A model structure for weakly horizontally invariant double categories
aa r X i v : . [ m a t h . A T ] A ug A MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANTDOUBLE CATEGORIES
LYNE MOSER, MARU SARAZOLA, AND PAULA VERDUGO
Abstract.
We construct another model structure on the category DblCat of doublecategories and double functors, Quillen equivalent to the model structure on DblCatdefined in a companion paper by the authors. The weak equivalences are still given bythe double biequivalences; the trivial fibrations are now the double functors that aresurjective on objects, full on horizontal and vertical morphisms, and fully faithful onsquares; and the fibrant objects are the weakly horizontally invariant double categories.We show that the functor H ≃ : 2Cat → DblCat, a more homotopical version of theusual horizontal embedding H , is right Quillen and homotopically fully faithful when con-sidering Lack’s model structure on 2Cat. In particular, H ≃ exhibits a fibrant replacementof H . Moreover, Lack’s model structure on 2Cat is right-induced along H ≃ from the modelstructure for weakly horizontally invariant double categories.We also show that this model structure is monoidal with respect to B¨ohm’s Gray tensorproduct. Finally, we prove a Whitehead Theorem characterizing the double biequivalencesbetween the fibrant objects as the double functors which admit a pseudo inverse up tohorizontal pseudo natural equivalences. Introduction
This paper aims to study and compare the homotopy theories of two related types of2-dimensional categories: 2 -categories and double categories . While 2-categories consist ofobjects, morphisms, and 2-cells, double categories admit two types of morphisms betweenobjects – horizontal and vertical morphisms – and their 2-cells are given by squares . Inparticular, a 2-category A can always be seen as a horizontal double category H A withonly trivial vertical morphisms. This assignment H gives a full embedding of 2-categoriesinto double categories.The category 2Cat of 2-categories and 2-functors admits a model structure, constructedby Lack in [10, 11]. In this model structure, the weak equivalences are the biequivalences;the trivial fibrations are the 2-functors which are surjective on objects, full on morphisms,and fully faithful on 2-cells; and all 2-categories are fibrant. Moreover, Lack gives a char-acterization of the cofibrant objects as the 2-categories whose underlying category is free.With this well-established model structure at hand, we raise the question of whether thereexists a homotopy theory for double categories which contains that of 2-categories.Several model structures for double categories were first constructed by Fiore and Paoliin [4], and by Fiore, Paoli, and Pronk in [5], but the homotopy theory of 2-categoriesdoes not embed in any of these homotopy theories for double categories. The first positive answer to this question is given by the authors in [13], and further related results appear inwork in progress by Campbell [2]. In [13], we construct a model structure on the categoryDblCat of double categories and double functors that is right-induced from two copies ofLack’s model structure on 2Cat; its weak equivalences are called the double biequivalences .This model structure is very well-behaved with respect to the horizontal embedding H :the functor H : 2Cat → DblCat is both left and right Quillen, and Lack’s model structureis both left- and right-induced along it. In particular, this says that Lack’s model structureon 2Cat is created by H from the model structure on DblCat of [13]. Moreover, the functor H is homotopically fully faithful, and it embeds the homotopy theory of 2-categories intothat of double categories in a reflective and coreflective way.Unsurprisingly, this model structure is not well-behaved with respect to the verticaldirection, as it is constructed with a pronounced horizontal bias. For example, trivial fi-brations, which are full on horizontal morphisms, are only surjective on vertical morphisms,and the free double category on two composable vertical morphisms is not cofibrant, asopposed to its horizontal analogue. As a further consequence of this asymmetry betweenthe horizontal and vertical directions, the model structure is not monoidal with respect tothe Gray tensor product for double categories defined by B¨ohm in [1], as we show in [13,Remark 7.4].The aim of this paper is to provide a new model structure on DblCat in Theorem 2.19with the same class of weak equivalences, i.e., the double biequivalences, but which is betterbehaved with respect to the vertical direction. We achieve this by adding the inclusion ⊔ → V of the two end-points into the vertical morphism to the class of cofibrations ofthe model structure in [13]. In particular, by making this inclusion into a cofibration, thetrivial fibrations will now be given by the double functors that are surjective on objects, fullon horizontal and vertical morphisms, and fully faithful on squares. The existence of thismodel structure was independently noticed at roughly the same time by Campbell [2]. Dueto the way we construct it, we directly get in Theorem 2.21 that the identity adjunctiongives a Quillen equivalence between this new model structure on DblCat and the originalone of [13]. Theorem A.
There is a cofibrantly generated model structure on
DblCat in which the weakequivalences are the double biequivalences defined in [13] , and the trivial fibrations are thedouble functors which are surjective on objects, full on horizontal and vertical morphisms,and fully faithful on squares. Moreover, it is Quillen equivalent to the model structureof [13] via the identity adjunction.
The improved symmetry of this model structure takes care of some of the issues posedabove. As mentioned before, the trivial fibrations are symmetric with respect to the hori-zontal and vertical directions, and moreover, the cofibrant objects can now be characterizedas the double categories whose underlying horizontal and vertical categories are free. Fur-thermore, this new model structure is monoidal, as we prove in Theorem 4.4.
Theorem B.
The model structure on
DblCat of Theorem A is monoidal with respect toB¨ohm’s Gray tensor product.
MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 3
Having added a cofibration, it is no longer the case that every double category is fibrant;instead, as we show in Theorem 2.20, the fibrant objects are given by what we call weaklyhorizontally invariant double categories. This condition requires that every vertical mor-phism in the double category can be lifted along horizontal equivalences at its source andtarget; see Definition 2.5. Such a notion is a 2-categorical analogue of Grandis’ horizontallyinvariant double categories (see [6, Theorem and Definition 4.1.7]), and in a similar vein to[6, Theorem 4.4.5], we obtain a Whitehead Theorem characterizing the weak equivalenceswith fibrant source in our model structure as the double functors which admit a pseudoinverse up to horizontal pseudo natural equivalences; see Theorem 5.1.
Theorem C (Whitehead Theorem for double categories) . Let A and B be double categoriessuch that A is weakly horizontally invariant. Then a double functor F : A → B is a doublebiequivalence if and only if there exists a pseudo double functor G : B → A together withhorizontal pseudo natural equivalences id A ≃ GF and F G ≃ id B . While the horizontal embedding H : 2Cat → DblCat remains a left Quillen and homo-topically fully faithful functor between Lack’s model structure and our new model structure,it is not right Quillen anymore. Indeed, the horizontal double category H A associated toa 2-category A is typically not weakly horizontally invariant; see Remark 3.12.In order to remedy this shortcoming, we consider instead a more homotopical version of H given by the functor H ≃ : 2Cat → DblCat. It sends a 2-category A to the double category H ≃ A , whose underlying horizontal 2-category is still A , but whose vertical morphismsare given by the adjoint equivalences of A . In particular, the inclusion H A → H ≃ A isa double biequivalence, as shown in Proposition 3.13, and therefore exhibits H ≃ A as afibrant replacement of H A in the model structure for weakly horizontally invariant doublecategories.In Theorem 3.6, we prove that H ≃ is a right Quillen functor, and that the derivedcounit is level-wise a biequivalence in 2Cat; therefore, H ≃ embeds the homotopy theory of2-categories into that of weakly horizontally invariant double categories in a reflective way.Furthermore, we show in Theorem 3.9 that H ≃ not only preserves, but also reflects weakequivalences and fibrations. Theorem D.
The adjunction
DblCat 2Cat L ≃ H ≃ ⊥ is a Quillen pair between Lack’s model structure on and the model structure on DblCat of Theorem A. Moreover, the derived counit of this adjunction is level-wise a biequivalence,and Lack’s model structure on is right-induced along H ≃ from the model structureon DblCat . While this new model structure is not as compatible with respect to the horizontalembedding as the original one defined in [13], its more symmetrical features provide a con-venient setting for applications. For example, it allows for a comparison between double
L. MOSER, M. SARAZOLA, AND P. VERDUGO categories and their ∞ -analogues. Namely, in [12], the first author shows that there is anerve functor from double categories into double ( ∞ , A is a double ( ∞ , A is weakly horizontally invariant. Furthermore, the homotopical horizontal em-bedding H ≃ can be used to restrict this nerve to a right Quillen and homotopically fullyfaithful functor from 2-categories into ( ∞ , H that one could hope for: it embeds the ho-motopy theory of 2-categories into that of double categories in a reflective and coreflectiveway, and it is such that a 2-functor F is a biequivalence (resp. cofibration, fibration) if andonly if H F is a double biequivalence (resp. cofibration, fibration) in DblCat. Furthermore,all double categories are fibrant in this first model structure.On the other hand, the new model structure is more symmetric with respect to the hor-izontal and vertical directions, and it is monoidal with respect to the Gray tensor producton DblCat. In addition, the Whitehead Theorem for double categories (see Theorem C) im-plies that the weak equivalences between fibrant objects in the model structure for weaklyhorizontally invariant double categories are precisely the double functors which admit a apseudo inverse up to horizontal pseudo natural equivalences. In particular, this notion ofequivalence resembles the notion of a biequivalence between 2-categories.Finally, both model structures are closely related, since we have a triangle of homotopi-cally fully faithful right Quillen functors 2Cat DblCatDblCat whi HH ≃ id ≃ QE ≃ filled by a natural transformation which is level-wise a double biequivalence. This impliesthat, in practice, one can choose the model for this homotopy theory whose features aremore convenient. Outline.
