A 2Cat-inspired model structure for double categories
aa r X i v : . [ m a t h . A T ] M a y A 2CAT-INSPIRED MODEL STRUCTURE FOR DOUBLECATEGORIES
LYNE MOSER, MARU SARAZOLA, AND PAULA VERDUGO
Abstract.
We construct a model structure on the category DblCat of double cate-gories and double functors. Unlike previous model structures for double categories, itrecovers the homotopy theory of 2-categories through the horizontal embedding functor H : 2Cat → DblCat, which is both left and right Quillen, and homotopically fully faithful.Furthermore, we show that Lack’s model structure on 2Cat is right-induced along H fromour model structure on DblCat. In addition, we obtain a 2Cat-enrichment of our modelstructure on DblCat, by using a variant of the Gray tensor product.Analogous statements hold for the category wkDblCat s of weak double categories andstrict double functors, whose homotopy theory recovers that of bicategories through thehorizontal embedding. Moreover, we show that the full embedding DblCat → wkDblCat s is a Quillen equivalence.Finally, under certain conditions, a characterization of our weak equivalences allowsus to prove a Whitehead theorem for double categories, which retrieves the Whiteheadtheorem for 2-categories as a special case. Contents
1. Introduction 2
I. The model structure
82. Double categorical preliminaries 83. Model structure for double categories 124. Quillen pairs between DblCat, 2Cat, and Cat 195. 2Cat-enrichment of the model structure on DblCat 256. Comparison with other model structures on DblCat 307. Model structure for weak double categories 32
II. Technical results
III. The Whitehead Theorem Introduction
In category theory as well as homotopy theory, we strive to find the correct notion of“sameness”, often with a specific context or perspective in mind. When working with cate-gories themselves, it is commonly agreed that having an isomorphism between categories ismuch too strong a requirement, and we instead concur that the right condition to demandis the existence of an equivalence of categories.There are many ways one can justify this in practice, but, at heart, it is due to thefact that the category Cat of categories and functors actually forms a 2-category, with2-cells given by the natural transformations. Therefore, instead of asking that a functor F : A → B has an inverse G : B → A such that their composites are equal to the identities,it is more natural to ask for the existence of natural isomorphisms id ∼ = F G and GF ∼ = id.In particular, this characterizes F as a functor that is essentially surjective on objects (i.e.,surjective up to an isomorphism) and fully faithful on morphisms.Ever since Quillen’s seminal work [17], and even more so in the last two decades, we havecome to expect that any reasonable notion of equivalence in a category should lend itself todefining the class of weak equivalences of a model structure. This is in fact the case of thecategorical equivalences: the category Cat can be endowed with a model structure, calledthe canonical model structure , in which the weak equivalences are precisely the equivalencesof categories.Going one dimension up and focusing on 2-categories, the 2-functors themselves nowform a 2-category, with higher cells given by the pseudo natural transformations, and theso-called modifications between them. We can then define a 2-functor F : A → B to bea biequivalence if it has an inverse G : B → A together with pseudo natural equivalencesid ≃ F G and GF ≃ id, i.e., equivalences in the corresponding 2-categories of 2-dimensionalfunctors. Note that this inverse G is in general a pseudo functor rather than a 2-functor.Furthermore, a Whitehead theorem for 2-categories [10, Theorem 7.4.1] is available, andcharacterizes the biequivalences as the 2-functors that are bi-essentially surjective on ob-jects (i.e., surjective up to an equivalence in the target 2-category), essentially full onmorphisms (i.e., full up to an invertible 2-cell), and fully faithful on 2-cells.As in the case of the equivalences of categories, the biequivalences of 2-categories are partof the data of a model structure. Indeed, in [11, 12], Lack defines a model structure on thecategory 2Cat of 2-categories and 2-functors in which the weak equivalences are preciselythe biequivalences; we henceforth refer to it as the Lack model structure . In particular,the canonical homotopy theory of categories embeds reflectively in this homotopy