ j • • ψ f ∼ = (2) for every vertical morphism u : i i ′ in I , F i GiF i ′ Gi ′ F i ′ Gi ′ ϕ i ϕ i ′ ψ i ′ • F u • Gu • • ϕ u µ i ′ F i GiF i = GiF i ′ Gi ′ . ϕ i ψ i ψ i ′ • •• F u • Guµ i ψ u In particular, we show that an equivalence in H [ I , A ] ps is precisely a horizontal pseudo-natural transformations whose squares components are weakly horizontally invertiblesquares. Lemma A.3.3. Let ϕ : F ⇒ G be a horizontal pseudo-natural transformation betweendouble functors F, G : I → A . Then ϕ is an equivalence in the -category H [ I , A ] ps if andonly if the square ϕ u : ( F u ϕ i ϕ i ′ Gu ) is weakly horizontally invertible, for every vertical mor-phism u : i i ′ in I . In particular, the horizontal morphism ϕ i : F i → Gi is a horizontalequivalence, for every object i ∈ I .Proof. Suppose first that ( ϕ, ψ, η, ǫ ) is an equivalence in the 2-category H [ I , A ] ps , i.e., wehave the data of horizontal pseudo-natural transformations ϕ : F ⇒ G and ψ : G ⇒ F to-gether with invertible modifications η : id F ∼ = ψϕ and ǫ : ϕψ ∼ = id G . By applying condition(2) of Definition A.3.2 to the modifications η and ǫ , we directly get that ( ϕ u , ψ u ) are weakinverses with respect to the horizontal equivalence data ( ϕ i , ψ i , η i , ǫ i ) and ( ϕ i ′ , ψ i ′ , η i ′ , ǫ i ′ ),for every vertical morphism u : i i ′ in A . This shows that every square ϕ u is weaklyhorizontally invertible.Now suppose that the square ϕ u : ( F u ϕ i ϕ i ′ Gu ) is weakly horizontally invertible, forevery vertical morphism u : i i ′ in I . For each object i ∈ I , let us fix a horizontaladjoint equivalence data ( ϕ i , ψ i , η i , ǫ i ). For each vertical morphism u : i i ′ in I , wedenote by ψ u : ( Gu ψ i ψ i ′ F u ) the unique weak inverse of ϕ u given by Lemma A.1.1 withrespect to the horizontal adjoint equivalence data ( ϕ i , ψ i , η i , ǫ i ) and ( ϕ i ′ , ψ i ′ , η i ′ , ǫ i ′ ).We define a horizontal pseudo-natural transformation ψ : G ⇒ F which is given by thehorizontal morphism ψ i : Gi → F i , at each object i ∈ I , the square ψ u : ( Gu ψ i ψ i ′ F u ), ateach vertical morphism u : i i ′ in I , and by the vertically invertible square ψ f Gi F i F jGi Gj F j ψ i F fψ j Gf • • ψ f ∼ = Gi = F i Gi GjGi Gi Gj F j ψ i ϕ i GfGf ψ j • • • ǫ i ∼ = e Gf F i F j Gj F j F f ϕ j ψ j • • • ϕ − f ∼ = e ψ j F iGi F j F j ψ i F f • • • • η j ∼ = e Ff e ψ i at each horizontal morphism f : i → j in I . We show that this data assemble into a hori-zontal pseudo-natural transformation ψ : G ⇒ F by verifying conditions (1)-(5) of Defini-tion A.3.1. We have (1), since ψ e i is the inverse of ϕ e i , which is unique by Lemma A.1.1and therefore must be equal to e ψ i . Condition (2) follows from the fact that the verticalcomposite of ψ u and ψ v , and the square ψ vu are both weak inverse of ϕ vu with respectto the horizontal adjoint equivalence data ( ϕ i , ψ i , η i , ǫ i ) and ( ϕ i ′′ , ψ i ′′ , η i ′′ , ǫ i ′′ ); they musttherefore be equal since such a weak inverse is unique by Lemma A.1.1. Conditions (3)and (4) follow from the definition of ψ f and the triangle identities for ( η i , ǫ i ), for each i ∈ I . The last condition follows from the definition of ψ f and condition (5) for the hori-zontal pseudo-natural transformation ϕ . Moreover, it is straightforward to check that thevertically invertible squares η i and ǫ i assemble into invertible modifications η : id F ∼ = ψϕ and ǫ : ϕψ ∼ = id G . This shows that ( ϕ, ψ, η, ǫ ) is an equivalence in H [ A , B ] ps . (cid:3) Appendix B. Explicit description of the nerves in lower dimensions In this appendix, we describe the nerve of the different double categories considered inthis paper in lower dimensions; namely, for 0 ≤ m, k ≤ ≤ n ≤ 2. The aim ofthese descriptions is to give the intuition that the space of the nerve at ( m, k ) = (0 , 0) isindeed the space of objects , the one at ( m, k ) = (1 , 0) the space of horizontal morphisms ,the one at ( m, k ) = (0 , 1) the space of vertical morphisms , and the one at ( m, k ) = (1 , space of squares of the double category. In Appendix B.1, we first describe the nerve N of a general double category. Then, in Appendix B.2, we describe the nerve NH ≃ of a2-category. Finally, in Appendix B.3, we also describe the nerve NH of a 2-category, inorder to compare it with its fibrant replacement NH ≃ .B.1. Nerve of a double category. Let A be a double category. We want to describethe 0-, 1-, and 2-simplices of the space ( NA ) m,k for 0 ≤ m, k ≤ Description B.1.1. By definition of N , we have that( NA ) m,k,n = DblCat(( V O ∼ ( k ) ⊗ O ∼ ( m )) ⊗ ^ O ( n ) , NA ) ∼ = 2Cat( ^ O ( n ) , H [ V O ∼ ( k ) ⊗ O ∼ ( m ) , A ] ps )Therefore we can describe the 0-, 1-, and 2-simplices of the space ( NA ) m,k as follows.(0) A 0-simplex in ( NA ) m,k is a double functor F : V O ∼ ( k ) ⊗ O ∼ ( m ) → A .(1) A 1-simplex in ( NA ) m,k is an adjoint equivalence in H [ V O ∼ ( k ) ⊗ O ∼ ( m ) , A ] ps , i.e.,by Lemma A.3.3, a horizontal pseudo-natural transformation V O ∼ ( k ) ⊗ O ∼ ( m ) A FGϕ DOUBLE ( ∞ , such that, the horizontal morphism ϕ i : F i → Gi is a horizontal adjoint equiva-lence, for each object i ∈ V O ∼ ( k ) ⊗ O ∼ ( m ), and the square ϕ u : ( F u ϕ i ϕ i ′ Gu ) isweakly horizontally invertible, for each vertical morphism u in V O ∼ ( k ) ⊗ O ∼ ( m ).In what follows, we call such a ϕ a horizontal pseudo-natural adjoint equiv-alence and we write ϕ : F ≃ = ⇒ G .(2) A 2-simplex is the data of three horizontal pseudo-natural adjoint equivalences ϕ : F ≃ = ⇒ G , ψ : G ≃ = ⇒ H , and θ : F ≃ = ⇒ H together with an invertible modification µ as follows. F G H ϕ ψθ ∼ = µ We first compute the space ( NA ) , , which is given by the space of objects . As expectedfrom the completeness condition being in the horizontal direction, its 0-simplices are givenby the objects, and its 1-simplices by the horizontal adjoint equivalences. Description B.1.2 ( m = 0, k = 0) . We describe the space ( NA ) , . First note that thedouble category V O ∼ (0) ⊗ O ∼ (0) = [0] is the terminal (double) category.(0) A 0-simplex in ( NA ) , is a double functor A : [0] → A , i.e., the data of an object A ∈ A .