The layout of this paper is as follows. In Section 2, we prove the existence ofthe desired model structure on DblCat, describing its weak equivalences and providingsets of generating cofibrations and generating trivial cofibrations. We also give explicitdescriptions of the cofibrations, fibrations, and trivial fibrations, and show that the fibrantobjects are precisely the weakly horizontally invariant double categories. Finally, we showthat this model structure is Quillen equivalent to the one constructed in [13].In Section 3 we introduce the functor H ≃ and show that it is right Quillen and homo-topically fully faithful. We also show that Lack’s model structure on 2Cat is right-inducedfrom our model structure on DblCat. We conclude this section by studying the behaviorof the embedding H with respect to this new model structure; in particular, we observe MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 5 that it is not right Quillen, but we show that, for any 2-category A , the double category H ≃ A provides a fibrant replacement for H A .In Section 4, we show that the model structure constructed in this paper is monoidalwith respect to B¨ohm’s Gray tensor product.Finally, Section 5 is devoted to the statement and proof of the Whitehead Theorem fordouble categories. Acknowledgements.
The authors would like to thank tslil clingman for sharing LaTeXcommands which greatly simplify the drawing of diagrams. We would also like to thankViktoriya Ozornova, J´erˆome Scherer, and Alexander Campbell for interesting discussionsrelated to the subject of this paper, and Martina Rovelli, Hadrian Heine, and Yuki Maeharafor useful answers to our questions.During the realization of this work, the first-named author was supported by the SwissNational Science Foundation under the project P1ELP2 188039.2.
The model structure
The aim of this section is to construct the desired model structure on DblCat. We willdo this by modifying the model structure of [13] so that the double functor ⊔ → V becomes a cofibration, and keeping the same class of weak equivalences. As we shall see,this results in a model structure that is Quillen equivalent to the one in [13], but in whichthe trivial fibrations are symmetric with respect to the horizontal and vertical directions.This paper assumes a certain familiarity with double categories, double functors, andrelated basic notions. For definitions and preliminary results, we refer the reader to [6], ormore concisely, to section 2 of [13].We begin by recalling two notions of weak invertibility for horizontal morphisms andsquares, first introduced in [13]. Definition 2.1.
A horizontal morphism a : A → B in a double category A is a horizontalequivalence if it is an equivalence in its underlying horizontal 2-category H A . Similarly,we define horizontal adjoint equivalences . Definition 2.2.
A square α : ( u ab v ) in a double category A is weakly horizontallyinvertible if there exists a square β : ( v a ′ b ′ u ) in A and four vertically invertible squares η a , η b , ǫ a , and ǫ b as in the pasting diagrams below. A B AA A a a ′ • • η a ∼ = A ′ B ′ bu v • • α A ′ ub ′ • β = A AA ′ A ′ u u • • id u A ′ B ′ A ′ b b ′ • • η b ∼ = L. MOSER, M. SARAZOLA, AND P. VERDUGO
B BB A B a ′ a • • ǫ a ∼ = B ′ B ′ v v • • id v = B A BB ′ A ′ B ′ a ′ ab ′ bv u v • • • β α B ′ B ′ • • ǫ b ∼ = We call β a weak inverse of α . Remark . In particular, the horizontal boundaries a and b of a weakly horizontally invert-ible square α as above are horizontal equivalences witnessed by the data ( a, a ′ , η a , ǫ a ) and( b, b ′ , η b , ǫ b ). We call them the horizontal equivalence data of α . Moreover, if ( a, a ′ , η a , ǫ a )and ( b, b ′ , η b , ǫ b ) are both horizontal adjoint equivalences, we call them the horizontal ad-joint equivalence data . Remark . Note that a horizontal equivalence can always be promoted to a horizontal adjoint equivalence, and a weakly horizontally invertible square can always be promotedto one with horizontal adjoint equivalence data, by [13, Lemma 2.19].We now introduce the notion of weakly horizontally invariant double categories, whichwill form the class of fibrant objects in our model structure.
Definition 2.5.
A double category A is weakly horizontally invariant if, for all horizon-tal equivalences a : A ≃ −→ C and c : A ′ ≃ −→ C ′ in A and every vertical morphism u ′ : C C ′ in A , there exists a vertical morphism u : A A ′ together with a weakly horizontallyinvertible square in A as depicted below. A CA ′ C ′ a ≃ c ≃ • u • u ′ ≃ Example . The class of weakly horizontally invertible double categories contains manyexamples of interest. For instance, one can easily check that the (flat) double category R elSet of relations of sets satisfies this condition. More relevantly, this class also containsthe double categories of quintets Q A and of adjunctions A dj A built from any 2-category A .A precise description of these double categories can be found in [6, § Definition 2.7.
Given double categories A and B , a double functor F : A → B is a doublebiequivalence if MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 7 (db1) for every object B ∈ B , there exists an object A ∈ A together with a horizontalequivalence B ≃ −→ F A ,(db2) for every horizontal morphism b : F A → F C in B , there exists a horizontal mor-phism a : A → C in A together with a vertically invertible square in B F A F CF A F C , bF a • • ∼ = (db3) for every vertical morphism v : B B ′ in B , there exists a vertical morphism u : A A ′ in A together with a weakly horizontally invertible square in B B F AB ′ F A ′ , ≃≃ • v • F u ≃ (db4) for every square in B of the form F A F CF A F C , F aF c • F u • F u ′ β there exists a unique square α : ( u ac u ′ ) in A such that F α = β .In order to construct the rest of the model structure, we propose sets of generatingcofibrations and of generating trivial cofibrations, and use a result due to Hovey thatprovides sufficient conditions for constructing a cofibrantly generated model category; forconvenience, we include it here. Definition 2.8.
Let I be a class of morphisms in a category C containing all small colimits. • A morphism in C is I -injective if it has the right lifting property with respect toevery morphism in I . The class of I -injective morphisms is denoted I -inj. • A morphism in C is an I -cofibration if it has the left lifting property with respectto every I -injective morphism. The class of I -cofibrations is denoted I -cof. • A morphism in C is a relative I -cell complex if it is a transfinite compositionof pushouts of morphisms in I . The class of relative I -cell complexes is denoted I -cell. Theorem 2.9 ([9, Theorem 2.1.19]) . Suppose C is a category with all small colimits andlimits. Suppose W is a subcategory of C , and I and J are sets of morphisms of C . Then L. MOSER, M. SARAZOLA, AND P. VERDUGO there is a cofibrantly generated model structure on C with I as the set of generating cofi-brations, J as the set of generating trivial cofibrations, and W as the subcategory of weakequivalences if and only if the following conditions are satisfied.(1) the subcategory W has the 2-out-of-3 property and is closed under retracts,(2) the domains of I are small relative to I -cell,(3) the domains of J are small relative to J -cell,(4) J -cell ⊆ W ∩ I -cof,(5) I -inj ⊆ W ∩ J -inj,(6) either W ∩ I -cof ⊆ J -cof, or
W ∩ J -inj ⊆ I -inj.
We now introduce our sets of generating cofibrations ( I ) and generating trivial cofibra-tions ( J ). Definition 2.10.
Let S be the double category containing a square, δ S be its boundary,and S be the double category containing two squares with the same boundaries.0 10 ′ ′ ; S = • • α ′ ′ ; δ S = • • ′ ′ . S = • • α α We define I to be the set containing the following double functors: • the unique morphism I : ∅ → , • the inclusion I : ⊔ → H , • the inclusion I : ⊔ → V , • the inclusion I : δ S → S , • the double functor I : S → S sending both squares in S to the non-trivial squareof S . Remark . In the model structure of [13], the double functor I is not a cofibration.Adding this as a generating cofibration makes the class of cofibrations symmetric withrespect to the horizontal and vertical directions, which was not the case for the modelstructure of [13]. Definition 2.12.
Let W be the double category consisting of a weakly horizontally invert-ible square with horizontal adjoint equivalence data, and W − be its double subcategorywhere we remove one of the vertical boundaries.0 10 ′ ′ ; W = ≃≃ • • ≃ ′ ′ . W − = ≃≃ • We define J to be the set containing the following double functors: • the inclusion J : → H E adj , where E adj is the 2-category containing an adjointequivalence, MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 9 • the inclusion J : H → H C inv , where C inv is the 2-category containing an invertible2-cell between parallel morphisms, and • the inclusion J : W − → W . Remark . For comparison, in the model structure of [13], the double functor J is nota cofibration. The addition of J as a (trivial) cofibration is what identifies the weaklyhorizontally invariant double categories as our class of fibrant objects, as we show in The-orem 2.20.The rest of this section is mostly devoted to proving the existence of a model structureon DblCat, cofibrantly generated by the sets I and J defined above, and with the doublebiequivalences as its class of weak equivalences. Before applying Hovey’s result, however,we provide a more explicit description of the classes I -inj, I -cof, and J -inj, for the case ofour sets I and J . These classes of morphisms will be, respectively, the trivial fibrations,cofibrations, and fibrations of our model structure. Proposition 2.14.
A double functor is in I -inj (i.e., a trivial fibration in our proposedmodel structure) if and only if it is surjective on objects, full on horizontal morphisms, fullon vertical morphisms, and fully faithful on squares.Proof. This is given by the right lifting property with respect to each generating cofibrationin I . (cid:3) Recall that there exists a functor U H : DblCat → Cat (resp. U V ), which sends a doublecategory to its underlying category of objects and horizontal (resp. vertical) morphisms. Proposition 2.15.
A double functor F : A → B is in I -cof (i.e., a cofibration in ourproposed model structure) if and only if the underlying functors U H F and U V F have theleft lifting property in Cat with respect to surjective on objects and full functors.Proof.
The proof proceeds just as [13, Proposition 4.7], with the evident modification forthe functor U V . (cid:3) Studying more closely what this lifting property means in Cat, one obtains the following.
Corollary 2.16.