theoryof 2-categories.In this paper, we consider another type of 2-dimensional objects, called double categories ,which have both horizontal and vertical morphisms between pairs of objects, related by2-dimensional cells called squares . These are more structured than 2-categories, in the sensethat a 2-category A can be seen as a horizontal double category H A with only trivial verticalmorphisms. As a consequence, the study of various notions of 2-category theory benefitsfrom a passage to double categories. For example, a 2-limit of a 2-functor F does not coincide with a 2-terminal object in the slice 2-category of cones, as shown in [2, Counter-example 2.12]. However, by considering the 2-functor F as a horizontal double functor H F ,we can see that a 2-limit of F is precisely a double terminal object in the slice doublecategory of cones over H F ; see [6, § A two 2-categories whose underlying categories are precisely the ones above. One isgiven by the underlying horizontal 2-category H A of objects, horizontal morphisms, andsquares with trivial vertical boundaries, and the other is given by the 2-category V A ofvertical morphisms, squares, and 2-cells as described in Definition 2.10. We then get anotion of double biequivalence : a double functor F : A → B such that the two induced2-functors H F and V F are biequivalences in 2Cat.A double biequivalence can alternatively be characterized as a double functor that ishorizontally bi-essentially surjective on objects, essentially full on horizontal morphisms,bi-essentially surjective on vertical morphisms, and fully faithful on squares. From thischaracterization and the fact that H H = id, we directly get that a 2-functor F : A → B isa biequivalence if and only if H F : H A → H B is a double biequivalence. This can be seenas a first step towards showing that the homotopy theory of 2-categories sits inside that ofdouble categories.More surprisingly, double biequivalences are similarly well-behaved with respect to otherdouble categories typically constructed from a 2-category A , and which have A itself astheir underlying horizontal 2-category; namely, the double category of quintets Q A (see[5, § A dj A (see [5, § F : A → B is a biequivalence if and only if the induced doublefunctor Q F : Q A → Q B is a double biequivalence, and similarly for A dj F .As further evidence supporting this notion of weak equivalence for double functors, weobtain a Whitehead theorem for double categories under an additional hypothesis. L. MOSER, M. SARAZOLA, AND P. VERDUGO
Theorem 11.14 (Whitehead Theorem for double categories) . Let A and B be doublecategories such that A is weakly horizontally invariant or B has only trivial vertical mor-phisms. Then a double functor F : A → B is a double biequivalence if and only if thereexists a pseudo double functor G : B → A together with horizontal pseudo natural equiva-lences id ≃ GF and F G ≃ id . Our first main result, Theorem 3.16, provides a model structure on the category ofdouble categories in which the weak equivalences are precisely the double biequivalences,and it is obtained as a right-induced model structure along ( H , V ) from two copies of theLack model structure on 2Cat × Theorem 3.16.
Consider the adjunction × , H ⊔ L ( H , V ) ⊥ where each copy of is endowed with the Lack model structure. Then the right-inducedmodel structure on DblCat exists. In particular, a double functor is a weak equivalence(resp. fibration) in this model structure if and only if it is a double biequivalence (resp. dou-ble fibration).
Since the Lack model structure on 2Cat is cofibrantly generated, we also get a cofibrantlygenerated model structure on DblCat from this construction. Moreover, every doublecategory is fibrant, since all objects are fibrant in 2Cat.This model structure on DblCat is defined in such a way that it is compatible with themodel structure on 2Cat. More precisely, the horizontal embedding H : 2Cat → DblCat isboth a left and a right Quillen functor, and it is homotopically fully faithful. This impliesthat the functor H embeds the homotopy theory of 2-categories in that of double categoriesin a reflective and coreflective way. Furthermore, the Lack model structure can be shownto be right-induced along H from our model structure on DblCat. Theorem 4.7.
The Lack model structure on is right-induced from the adjunction
DblCat 2Cat , L H ⊥ where DblCat is endowed with the model structure of Theorem 3.16.