(1) A 1-simplex in the space ( NA ) , is a horizontal pseudo-natural adjoint equivalence ϕ : A ≃ = ⇒ B , i.e., the data of a horizontal adjoint equivalence ϕ : A ≃ −→ C in A .(2) A 2-simplex in ( NA ) , is an invertible modification µ : θ ∼ = ψϕ between such hor-izontal pseudo-natural adjoint equivalences, i.e., the data of a vertically invertiblesquare in A A EA C E . • • ≃ θϕ ≃ ψ ≃ µ ∼ = We now turn our intention to the space of horizontal morphisms ( NA ) , . We observethat the squares appearing as n -simplices of this space all have trivial vertical boundaries.In particular, this prevents a completeness condition for ( NA ) , − for a general doublecategory. Description B.1.3 ( m = 1, k = 0) . We describe the space ( NA ) , . First note that V O ∼ (0) ⊗ O ∼ (1) = H [1] is the free double category on a horizontal morphism.(0) A 0-simplex in ( NA ) , is a double functor f : H [1] → A , i.e., the data of a hori-zontal morphism f : A → B in A .(1) A 1-simplex in the space ( NA ) , is a horizontal pseudo-natural adjoint equivalence ϕ : f ≃ = ⇒ g , i.e., the data of two horizontal adjoint equivalences ϕ : A ≃ −→ C and ϕ : B ≃ −→ D together with a vertically invertible square in A A C DA B D . ϕ ≃ gf ϕ ≃ • • ϕ ∼ = (2) A 2-simplex in ( NA ) , is an invertible modification µ : θ ∼ = ψϕ between suchhorizontal pseudo-natural adjoint equivalences, i.e., the data of two verticallyinvertible squares µ and µ in A satisfying the following pasting equality. A E FA C E F θ ≃ h • • • ≃ ϕ ≃ ψ hµ ∼ = e h A C D F • •• ≃ ϕ ≃ ψ g ψ ∼ = e ϕ A B D F • • • ≃ ϕ ≃ ψ f ϕ ∼ = e ψ A E FA = B E θ ≃ h • • θ ≃ f θ ∼ = A B D F • • • ≃ ϕ ≃ ψ f µ ∼ = e f We now compute the lower simplices of the space ( NA ) , – the space of vertical mor-phisms . As expected from the horizontally complete condition, its 0-simplices are given bythe vertical morphisms, and its 1-simplices by the weakly horizontally invertible squares. Description B.1.4 ( m = 0, k = 1) . We describe the space ( NA ) , . First note that V O ∼ (1) ⊗ O ∼ (0) = V [1] is the free double category on a vertical morphism.(0) A 0-simplex in ( NA ) , is a double functor u : V [1] → A , i.e., the data of a verticalmorphism u : A A ′ in A .(1) A 1-simplex in the space ( NA ) , is a horizontal pseudo-natural adjoint equivalence ϕ : u ≃ = ⇒ w , i.e., the data of two horizontal adjoint equivalences ϕ : A ≃ −→ C and ϕ ′ : A ′ ≃ −→ C ′ together with a weakly horizontally invertible square in A A ′ A CC ′ . ϕ ≃ • u ϕ ′ ≃ • w e ϕ ≃ (2) A 2-simplex in ( NA ) , is an invertible modification µ : θ ∼ = ψϕ between suchhorizontal pseudo-natural adjoint equivalences, i.e., the data of two verticallyinvertible squares µ and µ ′ in A satisfying the following pasting equality. A EA C E θ ≃ • • ϕ ≃ ψ ≃ µ ∼ = A ′ C ′ E ′ • u • w • y ≃ ϕ ′ ≃ ψ ′ e ϕ ≃ e ψ ≃ A = EA ′ E ′ θ ≃ • u ≃ θ ′ • y e θ ≃ A ′ C ′ E ′ ≃ ϕ ′ ≃ ψ ′ • • µ ′ ∼ = Finally, we consider the space of squares ( NA ) , . Description B.1.5 ( m = 1, k = 1) . We describe the space ( NA ) , . First note that V O ∼ (1) ⊗ O ∼ (1) = V [1] × H [1] is the free double category on a square.