A double functor F : A → B is in I -cof (i.e., a cofibration in our proposedmodel structure) 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 from theimage of U H A under U H F by freely adjoining objects and then morphisms betweenobjects, and(iii) the vertical underlying category U V B is a retract of a category obtained from theimage of U V A under U V F by freely adjoining objects and then morphisms betweenobjects. By applying this corollary to the unique double functor ∅ → A , we get the followingcharacterization of cofibrant objects in DblCat. Corollary 2.17.
A double category A is cofibrant if and only if its underlying horizontalcategory U H A and its underlying vertical category U V A are free. Finally, we characterize the morphisms in J -inj. Proposition 2.18.
A double functor F : A → B is in J -inj (i.e., a fibration in our proposedmodel structure) if and only if(i) for every horizontal equivalence b : B ≃ −→ F C in B , there exists a : A ≃ −→ C in A such that F a = b ,(ii) for every vertically invertible square β in B as below left, there exists a verticallyinvertible square α in A as below right such that F α = β , F A F CF A F C bF c • • β ∼ = A CA C ac • • α ∼ = (iii) for every diagram in A as below left, together with a weakly horizontally invertiblesquare β in B as below center, there exists a weakly horizontally invertible square α in A as below right such that F α = β . A CA ′ C ′ a ≃ a ′ ≃ • u ′ F A F CF A ′ F C ′ • v • F u ′ F a ≃ F a ′ ≃ β ≃ A CA ′ C ′ • u • u ′ a ≃ a ′ ≃ α ≃ Proof.
This is given by the right lifting property with respect to each generating trivialcofibration in J ; see Definition 2.12. (cid:3) We can now prove our first main result, establishing the existence of the desired modelstructure on DblCat.
Theorem 2.19.
There exists a cofibrantly generated model structure on
DblCat , whoseweak equivalences are the double biequivalences, and where sets of generating cofibrationsand generating trivial cofibrations are given by the sets I and J as in Definitions 2.10and 2.12.Proof. Let W denote the subcategory of DblCat whose morphisms are the double biequiv-alences; we shall prove that W , together with the sets I and J from Definitions 2.10and 2.12, satisfy the conditions of Theorem 2.9.First, recall that W is the class of weak equivalences of the model structure on DblCat of[13], and so it has the 2-out-of-3 property and is closed under retracts. Also, the categoryDblCat is locally presentable, and thus all objects are small.In order to show that J -cell ⊆ W ∩ I -cof, we can first note from Corollary 2.16 that J ⊆ I -cof. Since the class I -cof is defined through a left lifting property, it must be closedunder pushouts and transfinite compositions; then, J -cell ⊆ I -cof. MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 11
We now show that J -cell ⊆ W . Since all objects in DblCat are small, we can apply[9, Corollary 7.4.2] to the model structure of [13] and conclude that W is closed undertransfinite composition. It remains to show that pushouts of morphisms in J are in W .As J and J are trivial cofibrations in the model structure of [13], pushouts of these arein particular double biequivalences, and so it suffices to check the case of the morphism J .Let us then consider a pushout diagram ′ ′ ≃ f • v ≃ g ′ ′ • v • u ≃ f ≃ gα ≃ J AP . FF ′ I p The double category P is obtained from the double category A by freely adding the verticalmorphism F ′ u : F F ′ , and a weakly horizontally invertible square F ′ α : ( F ′ u F fF g
F v )together with its weak inverse F ′ α ′ . Then, the double functor I : A → P is the identity onobjects and on horizontal morphisms, and conditions (db1) and (db2) of Definition 2.7 aretrivially satisfied. For (db3), it suffices to show that it holds for F ′ u , which is immediatefrom the construction of P . Finally, for (db4), we can see that the double functor I is theidentity on squares, and so it is enough to show that the fullness condition holds for thesquare below left. However, since F ′ α and F ′ α ′ are weak inverses, their composite is equalto the pasting below right, which is made out of squares in A . F F F F ′ F ′ F ′ F f ′ F fF g ′ F g • F v • F ′ u • F vF ′ α ′ F ′ α F ′ = F ′ F F F F F F ′ F ′ F ′ • •• •• F v • F vF f ′ F fF g ′ F ge Fv ǫ Ff ∼ = ǫ − Fg ∼ = It remains to show that I -inj = W ∩ J -inj, which amounts to proving that a morphismis a trivial fibration if and only if it is a weak equivalence and a fibration. Since morphisms in I -inj are surjective on objects, full on horizontal and vertical morphisms, and fullyfaithful on squares by Proposition 2.14, they are also double biequivalences. Moreover,as we show above, we have that J ⊆ I -cof and therefore I -inj ⊆ J -inj. Finally, we showthat a morphism F ∈ W ∩ J -inj must belong to I -inj by verifying the conditions inProposition 2.14. To see that F is surjective on objects, let B ∈ B ; we know by (db1) inDefinition 2.7 that there exist A ∈ A and a horizontal equivalence b : B ≃ −→ F A . Then, by( i ) of Proposition 2.18, there exists a horizontal morphism a : C ≃ −→ A such that F a = b ;in particular, F C = B .To check F is full on horizontal morphisms, let b : F A → F C be one such morphism in B .By (db2), there exist a horizontal morphism c : A → C in A , and a vertically invertiblesquare β in B as below left. Then, condition ( ii ) in Proposition 2.18 gives a verticallyinvertible square α in A as below right such that F α = β ; in particular, F a = b . F A F CF A F C bF c • • β ∼ = A CA C ac • • α ∼ = To show that F is full on vertical morphisms, let v : F A F A ′ be one such morphismin B . By (db3), there exist a vertical morphism u ′ : C ′ C ′ and a weakly horizontallyinvertible square β in B as below left. By (db2), there exist horizontal morphisms a : A → C and a ′ : A ′ → C ′ in A , together with vertically invertible squares γ and γ ′ as displayed belowright. F A F CF A ′ F C ′ • v • F u ′ b ≃ b ′ ≃ β ≃ F A F CF A F CF A ′ F C ′ F A ′ F C ′ • v • F u ′ b ≃ b ′ ≃ F aF a ′ • •• • β ≃ γ ∼ = γ ′ ∼ = Then
F a and
F a ′ are horizontal equivalences since b and b ′ are, and thus so are a and a ′ ,as we will see in Lemma 5.11. Furthermore, by [12, Lemma A.2.1], we have that γ and γ ′ are weakly horizontally invertible. If we denote the pasting above right by δ , condition( iii ) of Proposition 2.18 gives a weakly horizontally invertible square α : ( u aa ′ u ′ ) in A suchthat F α = δ ; in particular, F u = v . This concludes our proof, as the final condition, fullyfaithfulness on squares, is precisely (db4). (cid:3) MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 13
As opposed to the case for the model structure in [13], not all double categories arefibrant here. However, our class of fibrant objects captures an important class of doublecategories, as we now show.
Theorem 2.20.
A double category A is fibrant in the model structure of Theorem 2.19 ifand only if it is weakly horizontally invariant.Proof. One needs to check the right lifting property of A → with respect to each trivialcofibration in J . Every double functor A → satisfies the right lifting property withrespect to J and J , since one can lift to the horizontal identity at an object in A and tothe vertical identity at a horizontal morphism in A . Then A → lifts against J if andonly if it is weakly horizontally invariant, by definition. (cid:3) We conclude this section with a comparison between the model structure on DblCatconstructed in Theorem 2.19, and the one in [13].
Theorem 2.21.
The identity adjunction
DblCat DblCat whi idid ⊥ is a Quillen equivalence, where DblCat denotes the model structure of [13] , and
DblCat whi denotes the model structure for weakly horizontally invariant double categories of Theo-rem 2.19.Proof.
We know that the left adjoint id preserves weak equivalences, as these are the samein both model structures. To see that it preserves cofibrations, it suffices to check thatdoes so for the generating cofibrations in DblCat, given in [13, Notation 4.2]. Since most ofthese are also generating cofibrations in DblCat whi , it is enough to check that ∅ → V is acofibration in DblCat whi , which is immediate from the description in Proposition 2.15. Thisshows that id ⊣ id is a Quillen adjunction; it is then clear that it is a Quillen equivalence,as both model structures have the same weak equivalences. (cid:3) The right Quillen functor H ≃ : 2Cat → DblCatIn this section we investigate the interaction between our model structure on DblCat andLack’s model structure on 2Cat constructed in [10], by looking at the full embedding functor H ≃ : 2Cat → DblCat, a modification of the horizontal full embedding H . We show that H ≃ is a right Quillen functor which is homotopically fully faithful; thus, the homotopy theoryof 2-categories is embedded in that of double categories in a reflective way. Moreover,Lack’s model structure is right-induced from the one introduced in Section 2; this showsthat Lack’s model structure is completely determined by the model structure on DblCat.We also compare the behavior of H and H ≃ with respect to this new model structure,and conclude this comparison by showing that for any 2-category A , the double category H ≃ A gives a fibrant replacement of H A . Definition 3.1.
The homotopical horizontal embedding is defined as the functor H ≃ : 2Cat → DblCat, which sends a 2-category A to the double category H ≃ A having thesame objects as A , the morphisms of A as horizontal morphisms, one vertical morphism( u, u ♯ , η, ǫ ) for each equivalence u in A and each choice of adjoint equivalence data, andsquares A CA ′ C ′ ac ( u, u ♯ , η, ǫ ) ≃ ( u ′ , u ′ ♯ , η ′ , ǫ ′ ) ≃ α given by the 2-cells α : u ′ a ⇒ cu in A . Although a vertical morphism always contains thewhole data of an adjoint equivalence, we often denote it by its left adjoint u . Remark . Note that every vertical morphism in the double category H ≃ A is a verticalequivalence , i.e., an equivalence in the underlying vertical 2-category. Remark . Given any 2-category A , it is possible to show that weakly horizontally in-vertible squares α : ( u ac u ′ ) in H ≃ A correspond to invertible 2-cells α : u ′ a ⇒ cu in A ,where a and c are equivalences in A (for details, see [12, Lemma A.2.3]). In particular, ifwe are given a boundary A CA ′ C ′ a ≃ c ≃ u ′ ≃ then there is an equivalence u : A ≃ −→ A ′ in A and an invertible 2-cell α : u ′ a ∼ = cu . Thisshows that H ≃ A is weakly horizontally invariant. Proposition 3.4.