As a consequence, the functor H preserves cofibrations, and creates weak equivalencesand fibrations.Having established a first compatibility of our model structure on DblCat with the Lackmodel structure on 2Cat, we want to further investigate their relation. Lack shows in [11]that the model structure on 2Cat is monoidal with respect to the Gray tensor product. Inthe double categorical setting, there is an analogous monoidal structure on DblCat given by the Gray tensor product constructed by B¨ohm in [1]. However, this monoidal structureis not compatible with our model structure on DblCat (see Remark 5.8), since it treats thevertical and horizontal directions symmetrically, while our model structure does not. Nev-ertheless, restricting this Gray tensor product for double categories in one of the variablesto 2Cat via H removes this symmetry and provides an enrichment of DblCat over 2Catthat is compatible with our model structure. More precisely, this enrichment is given bythe hom-2-categories of double functors, pseudo horizontal natural transformations, andmodifications between them. Theorem 5.11.
The model structure on
DblCat of Theorem 3.16 is a -enriched modelstructure, where the enrichment is given by [ − , − ] ps . The fact that pseudo horizontal natural transformations play a key role was to be ex-pected, since they are the type of transformations that detect our weak equivalences, asestablished in our version of the Whitehead theorem above.Just as the composition in 2-categories can be weakened to obtain the notion of bicate-gories , double categories also admit a weaker version, called weak double categories , wherehorizontal composition is associative and unital up to vertically invertible squares. A bicat-egory B can then be seen as a horizontal weak double category H w B . In [12], Lack showsthat the category Bicat s of bicategories and strict functors admits a model structure, inwhich the weak equivalences are again the biequivalences. Moreover, the full embeddingof 2-categories into bicategories induces a Quillen equivalence.Similarly, we endow the category wkDblCat s of weak double categories and strict doublefunctors with a model structure, whose weak equivalences are again the double biequiva-lences. This is right-induced from two copies of the Lack model structure on Bicat s alongan analogous adjunction to the one in Theorem 3.16. Theorem 7.5.
Consider the adjunction
Bicat s × Bicat s wkDblCat s , H w ⊔ L w ( H w , V w ) ⊥ where Bicat s is endowed with the Lack model structure. Then the right-induced modelstructure on wkDblCat s exists. In particular, a strict double functor is a weak equiva-lence (resp. fibration) in this model structure if and only if it is a double biequivalence(resp. double fibration). Just as in the strict case, the functor H w embeds the homotopy theory of bicategoriesin that of weak double categories in a reflective and coreflective way, and the Lack modelstructure for bicategories is right-induced from the model structure for weak double cate-gories along H w . Furthermore, the full embedding of DblCat into wkDblCat s is part of aQuillen equivalence, mirroring the Quillen equivalence between 2Cat and Bicat s of [12]. Theorem 7.17.
The adjunction
L. MOSER, M. SARAZOLA, AND P. VERDUGO
DblCat wkDblCat s SI ⊥ is a Quillen equivalence, where DblCat is endowed with the model structure of Theorem 3.16and wkDblCat s with the model structure of Theorem 7.5. The model structures on 2Cat, Bicat s , DblCat, and wkDblCat s interact as expected:we have a commutative square of right Quillen functors relating the model structuresintroduced in this paper and Lack’s model structures.2Cat Bicat s DblCat wkDblCat s I ≃ Q I ≃ Q H H w Outline.