(0) A 0-simplex in ( NA ) , is a double functor α : V [1] × H [1] → A , i.e., the data of asquare α in A DOUBLE ( ∞ , A ′ A BB ′ . f • u f ′ • vα (1) A 1-simplex in the space ( NA ) , is a horizontal pseudo-natural adjoint equivalence ϕ : α ≃ = ⇒ β , i.e., the data of four horizontal adjoint equivalences ϕ , ϕ , ϕ ′ , and ϕ ′ ,two vertically invertible squares ϕ and ϕ ′ , and two weakly horizontally invertiblesquares f ϕ and f ϕ in A fitting in the following pasting equality. A C DA B D ϕ ≃ g • • f ϕ ≃ ϕ ∼ = A ′ B ′ D ′ • u • v • xf ′ ≃ ϕ ′ α f ϕ ≃ A = C DA ′ C ′ D ′ ϕ ≃ g • u • w • xϕ ′ ≃ g ′ f ϕ ≃ β A ′ B ′ D ′ f ′ ≃ ϕ ′ • • ϕ ′ ∼ = (2) A 2-simplex in ( NA ) , is an invertible modification µ : θ ∼ = ψϕ between suchhorizontal pseudo-natural adjoint equivalences, i.e., the data of four verticallyinvertible squares in A A EA C E • • ≃ θ ϕ ≃ ψ ≃ µ ∼ = A EA C E • • ≃ θ ϕ ≃ ψ ≃ µ ∼ = A EA C E • • ≃ θ ′ ϕ ′ ≃ ψ ′ ≃ µ ′ ∼ = EAA C E • • ≃ θ ′ ϕ ′ ≃ ψ ′ ≃ µ ′ ∼ = such that • ( µ , µ ) satisfies the pasting equality as in Description B.1.3 (2) with respectto ϕ , ψ , and θ , • ( µ ′ , µ ′ ) satisfies the pasting equality as in Description B.1.3 (2) with respectto ϕ ′ , ψ ′ , and θ ′ , • ( µ , µ ′ ) satisfies the pasting equality as in Description B.1.4 (2) with respectto f ϕ , f ψ , and e θ , • ( µ , µ ′ ) satisfies the pasting equality as in Description B.1.4 (2) with respectto f ϕ , f ψ , and e θ .B.2. Nerve of a -category. By computing the nerve of a 2-category, we expect tosee the space of objects at ( m, k ) = (0 , space of morphisms at ( m, k ) = (1 , space of -cells at ( m, k ) = (1 , m, k ) = (0 , 1) should behomotopically the same as the space of objects.Let A be a 2-category. Recall that its nerve is given by the nerve of its associateddouble category H ≃ A . We therefore translate Descriptions B.1.2 to B.1.5 to this setting.In particular, we first obtain the space of objects ( NH ≃ A ) , , whose 0-simplices are the objects, and whose 1-simplices are the adjoint equivalences of A , as expected by thecompleteness condition. Description B.2.1 ( m = 0, k = 0) . We describe the space ( NH ≃ A ) , .(0) A 0-simplex in ( NH ≃ A ) , is the data of an object A ∈ A .(1) A 1-simplex in ( NH ≃ A ) , is the data of an adjoint equivalence A ≃ −→ C in A .(2) A 2-simplex in ( NH ≃ A ) , is the data of an invertible 2-cell as in the followingdiagram. A C E ≃ ≃≃∼ = As for the space of morphisms ( NH ≃ A ) , , we can see that the completeness conditionis now satisfied for ( NH ≃ A ) , − , since vertical morphisms are now adjoint equivalences in A and they therefore also appear in the horizontal direction. Description B.2.2 ( m = 1, k = 0) . We describe the space ( NH ≃ A ) , .(0) A 0-simplex in ( NH ≃ A ) , is the data of a morphism f : A → B in A .