The functor H ≃ is part of an adjunction DblCat 2Cat . L ≃ H ≃ ⊥ Proof.
In order to define the left adjoint L ≃ , first observe that any double category can beexpressed as a colimit involving only the double categories , H , V , and H × V , wherethe latter is the free double category on a square. Thus, it suffices to define L ≃ on thesefour double categories, as we can then set L ≃ A = colim i ∈ I L ≃ A i , for a double category A = colim i ∈ I A i with A i ∈ { , H , V , H × V } for every i ∈ I .Let L ≃ ( ) = , L ≃ ( H ) = , L ≃ ( V ) = E adj , and L ≃ ( H × V ) = A , where E adj isthe 2-category containing an adjoint equivalence, and A is the 2-category generated by thefollowing morphisms, adjoint equivalences, and 2-cell MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 15
A CA ′ C ′ . ≃ ≃ α One can then see that, for A i ∈ { , H , V , H × V } , we have2Cat( L ≃ A i , B ) ∼ = DblCat( A i , H ≃ B ) . (cid:3) Remark . One can show that L ≃ admits the following, more explicit, description. Givena double category A , L ≃ A is the 2-category with the same objects as A , a morphism foreach horizontal morphism in A , and a morphism for each vertical morphism in A whichwe formally make into an adjoint equivalence; i.e., we also add a formal inverse morphism,and the two necessary invertible 2-cells. Aside from these formal 2-cells added to createthe adjoint equivalences, we also have a 2-cell u ′ a ⇒ cu for each square in A of the form α : ( u ac u ′ ).As expected, the adjunction L ≃ ⊣ H ≃ is compatible with the model structures consid-ered; moreover, we show that the functor H ≃ is homotopically fully faithful. Theorem 3.6.
The adjunction
DblCat 2Cat L ≃ H ≃ ⊥ is a Quillen pair between Lack’s model structure and the model structure of Theorem 2.19.Furthermore, the counit and the derived counit are level-wise biequivalences, and thus H ≃ is homotopically fully faithful.Proof. We show that H ≃ is right Quillen.Suppose that F : A → B is a fibration in 2Cat. We need to show conditions ( i )-( iii ) ofProposition 2.18 for H ≃ F . First note that ( i ) and ( ii ) are satisfied by definition of F beinga fibration in 2Cat. It remains to show ( iii ). Consider a diagram in H ≃ A as below left,together with a weakly horizontally invertible square β in H ≃ B , i.e., an invertible 2-cell byRemark 3.3, as depicted below right. A CA ′ C ′ a ≃ c ≃ u ′ ≃ F A F CF A ′ F C ′ F a ≃ F c ≃ v ≃ F u ′ ≃ β ∼ = Let ( c ′ , c, η, ǫ ) be an adjoint equivalence data for c in A . We form the following pastingin B . F A F CF A ′ F C ′ F a ≃ F cv ≃ F u ′ ≃ F A ′ F c ′ ≃ β ∼ = F ǫ ∼ = Since F is a fibration in 2Cat, there exists an equivalence u : A ≃ −→ A ′ in A and an invertible2-cell α : c ′ u ′ a ∼ = u such that F u = v and F α = (
F ǫ ∗ v )( F c ′ ∗ β ). We set α to be thefollowing pasting. A CC ′ a ≃ C ′ A ′ u ′ ≃ u ≃ c ≃ c ′ α ∼ = η ∼ = Then, by the triangle identities for η and ǫ , we get that F α = β . This shows ( iii ) andproves that H ≃ F is a fibration in DblCat.Now suppose that F : A → B is a trivial fibration in 2Cat; by definition, we directlysee that H ≃ F is surjective on objects, full on horizontal morphisms, and fully faithful onsquares. Fullness on vertical morphisms for H ≃ F follows from the fact that a lift of anadjoint equivalence by a biequivalence is also an adjoint equivalence. This shows that H ≃ F is a trivial fibration in DblCat.It remains to show the claims regarding the (derived) counit. Let A be a 2-category. Itis not hard to see that the counit ǫ A : L ≃ H ≃ A → A is in fact a trivial fibration, using thedescription of H ≃ in Definition 3.1 and of L ≃ in Remark 3.5. As for the derived counit L ≃ ( H ≃ A ) cof L ≃ Q H ≃A −−−−−−→ L ≃ H ≃ A ≃ −→ ǫ A A , we now check that the following diagram commutes, L ≃ ( H ≃ A ) cof L ≃ H ≃ A ( L ≃ H ≃ A ) cof L ≃ Q H ≃ A Q L ≃ H ≃ A ≃ MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 17 where Q denotes the cofibrant replacements in 2Cat and DblCat. As a cofibrant doublecategory is characterized by having free underlying horizontal and vertical categories byCorollary 2.17, a cofibrant replacement of H ≃ A in DblCat can be built as follows: it hasthe same objects as H ≃ A ; there is a copy f for each morphism f in A and horizontalmorphisms are given by strings of f ’s; there is a copy u for each adjoint equivalence in A and vertical morphisms are given by strings of u ’s; and squares are given by squares of H ≃ A whose boundaries are the composites in H ≃ A of the strings. Then, applying L ≃ tothis cofibrant replacement gives the cofibrant replacement of L ≃ H ≃ A in 2Cat as describedby Lack in the proof of [10, Proposition 4.2]. We conclude that L ≃ Q H ≃ A is a biequivalencesince Q L ≃ H ≃ A is. (cid:3) Corollary 3.7.
The adjunction
DblCat 2Cat , L ≃ H ≃ ⊥ is a Quillen pair between Lack’s model structure on and the model structure on DblCat of [13] , whose derived counit is level-wise a biequivalence.Proof. The proof is obtained by composing the Quillen equivalence of Theorem 2.21 withthe Quillen pair of Theorem 3.6. (cid:3)
We saw that the derived counit of the adjunction L ≃ ⊣ H ≃ is level-wise a biequivalence.However, this adjunction is not expected to be a Quillen equivalence, since the homotopytheory of double categories should be richer than that of 2-categories. This is indeed thecase, as shown in the following remark. Remark . The components of the (derived) unit of the adjunction L ≃ ⊣ H ≃ are notdouble biequivalences. Indeed, since every 2-category is fibrant, we know that the counitand the derived counit agree on cofibrant double categories. Then, if we consider thecomponent η V : V → H ≃ L ≃ V of the unit at the cofibrant double category V , we seethat H ≃ L ≃ V has non-identity horizontal morphisms, given by the adjoint equivalencecreated by L ≃ from the unique vertical morphism of V , while V does not. Therefore η V cannot satisfy (db2).Nevertheless, the adjunction L ≃ ⊣ H ≃ exhibits a strong connection between Lack’smodel structure on 2Cat and our model structure on DblCat; indeed, the model structureon 2Cat is completely determined by the model structure on DblCat and the functor H ≃ . Theorem 3.9.
Lack’s model structure on is right-induced along the adjunction
DblCat 2Cat L ≃ H ≃ ⊥ from the model structure on DblCat of Theorem 2.19.
Proof.
We need to show that, given a 2-functor F : A → B , then F is a fibration (resp.biequivalence) in 2Cat if and only if H ≃ F is a fibration (resp. double biequivalence) inDblCat. Since H ≃ is right Quillen, we know it preserves fibrations and trivial fibrations.Moreover, since all 2-categories are fibrant, by Ken Brown’s Lemma (see [9, Lemma 1.1.12]),the functor H ≃ preserves all weak equivalences. Therefore, if F is a fibration (resp. biequiv-alence), then H ≃ F is a fibration (resp. double biequivalence).Now suppose that H ≃ F is a fibration in DblCat. Then conditions ( i ) and ( ii ) of Propo-sition 2.18 say that F is a fibration in 2Cat.Finally, suppose that H ≃ F is a double biequivalence. By (db1) and (db2) of Defi-nition 2.7, we have that F is bi-essentially surjective on objects and essentially full onmorphisms. Fully faithfulness on 2-cells follows from applying (db4) of Definition 2.7 to asquare with trivial vertical boundaries. This shows that F is a biequivalence. (cid:3) Remark . Given the similarities between the results obtained with the functors H ≃ here and H in [13] (notably, [13, Theorems 6.4 and 6.7]), one could ask whether H ≃ is bothright and left Quillen, as is H . We answer this question negatively by showing that thefunctor H ≃ is not a left adjoint.Indeed, consider the 2-category E adj , containing the data of an adjoint equivalence, i.e.,two opposite morphisms f and g and two invertible 2-cells η and ǫ satisfying the triangleidentities. One can obtain E adj as the pushout of the span B ← A → C , where A is the2-category containing f, g, η and ǫ but no relations between them, and B and C each containonly one of the triangle identities. One can see that the images of A , B , and C under H ≃ do not have non-trivial vertical morphisms since they do not contain adjoint equivalences,whereas H ≃ E adj does. This reveals that H ≃ does not preserve pushouts.Continuing in the spirit of comparing the behavior of the functors H and H ≃ , we observethe following. Corollary 3.11.