This paper is divided in three parts. The first part, comprising Sections 2to 7, contains all of the key information about our model structure. This includes itsconstruction, together with characterizations of the relevant classes of maps, as well asits enrichment over 2Cat. We also treat here the case of weak double categories. Tofacilitate the reading of this paper, we postpone lengthy technical proofs to the secondpart, containing Sections 8 to 10. The reader can be assured that no important resultsare introduced in the second part, and thus it can be omitted, if they are willing to trustthe claims made in the first part. The third part is self-contained (except for the use ofDefinition 3.5, which introduces double biequivalences), and addresses the statement andproof of a Whitehead theorem for double categories.Let us provide a more detailed outline of the paper. In Section 2, we introduce the doublecategorical notions that will be used. In particular, we present the different adjunctions ofinterest between 2Cat and DblCat. After recalling the important features of the Lack modelstructure on 2Cat at the beginning of Section 3, we use it to construct a model structureon DblCat which is right-induced from 2Cat × defined by Fiore, Paoli, and Pronk in [3]. We show that the adjunction given by the identityfunctors on DblCat is not a Quillen pair between our model structure and any of the modelstructures of [3].To conclude this first part, in Section 7 we investigate the case of weak double categories.In analogy to the strict case, we get a model structure that is related by a Quillen pair tothat of bicategories. Moreover, we show that the full embedding of DblCat into wkDblCat s is the right part of a Quillen equivalence between the two model structures established inthis paper.In the second part, we first complete the proof of the main result, Theorem 3.16, byconstructing a path object in double categories in Section 8. We then prove in Section 9 thecharacterizations of weak equivalences and fibrations stated in Section 3, which are given bythe notions of double biequivalences and double fibrations . Finally, in Section 10, we providea better description of the generating (trivial) cofibrations, and give a characterization ofthe cofibrations in our model structure. As a corollary, we get the characterization ofcofibrant objects stated in Section 3.As mentioned above, the last part addresses the Whitehead theorem for double cate-gories, Theorem 11.14. Acknowledgements.
The authors would like to thank Martina Rovelli for reading anearly version of this paper and providing many helpful comments; especially, for suggestingat the beginning of this project that we could induce the model structure from two copiesof 2Cat. The authors are also grateful to tslil clingman, for suggesting a construction thatbecame our functor V : DblCat → L. MOSER, M. SARAZOLA, AND P. VERDUGO
Part I. The model structure Double categorical preliminaries
In this section, we recall the basic notions about double categories, and also introducenon-standard definitions and terminology that will be used throughout the paper. Thereader familiar with double categories may wish to jump directly to Definition 2.10.
Definition 2.1. A double category A consists of(i) objects A , B , C , . . . ,(ii) horizontal morphisms a : A → B with composition denoted by b ◦ a or ba ,(iii) vertical morphisms u : A A ′ with composition denoted by v • u or vu ,(iv) squares (or cells) α : ( u ab v ) of the form A BA ′ B ′ abu v • • α with both horizontal composition along their vertical boundaries and vertical com-position along their horizontal boundaries, and(v) horizontal identities id A : A → A and vertical identities e A : A A for each ob-ject A , vertical identity squares e a : (id A aa id B ) for each horizontal morphism a : A → B , horizontal identity squares id u : ( u id A id A ′ u ) for each vertical morphism u : A A ′ , and identity squares (cid:3) A = id e A = e id A for each object A ,such that all compositions are unital and associative, and such that the horizontal andvertical compositions of squares satisfy the interchange law. Definition 2.2.
Let A and B be double categories. A double functor F : A → B consistsof maps on objects, horizontal morphisms, vertical morphisms, and squares, which arecompatible with domains and codomains and preserve all double categorical compositionsand identities strictly.The category of double categories is cartesian closed, and therefore there is a doublecategory whose objects are the double functors. We describe the horizontal morphisms,vertical morphisms, and squares of this double category. Definition 2.3.
Let
F, G, F ′ , G ′ : A → B be four double functors.A horizontal natural transformation h : F ⇒ G consists of(i) a horizontal morphism h A : F A → GA in B , for each object A ∈ A , and(ii) a square h u : ( F u h A h A ′ Gu ) in B , for each vertical morphism u : A A ′ in A ,such that the assignment of squares is functorial with respect to the composition of ver-tical morphisms, and these data satisfy a naturality condition with respect to horizontalmorphisms and squares. Similarly, a vertical natural transformation r : F ⇒ F ′ consists of(i) a vertical morphism r A : F A F ′ A in B , for each object A ∈ A , and(ii) a square r a : ( r A F aF ′ a r B ) in B , for each horizontal morphism a : A → B in A ,satisfying transposed conditions.Given another horizontal natural transformation k : F ′ ⇒ G ′ and another vertical naturaltransformation s : G ⇒ G ′ , a modification µ : ( r hk s ) consists of(i) a square µ A : ( r A h A k A s A ) in B , for each object A ∈ A ,satisfying horizontal and vertical coherence conditions with respect to the squares of thetransformations h , k , r , and s .See [5, § Definition 2.4.