(1) A 1-simplex in ( NH ≃ A ) , is the data of two adjoint equivalences and an invertible2-cell in A as in the following diagram. A CB D ≃≃ f g ∼ = (2) A 2-simplex in ( NH ≃ A ) , is the data of two invertible 2-cells filling the trianglesof the following pasting equality. A C E ≃ ≃≃∼ = B D F f g h ≃ ≃∼ = ∼ = A = B D FE ≃≃ f h ≃ ≃∼ = ∼ = The space ( NH ≃ A ) , is actually given by the space of adjoint equivalences . Since the“free-living adjoint equivalence” is biequivalent to the point, this space can be interpretedas “homotopically the same” as the space of objects. Description B.2.3 ( m = 0, k = 1) . We describe the space ( NH ≃ A ) , .(0) A 0-simplex in ( NH ≃ A ) , is the data of an adjoint equivalence u : A ≃ −→ A ′ in A .(1) A 1-simplex in ( NH ≃ A ) , is the data of an invertible 2-cell as in the followingdiagram, by Lemma A.2.3. A CA ′ C ′ ≃≃ u ≃ w ≃ ∼ = (2) A 2-simplex in ( NH ≃ A ) , is the data of two invertible 2-cells filling the trianglesof the following pasting equality. DOUBLE ( ∞ , A C E ≃ ≃≃∼ = A ′ C ′ E ′ u ≃ w ≃ y ≃ ≃ ≃∼ = ∼ = A = A ′ C ′ E ′ E ≃≃ u ≃ y ≃ ≃ ≃∼ = ∼ = Finally, we compute the space of -cells ( NH ≃ A ) , . Although its 0-simplices are notprecisely the 2-cells of A , homotopically they give the right notion as the vertical mor-phisms u and v in the square below are adjoint equivalences. Description B.2.4. We describe the space ( NH ≃ A ) , .(0) A 0-simplex in ( NH ≃ A ) , is the data of a 2-cell in A as in the following diagram. A BA ′ B ′ ff ′ u ≃ v ≃ α (1) A 1-simplex in ( NH ≃ A ) , is the data of four adjoint equivalences and four invert-ible 2-cells in A as in the following diagram. A BC D f ≃≃ g ∼ = A ′ B ′ D ′ u ≃ v ≃ x ≃ f ′ ≃ α ∼ = A C = A ′ C ′ B ′ D ′ D ≃ gu ≃ w ≃ x ≃ ≃ g ′ f ′ ≃∼ = β ∼ = (2) A 2-simplex in ( NH ≃ A ) , is the data of four invertible 2-cells filling trianglessatisfying relations as described in Description B.2.2 (2) an Description B.2.3 (2).B.3. Nerve of a horizontal double category. Finally, we want to compute the nerve ofa horizontal double category H A , where A is a 2-category, in order to compare it with thenerve NH ≃ A described above. Since H A and H ≃ A have the same underlying horizontal 2-category, namely A itself, then the spaces ( NH A ) , and ( NH A ) , are equal to the spaces( NH ≃ A ) , and ( NH ≃ A ) , and they can therefore be described as in Descriptions B.2.1and B.2.2, respectively. In particular, they are the desired space of objects and space ofmorphisms .We now turn our intention to the space ( NH A ) , . Unlike ( NH ≃ A ) , , this space hasas 0-simplices the objects of A . This prohibits a completeness condition in the verticaldirection since equalities are not homotopically good enough. Description B.3.1 ( m = 0, k = 1) . We describe the space ( NH A ) , .(0) A 0-simplex in ( NH A ) , is the data of an object A ∈ A .(1) A 1-simplex in ( NH A ) , is the data of an invertible 2-cell as in the followingdiagram, by Lemma A.2.1. A C ≃≃∼ = (2) A 2-simplex in ( NH A ) , is the data of two invertible 2-cells filling the trianglesof the following pasting equality. A C E ≃ ≃≃∼ = ≃ ≃∼ = ∼ = A = C E ≃≃≃ ≃∼ = ∼ = Finally, we compute the space of -cells ( NH A ) , , which appears to have preciselythe 2-cells of A as 0-simplices. 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