The adjunction H H ⊥ is a Quillen pair between Lack’s model structure on and the model structure on DblCat of Theorem 2.19, whose derived counit is level-wise a biequivalence.Proof.
The proof is obtained by composing the Quillen pair of [13, Theorem 6.2] and theQuillen equivalence of Theorem 2.21. (cid:3)
Remark . Note that the functor H is not right Quillen (as opposed to the case whereDblCat is endowed with the model structure of [13]; see [13, Theorem 6.4]), since thedouble category H A given by a 2-category A is generally not fibrant in the model structureof Theorem 2.19. Indeed, whenever the 2-category A contains an equivalence, its associatedhorizontal double category H A cannot be weakly horizontally invariant. To illustrate this,consider the 2-category E adj containing an adjoint equivalence. Then the diagram MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 19 ≃ • cannot be completed to a weakly horizontally invertible square in H E adj .As noted above, most double categories of the form H A are not weakly horizontallyinvariant. The next result says that H ≃ A provides a fibrant replacement of H A . Proposition 3.13.
Let A be a -category. Then the inclusion R A : H A → H ≃ A is a doublebiequivalence. In particular, this exhibits H ≃ A as a fibrant replacement of H A in the modelstructure on DblCat for weakly horizontally invariant double categories of Theorem 2.19.Proof.
We show (db1-4) of Definition 2.7. The inclusion R A : H A → H ≃ A is the identityon underlying horizontal categories, so that we directly have (db1) and (db2). Let u be avertical morphism in H ≃ A , i.e., an adjoint equivalence u : A ≃ −→ A ′ . Then the square A A ′ A ′ A ′ u ≃ u ≃ R A id A ′ id u ∼ = is weakly horizontally invertible in H ≃ A by Remark 3.3, which shows (db3). Finally, fullyfaithfulness on squares of (db4) follows from the fact that boundaries in the image of R A must have trivial vertical morphisms, and therefore a square with this boundary must comefrom a square in H A .This implies that H ≃ A is a fibrant replacement of H A , using the fact that H ≃ A is weaklyhorizontally invariant by Remark 3.3, and that these are precisely the fibrant objects inDblCat, as seen in Theorem 2.20. (cid:3) Compatibility with the monoidal structure
In this section, we consider the monoidal structure on DblCat given by the Gray tensorproduct, and show that it is compatible with our model structure; in other words, we provethat we have a monoidal model category . The structure of the proof mirrors that of Lack’sin [10, § Proposition 4.1 ([1, § . There is a closed symmetric monoidal structure on
DblCat given by the Gray tensor product ⊗ Gr : DblCat × DblCat → DblCat such that, for all double categories A , B , and C , there is an isomorphism DblCat( A ⊗ Gr B , C ) ∼ = DblCat( A , [ B , C ] ps ) , natural in A , B and C , where [ B , C ] ps is the double category of double functors from B to C , horizontal pseudo natural transformations, vertical pseudo natural transformations, andmodifications. Notation 4.2.
Let i : A → B and j : A ′ → B ′ be double functors. We write i (cid:3) j for theirpushout-product i (cid:3) j : A ⊗ Gr B ′ a A ⊗ Gr A ′ B ⊗ Gr A ′ → B ⊗ Gr B ′ with respect to the Gray tensor product ⊗ Gr on DblCat.Explicitly, following [1, § A ⊗ Gr B of the double categories A and B can be described as the double category given by the following data. • The objects are pairs (
A, B ) of objects A ∈ A and B ∈ B . • The horizontal morphisms are of two kinds: pairs (
A, b ) : (
A, B ) → ( A, D ) where A is an object in A and b : B → D is a horizontal morphism in B , and pairs( a, B ) : ( A, B ) → ( C, B ) where B is an object in B and a : A → C is a horizontalmorphism in A . • Similarly, the vertical morphisms are given by pairs (
A, v ) and ( u, B ) with A and B being objects in A and B respectively, and u and v vertical morphisms in A and B respectively. • There are six kinds of squares: the ones determined by an object B ∈ B and asquare α : ( u ac u ′ ) in A as shown below left, the ones given by an object A ∈ A anda square β : ( v bd v ′ ) in B as below right,( A, B ) (
C, B )( A ′ , B ) ( C ′ , B ) ( a, B )( c, B ) • ( u, B ) • ( u ′ , B )( α, B ) ( A, B ) (
A, D )( A, B ′ ) ( A, D ′ ) ( A, b )( A, d ) • ( A, v ) • ( A, v ′ )( A, β ) the squares determined by a horizontal morphism b in B and a vertical morphism u in A as displayed below left, and the ones given by a horizontal morphism a in A and a vertical morphism v in B as below right,( A, B ) (
A, D )( A ′ , B ) ( A ′ , D ) ( A, b )( A ′ , b ) • ( u, B ) • ( u, D )( u, b ) ( A, B ) (
C, B )( A, B ′ ) ( C, B ′ ) ( a, B )( a, B ′ ) • ( A, v ) • ( C, v )( a, v ) vertically invertible squares determined by horizontal morphisms a in A and b in B , as shown below, MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 21 ( A, B ) (
C, B ) (
C, D )( A, B ) (
A, D ) (
C, D ) ( a, B ) ( C, b )( A, b ) ( a, D ) • • ( a, b ) ∼ = and horizontally invertible squares, given by vertical morphisms u in A and v in B ,as below. ( A, B )( A, B ′ )( A ′ , B ′ ) ( A, B )( A ′ , B )( A ′ , B ′ ) • ( A, v ) • ( u, B ′ ) • ( u, B ) • ( A ′ , v ) ∼ =( u, v ) We also add all the possible compositions of the data above, subject to the associativityand unitality conditions, the middle four interchange law, and equations analogous to [1,Diagrams (2.6)-(2.8)].In order to establish the monoidality of our model structure, we need the following result,analogous to [10, Lemma 7.4].
Lemma 4.3.
Let F : A → B be a double biequivalence and X be a double category. Thenthe double functor id X ⊗ Gr F : X ⊗ Gr A → X ⊗ Gr B is a double biequivalence.Proof. We show that conditions (db1-4) of Definition 2.7 hold.For (db1), given an object (
X, B ) ∈ X ⊗ Gr B , we can use (db1) for F and obtain an object A ∈ A and a horizontal equivalence b : B ≃ −→ F A in B . Taking the pair ( X, A ) togetherwith the horizontal equivalence (
X, b ) : (
X, B ) ≃ −→ ( X, F A ) we obtain (db1) for id X ⊗ Gr F .We proceed with (db2). Given a horizontal morphism ( X, b ) : (
X, F A ) → ( X, F C ),the required condition is satisfied by the horizontal morphism (
X, a ) : (
X, A ) → ( X, C ) in X ⊗ Gr A together with the vertically invertible square ( X, ψ ) in X ⊗ Gr B ( X, F A ) (
X, F C )( X, F A ) (
X, F C ) ( X, b )( X, F a ) • • ( X, ψ ) ∼ = determined by the horizontal morphism a : A → C in A and the vertically invertible square ψ in B obtained by (db2) of F .If we start with a horizontal morphism of the form ( x, F A ) : ( X, F A ) → ( Y, F A ), it isenough to observe that ( x, F A ) = (id X ⊗ Gr F )( x, A ). We now prove (db3). Again, we will divide the proof in two cases, depending on the typeof vertical morphism we have. For a vertical morphism (
X, v ) : (
X, B ) (
X, B ′ ) in X ⊗ Gr B ,the required condition is satisfied by the vertical morphism ( X, u ) : (
X, A ) (
X, A ′ ) in X ⊗ Gr A and the weakly horizontally invertible square ( X, ϕ ) in X ⊗ Gr B ( X, B ) (
X, F A )( X, B ′ ) ( X, F A ′ ) ≃≃ • ( X, v ) • ( X, F u )( X, ϕ ) ≃ determined by the vertical morphism u : A A ′ in A and the weakly horizontally invertiblesquare ϕ in B obtained by (db3) of F .When we start with a vertical morphism in X ⊗ Gr B of the form ( w, B ) : ( X, B ) ( X ′ , B ),we use (db1) for F and obtain an object A ∈ A together with a horizontal equivalence b : B ≃ −→ F A in B . We conclude the proof of (db3) by considering the vertical morphism( w, A ) : ( X, A ) ( X ′ , A ) in X ⊗ Gr A together with the square ( w, b ) in X ⊗ Gr B as depictedbelow, which can be checked to be weakly horizontally invertible.( X, B ) (
X, F A )( X ′ , B ) ( X ′ , F A ) ( X, b ) ≃ ( X ′ , b ) ≃ • ( w, B ) • ( w, F A )( w, b ) ≃ Finally, we show (db4). For this we need to show that for any square δ in X ⊗ Gr B whoseboundary is in the image of (id X ⊗ Gr F ), there is a unique square γ in X ⊗ Gr A filling thepreimage of the boundary such that (id X ⊗ Gr F ) γ = δ . First, let δ be of the form ( χ, F A )for a square χ in X ; then γ is given by ( χ, A ). If δ is of the form ( X, β ) for a square β in B whose boundary is in the image of F , then γ is given by ( X, α ), where α is the uniquesquare in A such that F α = β given by (db4) of F . If δ is of the form ( w, F a ) for a verticalmorphism w in X and a horizontal morphism a in A , then γ is given by ( w, a ). Similarly, if δ is of the form ( x, F u ) for a horizontal morphism x in X and a vertical morphism u in A ,then γ is given by ( x, u ). Finally, if δ is of the form( X, F A ) (
Y, F A ) (
Y, F C )( X, F A ) (
X, F C ) (
Y, F C ) ( x, F A ) ( Y, F a )( X, F a ) ( x, F C ) • • ( x, F a ) ∼ = MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 23 for horizontal morphisms x in X and a in A , then γ is given by the square ( x, a ); similarly,if δ is of the form ( w, F u ) for vertical morphisms w in X and u in A , then γ is given by( w, u ). (cid:3) We now proceed to show the main result of this section.