Let A and B be double categories. We define the double category [ A , B ]whose(i) objects are the double functors A → B ,(ii) horizontal morphisms are the horizontal natural transformations,(iii) vertical morphisms are the vertical natural transformations, and(iv) squares are the modifications. Proposition 2.5 ([3, Proposition 2.11]) . For every double category A , there is an adjunc-tion DblCat DblCat − × A [ A , − ] ⊥ . As mentioned in the introduction, there is a full horizontal embedding of the categoryof 2-categories into that of double categories. This is given by the following functor.
Definition 2.6.
We define the functor H : 2Cat → DblCat. It takes a 2-category A tothe double category H A having the same objects as A , the morphisms of A as horizontalmorphisms, only identities as vertical morphisms, and squares AA BB ab • • α given by the 2-cells α : a ⇒ b in A . It sends a 2-functor F : A → B to the double functor H F : H A → H B that acts as F does on the corresponding data.The functor H admits a right adjoint given by the following. Definition 2.7.
We define the functor H : DblCat → A toits underlying horizontal 2-category H A , i.e., the 2-category whose objects are the objects of A , whose morphisms are the horizontal morphisms of A , and whose 2-cells α : a ⇒ b aregiven by the squares in A of the form AA BB . ab • • α It sends a double functor F : A → B to the 2-functor H F : H A → H B that acts as F doeson the corresponding data. Proposition 2.8 ([3, Proposition 2.5]) . The functors H and H form an adjunction . H H ⊥ Moreover, the unit η : id ⇒ H H is the identity.Remark . Similarly, we can define a functor V : 2Cat → DblCat, sending a 2-category toits associated vertical double category with only trivial horizontal morphisms, and a functor V : DblCat → V ⊣ V .We now introduce a functor that extracts, from a double category, a 2-category whoseobjects and morphisms are the vertical morphisms and squares; this is the functor V mentioned in the introduction. In order to do this, we need the category V , where isthe (2-)category { → } . This double category V contains exactly one vertical morphism. Definition 2.10.
We define the functor V : DblCat → H [ V , − ].More explicitly, it sends a double category A to the 2-category V A = H [ V , A ] given bythe following data.(i) An object in V A is a vertical morphism u : A A ′ in A .(ii) A morphism ( a, b, α ) : u → v is a square in A A BA ′ B ′ . abu v • • α (iii) Composition of morphisms is given by the horizontal composition of squares in A .(iv) A 2-cell ( σ , σ ) : ( a, b, α ) ⇒ ( c, d, β ) consists of two squares in A A BA B ac • • σ A ′ B ′ A ′ B ′ bd • • σ such that the following pasting equality holds. A BA BA ′ B ′ acu vd • •• • σ β = A BA ′ B ′ A ′ B ′ abu vd • •• • ασ (v) Horizontal and vertical compositions of 2-cells are given by the componentwisehorizontal and vertical compositions of squares in A . Proposition 2.11.
The functor V has a left adjoint L L V⊥ given by L = H ( − ) × V .Proof. By definition, the functor V : DblCat → [ V , − ] H Since [ V , − ] has a left adjoint − × V given by Proposition 2.5, and H has a left adjoint H given by Proposition 2.8, it follows that V has a left adjoint given by the composite of thetwo left adjoints, namely, L = H ( − ) × V . (cid:3) We conclude this section by introducing notions of weak invertibility for horizontal mor-phisms and squares, together with some technical results that will be of use later in thepaper. We do not prove these results here, but proofs will be provided in forthcoming workby the first author [15].
Definition 2.12.
A horizontal morphism a : A → B in a double category A is a horizontalequivalence if it is an equivalence in the 2-category H A . Definition 2.13.
A square α : ( u ab v ) in a double category A is weakly horizontallyinvertible if it is an equivalence in the 2-category V A . In other words, the square α is weakly horizontally invertible if there exists a cell β : ( v a ′ b ′ u ) in A and four verticallyinvertible cells 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 ∼ = 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 horizontallyinvertible 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 these tuples the horizontal equivalence data of ( α, β ).Since an equivalence in a 2-category can always be promoted to an adjoint equivalence,we get the following result. Lemma 2.15.
Every horizontal equivalence can be promoted to a horizontal adjoint equiv-alence. Similarly, every weakly horizontally invertible square can be promoted to one withhorizontal adjoint equivalence data.
Finally, we conclude with two results concerning weakly horizontally invertible cells.
Lemma 2.16. [15]
Given a weakly horizontally invertible square α : ( u ab v ) and two hori-zontal adjoint equivalences ( a, a ′ , η a , ǫ a ) and ( b, b ′ , η b , ǫ b ) , there exists a unique weak inverse β : ( v a ′ b ′ u ) of α with respect to these horizontal adjoint equivalences. Lemma 2.17. [15]
A square whose horizontal boundaries are horizontal equivalences, andwhose vertical boundaries are identities, is weakly horizontally invertible if and only if it isvertically invertible.Remark . It follows that, for a 2-category A , a weakly horizontally invertible squarein the double category H A corresponds to an invertible 2-cell in A .3. Model structure for double categories
This section contains our first main result, which proves the existence of a model struc-ture on DblCat that is right-induced along the functor ( H , V ) : DblCat → × where both copies of 2Cat are endowed with the Lack model structure. After recalling themain features of the Lack model structure on 2Cat in Section 3.1, we define in Section 3.2notions of double biequivalences and double fibrations in DblCat, which extend the notionsof biequivalences and fibrations in 2Cat. These appear to be exactly the weak equivalencesand fibrations of the right-induced model structure mentioned above. We then prove, us-ing a result inspired by the Quillen Path Object Argument, that this right-induced modelstructure on DblCat exists. Moreover, as the model structure on 2Cat × Lack model structure on . We start by recalling the relevant classes of mapsin Lack’s model structure on 2Cat; see [11, 12]. The weak equivalences are given by the biequivalences , and we refer to the fibrations in this model structure as
Lack fibrations . Definition 3.1.
Given 2-categories A and B , a 2-functor F : A → B is a biequivalence if (b1) for every object B ∈ B , there exist an object A ∈ A and an equivalence B ≃ −→ F A ,(b2) for every morphism b : F A → F C in B , there exist a morphism a : A → C in A andan invertible 2-cell b ∼ = F a , and(b3) for every 2-cell β : F a ⇒ F c in B , there exists a unique 2-cell α : a ⇒ c in A suchthat F α = β . Definition 3.2.
Given 2-categories A and B , a 2-functor F : A → B is a
Lack fibration if (f1) for every equivalence b : B ≃ −→ F C in B , there exists an equivalence a : A ≃ −→ C in A such that F a = b , and(f2) for every morphism c : A → C in A and every invertible 2-cell β : b ∼ = F c , thereexists an invertible 2-cell α : a ∼ = c in A such that F α = β .There exists a model structure on 2Cat determined by the classes above. Theorem 3.3 ([12, Theorem 4]) . There is a cofibrantly generated model structure on ,called the
Lack model structure , in which the weak equivalences are the biequivalences andthe fibrations are the Lack fibrations.Remark . Note that every 2-category is fibrant in the Lack model structure.3.2.
Constructing the model structure for
DblCat . We define double biequivalencesin DblCat inspired by the characterization of biequivalences in 2Cat in terms of 2-functorsthat are bi-essentially surjective on objects, essentially full on morphisms, and fully faithfulon 2-cells. Our convention of regarding 2-categories as horizontal double categories justifiesthe choice of directions when emulating this characterization of biequivalences in the con-text of double categories. Thus, a double biequivalence will be required to be bi-essentiallysurjective on objects up to a horizontal equivalence (see Definition 2.12), essentially fullon horizontal morphisms, and fully faithful on squares. However, this does not take into account the vertical structure of double categories, and so we need to add a condition ofbi-essential surjectivity on vertical morphisms given up to a weakly horizontally invertiblesquare (see Definition 2.13).