Theorem 4.4.
The model structure on
DblCat of Theorem 2.19 is monoidal with respectto the Gray tensor product.Proof.
We begin by showing that whenever i and j are cofibrations, the pushout-product i (cid:3) j is also a cofibration; it is enough to consider the cases when i, j ∈ { I , I , I , I , I } .Moreover, since the Gray tensor product is symmetric, if we show the result for i (cid:3) j , thenit is also true for j (cid:3) i . First note that I (cid:3) j ∼ = j , which proves the cases involving I .To show the cases involving I or I , we observe the following three facts: the functors U H , U V : DblCat → Cat preserve pushouts, since they are left adjoints (see [13, Remarks4.5 and 4.6]); U H ( I ) , U H ( I ) , U V ( I ) , and U V ( I ) are identities; and U H ( A ⊗ Gr B ) (resp. U V ( A ⊗ Gr B )) only depends on U H ( A ) and U H ( B ) (resp. U V ( A ) and U V ( B )). Thenit follows that i (cid:3) j is such that U H ( i (cid:3) j ) and U V ( i (cid:3) j ) are isomorphisms of categories,and thus i (cid:3) j is a cofibration by Proposition 2.15, if either i or j is in { I , I } .We now verify the cases I (cid:3) I , I (cid:3) I , and I (cid:3) I . One can check that I (cid:3) I is theboundary inclusion δ ( H ⊗ Gr H ) → H ⊗ Gr H , which is a cofibration by Corollary 2.16,since it is the identity on underlying horizontal and vertical categories. Similarly, one canshow that I (cid:3) I and I (cid:3) I are cofibrations, as they are given by the boundary inclusions δ ( V ⊗ Gr V ) → V ⊗ Gr V and δ ( H ⊗ Gr V ) → H ⊗ Gr V , respectively.It remains to show that if i and j are cofibrations and one of them is a double biequiv-alence, then i (cid:3) j is also a double biequivalence. Suppose that i : A → B is a cofibrationwhere A is cofibrant and assume without lost of generality that j is a trivial cofibration.Consider the pushout diagram below A ⊗ Gr C A ⊗ Gr DB ⊗ Gr C P B ⊗ Gr D . id A ⊗ Gr ji ⊗ Gr id C k i (cid:3) j id B ⊗ Gr j i ⊗ Gr id D p Since A is cofibrant and j is a cofibration, we know that ( ∅ → A ) (cid:3) j = id A ⊗ Gr j is alsoa cofibration by the above. Since j is a trivial cofibration by assumption, then the doublefunctor id A ⊗ Gr j is also a double biequivalence by Lemma 4.3. We then conclude thatid A ⊗ Gr j is a trivial cofibration, and therefore so is k since trivial cofibrations are stableunder pushout. Lemma 4.3 also guarantees that id B ⊗ Gr j is a double biequivalence, andthen by the 2-out-of-3 property we conclude that so is i (cid:3) j . Since all generating trivial cofibrations of Definition 2.12 have cofibrant source by Corollary 2.17, this proves thetheorem. (cid:3) Remark . In particular, we can restrict the Gray tensor product to 2Cat on the secondcoordinate via the embedding H : 2Cat → DblCat, and get a tensoring functor ⊗ : DblCat × → DblCatgiven by A ⊗ B = A ⊗ Gr H B . The category DblCat is enriched over 2Cat with hom 2-categories given by H [ − , − ] ps , and this enrichment is both tensored and cotensored, withtensors given by ⊗ (see [13, Proposition 7.6]). Since H is left Quillen by Corollary 3.11,Theorem 4.4 implies that the model structure on DblCat is a 2Cat-enriched model struc-ture. 5. A Whitehead Theorem for double categories
In this section, we show a Whitehead Theorem for double categories, which characterizesthe double biequivalences between fibrant objects as the double functors that admit apseudo inverse up to horizontal pseudo natural equivalence. Such a statement is reminiscentof the Whitehead Theorem for 2-categories: a 2-functor F : A → B is a biequivalence ifand only if there exists a pseudo functor G : B → A together with two pseudo naturalequivalences id A ≃ GF and F G ≃ id B . It therefore gives a more natural descriptionof our double biequivalences when restricting to the weakly horizontally invariant doublecategories.Our Whitehead Theorem can be seen as a 2-categorical version of Grandis’ statement [6,Theorem 4.4.5]. Under the hypothesis that the double categories involved are horizontallyinvariant – defined analogously to the weakly horizontally invariant double categories withequivalences replaced by the stronger notion of isomorphisms; see [6, Theorem and Defi-nition 4.1.7] –, Grandis characterizes the double functors F such that U H F and U V F areboth equivalences of categories as the ones which admit a pseudo inverse up to horizontalnatural isomorphism .In the theorem below, whose proof will be the content of this section, it is actuallyenough to require that the source be weakly horizontally invariant. Theorem 5.1 (Whitehead Theorem) . Let A and B be double categories such that A isweakly horizontally invariant. Then a double functor F : A → B is a double biequivalence ifand only if there exists a pseudo double functor G : B → A together with horizontal pseudonatural equivalences id A ≃ GF and F G ≃ id B . This retrieves a formulation of the usual Whitehead Theorem for model categories (see[3, Lemma 4.24]) in our setting; such a result characterizes the weak equivalences betweencofibrant-fibrant objects in a model structure as the homotopy equivalences.
Remark . Just as in [13, Lemma 5.10], one can check that the homotopy equivalencesbetween cofibrant-fibrant double categories in the model structure of Theorem 2.19 aregiven by the double functors F : A → B such that there exists a double functor G : B → A together with horizontal pseudo natural equivalences id A ≃ GF and F G ≃ id B . Indeed, a MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 25 path object for our model structure here can also given by [ H E adj , − ] ps ; see [13, Definition3.17 and Proposition 3.18] for details.Let us now introduce what we mean by a pseudo double functor. Definition 5.3. A pseudo double functor G : B → A consists of maps on objects, hor-izontal morphisms, vertical morphisms, and squares, compatible with sources and targets,which preserve(i) horizontal compositions and identities up to coherent vertically invertible squares GB GC GDGB GD
Gb GdG ( db ) • • Φ b,d ∼ = GB GBGB GB G id B • • Φ B ∼ = for all B ∈ B , and all composable horizontal morphisms b : B → C and d : C → D in B ,(ii) vertical compositions and identities up to coherent horizontally invertible squares GB GBGB ′ Gv • GB ′′ GB ′′ Gv ′ G ( v ′ v ) • • Ψ v,v ′ ∼ = GB GBGB GB G id B • • Ψ B ∼ = for all B ∈ B , and all composable vertical morphisms v : B B ′ and v ′ : B ′ B ′′ in B .For a detailed description of the coherences, the reader can see [6, Definition 3.5.1].The pseudo double functor G is said to be normal if the squares Φ B and Ψ B areidentities for all B ∈ B . Remark . There are notions of horizontal pseudo natural transformations between (nor-mal) pseudo double functors, and modifications between them (with trivial vertical bound-aries). These are defined analogously to [6, § §
3] under the name of pointwise equivalences . Definition 5.5.
Let
F, G : A → B be (normal) pseudo double functors. A horizontalpseudo natural equivalence h : F ⇒ G is an equivalence in the 2-category of (normal)pseudo double functors A → B , horizontal pseudo natural transformations, and modifica-tions with trivial vertical boundaries.Equivalently, the horizontal pseudo natural equivalences can be described as follows. Lemma 5.6.
Let
F, G : A → B be (normal) pseudo double functors. A horizontal pseudonatural transformation h : F ⇒ G is a horizontal pseudo natural equivalence if and only if(i) the horizontal morphism h A : F A ≃ −→ GA is a horizontal equivalence, for everyobject A ∈ A , and(ii) the square h u : ( F u h A h A ′ Gu ) is weakly horizontally invertible, for every verticalmorphism u : A A ′ in A .Proof. The proof is analogous to the one of [12, Lemma A.3.3]. Another proof can be foundin [7, Theorem 4.4]. (cid:3)
We now introduce a notion of horizontal biequivalence for a double functor which admitsa pseudo inverse up to horizontal pseudo natural equivalence.
Definition 5.7.
A double functor F : A → B is a horizontal biequivalence if thereexist a pseudo double functor G : B → A and horizontal pseudo natural equivalences η : id A ⇒ GF and ǫ : F G ⇒ id B . Remark . Let F : A → B be a double functor. If F is a horizontal biequivalence, itsdata ( G, η, ǫ ) can always be promoted to the following data:(i) a normal pseudo double functor G : B → A ,(ii) a horizontal pseudo natural adjoint equivalence( η : id A ⇒ GF, η ′ : GF ⇒ id A , λ : id ∼ = η ′ η, κ : ηη ′ ∼ = id) , where λ and κ satisfy the triangle identities,(iii) a horizontal pseudo natural adjoint equivalence( ǫ : F G ⇒ id B , ǫ ′ : id B ⇒ F G, µ : id ∼ = ǫ ′ ǫ, ν : ǫǫ ′ ∼ = id) , where µ and ν satisfy the triangle identities,(iv) two invertible modifications Θ : id F ∼ = ǫ F ◦ F η and Σ : id G ∼ = Gǫ ◦ η G , expressingthe triangle (pseudo-)identities for η and ǫ .This follows from the fact that a pseudo double functor can always be promoted to a normalone, and from a result by Gurski [8, Theorem 3.2], saying that a biequivalence can alwaysbe promoted to a biadjoint biequivalence.Theorem 5.1 now amounts to showing that a double functor whose source is weaklyhorizontally invariant is a double biequivalence if and only if it is a horizontal biequivalence.However, it is always true that a horizontal biequivalence is a double biequivalence; noadditional hypothesis is needed here. In order to prove this first result, we need thefollowing lemma. Lemma 5.9.