Definition 3.5.
Given double categories A and B , a double functor F : A → B is a doublebiequivalence if(db1) for every object B ∈ B , there exist an object A ∈ A and a horizontal equivalence B ≃ −→ F A ,(db2) for every horizontal morphism b : F A → F C in B , there exist a horizontal morphism a : A → C in A and 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 exist a vertical morphism u : A A ′ in A and 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 cF u F u ′ • • β there exists a unique square α : ( u ac u ′ ) in A such that F α = β . Remark . In 2Cat, one can prove that 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 ≃ GF and F G ≃ id. Under certain hypotheses, we can show a similarcharacterization of double biequivalences using pseudo horizontal natural equivalences.This is done in Section 11.Similarly to the definition of double biequivalence, we take inspiration from the Lackfibrations to define a notion of double fibrations . Definition 3.7.
Given double categories A and B , a double functor F : A → B is a doublefibration if (df1) for every horizontal equivalence b : B ≃ −→ F C in B , there exists a horizontal equiv-alence a : A ≃ −→ C in A such that F a = b ,(df2) for every horizontal morphism c : A → C in A and for every vertically invertiblesquare β : ( e F A bF c e F C ) in B as depicted below left, there exists a vertically invert-ible square α : ( e A ac e C ) in A as depicted below right such that F α = β , F A F CF A F C bF c • • β ∼ = A CA C ac • • α ∼ = (df3) for every vertical morphism u ′ : C C ′ in A and every weakly horizontally in-vertible square β : ( v ≃≃ F u ′ ) in B as depicted below left, there exists a weaklyhorizontally invertible square α : ( u ≃≃ u ′ ) in A as depicted below right such that F α = β . B F CB ′ F C ′ ≃≃ v F u ′ • • β ≃ A CA C ≃≃ u u ′ • • α ≃ By requiring that a double functor is both a double biequivalence and a double fibration,we get a notion of double trivial fibration . Definition 3.8.
Given double categories A and B , a double functor F : A → B is a doubletrivial fibration if(dt1) for every object B ∈ B , there exists an object A ∈ A such that B = F A ,(dt2) for every horizontal morphism b : F A → F C in B , there exists a horizontal mor-phism a : A → C such that b = F a ,(dt3) for every vertical morphism v : B B ′ in B , there exists a vertical morphism u : A A ′ in A such that v = F u , and(dt4) for every square in B of the form F A F CF A ′ F C ′ , F aF cF u F u ′ • • β there exists a unique square α : ( u ac u ′ ) in A such that F α = β . Remark . Note that (dt2) says that a double trivial fibration is full on horizontal mor-phisms, while (dt3) says that a double trivial fibration is only surjective on vertical mor-phisms.
We can characterize double biequivalences and double fibrations through biequivalencesand Lack fibrations in 2Cat. Recall the functors H , V : DblCat → H and V are biequivalences or Lack fibrations. This isintuitively sound, since horizontal equivalences and weakly horizontally invertible squareswere defined to be the equivalences in the 2-categories induced by H and V , respectively.We state these characterizations here, and defer their proofs to Section 9. Proposition 3.10.
Given double categories A and B , a double functor F : A → B is adouble biequivalence in DblCat if and only if H F : H A → H B and V F : V A → V B arebiequivalences in . Proposition 3.11.
Given double categories A and B , a double functor F : A → B is adouble fibration in DblCat if and only if H F : H A → H B and V F : V A → V B are Lackfibrations in . As a corollary, we get a similar characterization for double trivial fibrations.
Corollary 3.12.