The data of Remark 5.8 induces an invertible modification θ : F η ′ ∼ = ǫ F .Proof. Given an object A ∈ A , we define the data of θ at A to be the vertically invertiblesquare MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 27
F GF A F AF GF A F A
F η ′ A ǫ FA • • θ A ∼ = F GF A F A F AF GF A F A F GF A F AF GF A F GF A F A .= F η ′ A F η ′ A F η A ǫ FA ǫ FA •• • •• • F κ A ∼ = e ǫ FA e Fη ′ A Θ A ∼ = The proof of horizontal and vertical coherences for θ is a standard check that stems fromthe constructions of the squares θ A and from the horizontal and vertical coherences of themodifications F κ : (
F η )( F η ′ ) ∼ = id and Θ : id ∼ = ǫ F ◦ F η . (cid:3) Proposition 5.10. If F : A → B is a horizontal biequivalence, then F is a double biequiv-alence.Proof. We proceed to check that F satisfies (db1-4) of Definition 2.7. Let ( F, G, η, ǫ ) bethe data of a horizontal adjoint biequivalence as in Remark 5.8.We first show (db1). For every object B ∈ B , we want to find an object A ∈ A and ahorizontal equivalence B ≃ −→ F A in B . Setting A = GB , we have that ǫ ′ B : B ≃ −→ F GB = F A gives such a horizontal equivalence.We now show (db2). Let b : F A → F C be a horizontal morphism in B . We want to finda horizontal morphism a : A → C in A and a vertically invertible square in B F A F CF A F C . bF a • • ∼ = Let a : A → C be the composite A GF A GF C C ; η A Gb η ′ C we then have a vertically invertible square as desired, F A F A F CF A F GF A F A F CF A F GF A F GF C F CF A F GF A F GF C F C bF η A ǫ FA bF η A F Gb ǫ FC F η A F Gb F η ′ C • • •• • •• • • Θ A ∼ = e b e Fη A ǫ b ∼ = e ( FGb )( Fη A ) θ − C ∼ = where θ C is the component at C of the invertible modification θ of Lemma 5.9.We now show (db3). Let v : B B ′ be a vertical morphism in B . We want to find avertical morphism u : A A ′ in A and a weakly horizontally invertible square in B B F AB ′ F A ′ . ≃≃ v F u • • ≃ Let u : A A ′ be the vertical morphism Gv : GB GB ′ . Then ǫ ′ v gives the desired weaklyhorizontally invertible square. B F GBB ′ F GB ′ ≃ ǫ ′ B ≃ ǫ ′ B ′ v F Gv • • ǫ ′ v ≃ We finally show (db4). Let β be a square in B of the form F A F CF A ′ F C ′ . F aF cF u F u ′ • • β We want to show that there exists a unique square α : ( u ac u ′ ) in A such that F α = β .Define α to be the square given by the following pasting. MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 29
A CA ′ C ′ acu u ′ • • α = A A C a A GF A A C η A η ′ A a • • • µ A ∼ = e a A GF A GF C C η A GF a η ′ C • • • η ′ a ∼ = e η A A ′ GF A ′ GF C ′ C ′ η A ′ GF c η ′ C ′ u GF u GF u ′ u ′ • • • • η u Gβ η ′ u ′ A ′ GF A ′ A ′ C ′ η A ′ η ′ A ′ c • • • η ′ c − ∼ = e η A ′ A ′ A ′ C ′ c • • • µ − A ′ ∼ = e c The thorough reader might check that
F α = β by completing the following steps. Firsttransform F η ′ u ′ by using the invertible modification θ : F η ′ ∼ = ǫ F of Lemma 5.9; then apply,in the given order: the horizontal coherence of the modification F ν : (
F η ′ )( F η ) ∼ = id, thehorizontal coherence of the modification Θ : id ∼ = ǫ F ◦ F η , the triangle identity for ( µ, ν ),the compatibility of ǫ F : F GF ⇒ F with F Gβ and β , and finally the horizontal coherenceof the modification Θ : id ∼ = ǫ F ◦ F η .Suppose now that α ′ : ( u ac u ′ ) is another square in A such that F α ′ = β . If we replace Gβ with GF α ′ in the pasting diagram above, then it follows from the compatibility of η ′ : GF ⇒ id A with GF α ′ and α ′ , and the vertical coherence of the modification µ : id ∼ = η ′ η ,that this pasting is also equal to α ′ . Therefore, we must have α = α ′ . This proves boththe existence and unicity required in (db4). (cid:3) The second part of this section is largely devoted to proving the direct implicationof Theorem 5.1. It is not true in general that a double biequivalence is a horizontalbiequivalence, unless we impose an additional condition on the source or on the target.Here, we require the source to be fibrant. In [13, Theorem 5.15] we provide anotherWhitehead Theorem, where the target satisfies a condition related to cofibrancy in themodel structure of [13].The following is a technical lemma, needed for the proof of the next proposition. It showsthat the lift along a double biequivalence of a horizontal equivalence is again a horizontalequivalence.
Lemma 5.11.
Let F : A → B be a double biequivalence, and let b : F A ≃ −→ F C be ahorizontal equivalence in B . Then any horizontal morphism a : A → C in A such that thereexists a vertically invertible square β in B as in the diagram below F A F CF A F C b ≃ F a • • β ∼ = is a horizontal equivalence in A .Proof. Let ( b, b ′ , η, ǫ ) be the data of a horizontal equivalence. By (db2) of Definition 2.7,there exists a horizontal morphism a ′ : C → A in A together with a vertically invertiblesquare β ′ in B . F C F AF C F A b ′ ≃ F a ′ • • β ′ ∼ = Let us denote by Λ η the pasting shown below right, and by Λ ǫ the pasting shown belowleft. F A F AF A F C F A b b ′ • • η ∼ = F A F C F A
F a F a ′ • • • β ∼ = β ′ ∼ = F C F CF C F A F C b ′ b • • ǫ ∼ = F C F A F C
F a F a ′ • • • β − ∼ = β ′− ∼ = By (db4) of Definition 2.7, there exist unique vertically invertible squares η and ǫ in A A C AA A a a ′ • • η ∼ = C A CC C a ′ a • • ǫ ∼ = such that F η = Λ η and F ǫ = Λ ǫ . This provides the data of a horizontal equivalence( a, a ′ , η, ǫ ). (cid:3) We now prove a converse of Proposition 5.10, under the additional assumption that oursource double category is weakly horizontally invariant.
Proposition 5.12.
Let F : A → B be a double biequivalence, where the double category A is weakly horizontally invariant. Then F is a horizontal biequivalence. MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 31
Proof.
We simultaneously define the pseudo double functor G : B → A and the horizontalpseudo natural transformation ǫ : F G ⇒ id B . G and ǫ on objects. Let B ∈ B be an object. By (db1) of Definition 2.7, there existan object A ∈ A and a horizontal equivalence b : F A ≃ −→ B in B . We set GB := A and ǫ B := b : F GB ≃ −→ B , and also fix a horizontal equivalence data ( ǫ B , ǫ ′ B , µ B , ν B ). G and ǫ on horizontal morphisms. Now let b : B → C be a horizontal morphismin B . By (db2) of Definition 2.7, there exist a horizontal morphism a : GB → GC in A anda vertically invertible square ǫ b as in F GB B C F GCF GB F GC . ǫ B b ǫ ′ C F a • • ǫ b ∼ = We set Gb := a : GB → GC and ǫ b to be the square given by the following pasting. F GB B CF GB F GC C ǫ B bF Gb ǫ C • • ǫ b ∼ = = F GB B C CF GB B C F GC C ǫ B b • • • e bǫ B ν − C ∼ = F GB F GC C ǫ B b ǫ ′ C ǫ C F Gb ǫ C • • • ǫ b ∼ = e ǫ C If b = id B , we can choose G id B := id GB and ǫ id B := µ − B . Then ǫ id B = e ǫ B by the triangleidentities for ( µ B , ν B ). Horizontal coherence.
Given horizontal morphisms b : B → C and d : C → D in B ,we define the vertically invertible comparison square between Gd ◦ Gb and G ( db ) as follows.Let us denote by Θ b,d the following pasting. F GB F GC F GD
F Gb F Gd
F GB B C F GC C D F GD ǫ B b ǫ ′ C ǫ C d ǫ ′ D • • • ǫ − b ∼ = ǫ − d ∼ = F GB B C C D F GD ǫ B b d ǫ ′ D • • • • e bǫ B e ǫ ′ D d ν C ∼ = F GB F GD
F G ( db ) • • ǫ db ∼ = Then, by (db4) of Definition 2.7, there exists a unique vertically invertible square
GB GC GDGB GD
Gb GdG ( db ) • • Φ b,d ∼ = such that F Φ b,d = Θ b,d . In particular, one can check that, with this definition of Φ b,d , thesquares ǫ b , ǫ d , and ǫ db satisfy the following pasting equality. B F GB F GC F GD ǫ B F Gb F Gd
B F GB F GD ǫ B F G ( db ) • • • F Φ b,d ∼ = e ǫ B B C D F GD b d ǫ D • • ǫ db ∼ = = B F GB F GC F GD ǫ B F Gb F Gd
B C F GC F GD b ǫ C F Gd • • • ǫ b ∼ = e FGd
B C D F GD b d ǫ D • • • ǫ d ∼ = e b G and ǫ on vertical morphisms. Now let v : B B ′ be a vertical morphism in B .By (db3) of Definition 2.7, there exist a vertical morphism u ′ : A A ′ and a weaklyhorizontally invertible square γ v as in B F AB ′ F A ′ . b ≃ d ≃ v F u ′ γ v ≃ • • MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 33
If we consider the horizontal equivalences bǫ B : F GB ≃ −→ F A and dǫ B ′ : F GB ′ ≃ −→ F A ′ , thereexist horizontal morphisms a : GB → A and c : GB ′ → A ′ in A and vertically invertiblesquares γ b and γ d as depicted below. F GB B F AF GB F A ǫ B ≃ ≃ bF aγ b ∼ = • • F GB ′ B ′ F A ′ F GB ′ F A ′ ǫ B ′ ≃ ≃ dF c • • γ d ∼ = By Lemma 5.11, we have that a : GB ≃ −→ A and c : GB ′ ≃ −→ A ′ are horizontal equiva-lences in A ; thus, since A is weakly horizontally invariant, there exist a vertical morphism u : GB GB ′ and a weakly horizontally invertible square GB AGB ′ A ′ . a ≃ c ≃ u u ′ α v ≃ • • We set Gv := u : GB GB ′ . To define the weakly horizontally invertible square ǫ v , letus first fix a weak inverse γ ′ v of γ v with respect to some horizontal equivalences ( b, b ′ , λ, κ )and ( d, d ′ , λ ′ , κ ′ ). We set ǫ v to be the square given by the following pasting. F GB BF GB ′ B ′ ǫ B ǫ B ′ F Gv vǫ v • • = F GB B B ǫ B F GB B F A B ǫ B b b ′ • • • e ǫ B λ ∼ = F GB F A B
F a b ′ • • • e b ′ γ b ∼ = F GB ′ F A ′ B ′ F Gv F u ′ vF c d ′ • • • γ ′ v ≃ F α v ≃ F GB ′ B ′ F A ′ B ′ ǫ B ′ d d ′ • • • e d ′ γ − d ∼ = F GB ′ B ′ B ′ ǫ B ′ • • • e ǫ B ′ λ ′− ∼ = Note that all the squares in the pasting are weakly horizontally invertible by [12, LemmaA.2.1], and thus so is ǫ v . We write ǫ ′ v for its unique weak inverse with respect to the horizontal adjoint equivalences ( ǫ B , ǫ ′ B , µ B , ν B ) and ( ǫ B ′ , ǫ ′ B ′ , µ B ′ , ν B ′ ), as given by [12,Lemma A.1.1].If v = e B , we can choose Ge B := e GB and γ e B := e ǫ B . Then α e B can be chosen to bethe identity square at the object GB and we get ǫ e B = e ǫ B . Vertical coherence.
Given vertical morphisms v : B B ′ and v ′ : B ′ B ′′ in B , wedefine the horizontally invertible comparison square between Gv ′ • Gv and G ( v ′ v ) as follows.Let us denote by Ω v,v ′ the following pasting. F GB F GBF GB B F GB ǫ B ǫ ′ B • • µ B ∼ = F GB ′ B ′ F Gv vǫ B ′ • • ǫ v F GB ′′ B ′′ F GB ′′ F Gv ′ v ′ F G ( v ′ v ) ǫ B ′′ ǫ ′ B ′′ • • • ǫ v ′ ǫ ′ v ′ v F GB ′′ F GB ′′ • • µ − B ′′ ∼ = Note that this square is horizontally invertible, since it is weakly horizontally invertible andits horizontal boundaries are identities. By (db4) of Definition 2.7, there exists a uniquehorizontally invertible square Ψ v,v ′ as depicted below left such that F Ψ v,v ′ = Ω v,v ′ . Inparticular, one can check that, with this definition of Ψ v,v ′ , the squares ǫ v , ǫ v ′ and ǫ v ′ v satisfy the pasting equality below right. GB GBGB ′ Gv • GB ′′ GB ′′ Gv ′ G ( v ′ v ) • • Ψ v,v ′ ∼ = F GB F GBF GB ′ F Gv • F GB ′′ F GB ′′ F Gv ′ F G ( v ′ v ) • • F Ψ v,v ′ ∼ = BB ′′ ǫ B ǫ B ′′ v ′ v • ǫ v ′ v = F GB B ǫ B F GB ′ B ′ F Gv vǫ B ′ • • ǫ v F GB ′′ B ′′ F Gv ′ v ′ ǫ B ′′ • • ǫ v ′ G on squares. Let β : ( v bd v ′ ) be a square in B . Let us denote by δ the followingpasting. MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 35
F GB F GC
F Gb
F GB B C F GC ǫ B b ǫ ′ C • • ǫ − b ∼ = F GB ′ B ′ C ′ F GC ′ F Gv v v ′ F Gv ′ ǫ B ′ d ǫ ′ C ′ • • • • ǫ v β ǫ ′ v ′ F GB ′ F GC ′ F Gd • • ǫ d ∼ = Then, by (db4) of Definition 2.7, there exists a unique square
GB GCGB ′ GC ′ GbGdGv Gv ′ α • • such that F α = δ . We set Gβ := α : ( Gv GbGd Gv ′ ).Let b : B → C be a horizontal morphism in B , and β = e b : ( e B bb e C ). Then we havethat δ = e F Gb , since ǫ e B = e ǫ B and ǫ ′ e C = e ǫ C , and the unique square α : ( e GB GbGb e GC ) suchthat F α = e F Gb is given by e Gb . Therefore, Ge b = e Gb .Now let v : B B ′ be a vertical morphism in B , and β = id v : ( v id B id B ′ v ). Then we havethat δ = id F Gv , since ǫ − B = µ B and ǫ id B ′ = µ − B ′ and ǫ ′ v is the weak inverse of ǫ B withrespect to the horizontal adjoint equivalence data ( ǫ B , ǫ ′ B , µ B , ν B ) and ( ǫ B ′ , ǫ ′ B ′ , µ B ′ , ν B ′ ).The unique square α : ( Gv id GB id GB ′ Gv ) such that F α = id
F Gv is given by id Gv . Therefore, G id v = id Gv . Naturality and adjointness of ǫ and ǫ ′ . The assignment of G on squares is naturalwith the data of ǫ B , ǫ b and ǫ v , and therefore the latter assemble into a horizontal pseudonatural equivalence ǫ : F G ⇒ id B . Moreover, since ( ǫ B , ǫ ′ B , µ B , ν B ) are horizontal adjointequivalences, the data of ǫ ′ B , ǫ ′ b and ǫ ′ v also assemble into a horizontal pseudo naturalequivalence ǫ ′ : id B ⇒ F G , where ǫ ′ b is defined in a similar manner as ǫ b was. In particular, ǫ : F G ⇒ id B and ǫ ′ : id B ⇒ F G are adjoint equivalences, where the invertible modificationsare given by µ : id ∼ = ǫ ′ ǫ and ν : ǫǫ ′ ∼ = id.It remains to define the horizontal pseudo natural equivalence η : id A ⇒ GF . For thispurpose, we use the horizontal pseudo natural equivalence ǫ ′ : id B ⇒ F G . η on objects. Let A ∈ A , and consider the horizontal equivalence ǫ ′ F A : F A ≃ −→ F GF A .By (db2) of Definition 2.7, there exist a horizontal morphism a : A → GF A and a verticallyinvertible square
F A F GF AF A F GF A . ǫ ′ FA F a • • ρ A ∼ = We set η A := a : A → GF A . Note that η A : A ≃ −→ GF A is a horizontal equivalence byLemma 5.11. η on horizontal morphisms. Let a : A → C be a horizontal morphism in A . Wedenote by ψ a the following pasting. F A F GF A F GF C
F η A F GF a
F A F GF A F GF C ǫ ′ FA F GF a • • • ρ − A ∼ = e FGFa
F A F C F GF C
F a ǫ ′ FC • • ǫ ′ Fa ∼ = F A F C F GF C
F a F η C • • • ρ C ∼ = e Fa By (db4) of Definition 2.7, there exists a unique vertically invertible square
A GF A GF CA C GF C η A GF aa η C • • α ∼ = such that F α = ψ a ; let η a := α . η on vertical morphisms. Let u : A A ′ be a vertical morphism in A . We denoteby ψ u the following pasting. MODEL STRUCTURE FOR WEAKLY HORIZONTALLY INVARIANT DOUBLE CATEGORIES 37
F A F GF AF A F GF A
F η A ǫ ′ FA • • ρ − A ∼ = F A ′ F GF A ′ ǫ ′ FA ′ F u F GF u • • ǫ ′ Fu ≃ F A ′ F GF A ′ F η A ′ • • ρ A ′ ∼ = Note that all the squares in the pasting are weakly horizontally invertible by [12, LemmaA.2.1], and thus so is ψ u . By (db4) of Definition 2.7, there exists a unique weakly horizon-tally invertible square A GF AA ′ GF A ′ η A η A ′ u GF uγ • • such that F γ = ψ u ; let η u := γ . Naturality of η . Since ǫ ′ : id B ⇒ F G is a horizontal pseudo natural transformation,then η A , η a , and η u assemble into a horizontal pseudo natural transformation η : id A ⇒ GF . Note that η is a horizontal pseudo natural equivalence, because η A are horizontalequivalences and η u are weakly horizontally invertible squares. Moreover, ρ : ǫ ′ F ∼ = F η gives the data of an invertible modification. (cid:3)
Proof of Theorem 5.1 (Whitehead Theorem).
This follows directly from Propositions 5.10and 5.12. (cid:3)
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