Given double categories A and B , a double functor F : A → B is a doubletrivial fibration in DblCat if and only if H F : H A → H B and V F : V A → V B are trivialfibrations in the Lack model structure on .Proof. It is a routine exercise to check that a double functor that is both a double biequiva-lence and a double fibration is precisely a double trivial fibration as defined in Definition 3.8.Therefore, this follows directly from Propositions 3.10 and 3.11. (cid:3)
In order to build a model structure on DblCat with these classes of morphisms as itsweak equivalences and (trivial) fibrations, we make use of the notion of right-induced modelstructure . Given a model category M and an adjunction M N LR ⊥ (3.13)we can, under certain conditions, induce a model structure on N along the right adjoint R ,in which a weak equivalence (resp. fibration) is a morphism F in N such that RF is a weakequivalence (resp. fibration) in M .Propositions 3.10 and 3.11 suggest that the model structure on DblCat we desire, withdouble biequivalences as the weak equivalences and double fibrations as the fibrations,corresponds to the right-induced model structure, if it exists, along the adjunction2Cat × H ⊔ L ( H , V ) ⊥ where each copy of 2Cat is endowed with the Lack model structure. To prove the existenceof this model structure, we use results by Garner, Hess, K¸edziorek, Riehl, and Shipleyin [4, 8]. In particular, we use the following theorem, inspired by the original Quillen PathObject Argument [17]. Theorem 3.14.
Let M be an accessible model category, and let N be a locally presentablecategory. Suppose we have an adjunction L ⊣ R between them as in (3.13). Supposemoreover that every object is fibrant in M and that, for every X ∈ N , there exists afactorization X W −→ Path( X ) P −→ X × X of the diagonal morphism in N such that RP is a fibration in M and RW is a weakequivalence in M . Then the right-induced model structure on N exists.Proof. This follows directly from [16, Theorem 6.2], which is the dual of [8, Theorem 2.2.1].Indeed, if every object in M is fibrant, then the underlying fibrant replacement of condi-tions (i) and (ii) of [16, Theorem 6.2] are trivially given by the identity. (cid:3) Our strategy is then to construct a path object P A for a double category A together withdouble functors W and P factorizing the diagonal morphism A → A × A , such that theirimages under ( H , V ) give a weak equivalence and a fibration in 2Cat × W is a double biequivalence and P is a double fibration. This construction is inspired by the path object construction in 2Catof [11, Proof of Theorem 5.1], and its statement is summarized in the following technicalresult, whose proof is the content of Section 8. Proposition 3.15.
For every double category A , there exists a double category P A togetherwith a factorization of the diagonal double functor A W −→ P A P −→ A × A such that W is a double biequivalence and P is a double fibration. We are finally ready to prove the existence of the right-induced model structure onDblCat along the adjunction H ⊔ L ⊣ ( H , V ). Theorem 3.16.
Consider the adjunction × , H ⊔ L ( H , V ) ⊥ where each copy of is endowed with the Lack model structure. Then the right-inducedmodel structure on DblCat exists. In particular, a double functor is a weak equivalence(resp. fibration) in this model structure if and only if it is a double biequivalence (resp. dou-ble fibration).
Proof.
We first describe the weak equivalences and fibrations in this model structureon DblCat. The weak equivalences (resp. fibrations) in the right-induced model structureon DblCat are given by the double functors F such that ( H , V ) F is a weak equivalence(resp. fibration) in 2Cat × H F and V F are biequiva-lences (resp. Lack fibrations) in 2Cat. Then it follows from Propositions 3.10 and 3.11 thatthe weak equivalences (resp. fibrations) in DblCat are precisely the double biequivalences(resp. double fibrations).We now prove the existence of the model structure. For this purpose, we want to applyTheorem 3.14 to our setting. First note that 2Cat and DblCat are locally presentable,and that the Lack model structure on 2Cat is cofibrantly generated. In particular, thisimplies that the product 2Cat × × A , Proposition 3.15 gives a factorization A W −→ P A P −→ A × A such that W is a double biequivalence and P is a double fibration. By Theorem 3.14, thisproves that the right-induced model structure along ( H , V ) on DblCat exists. (cid:3) Remark . Note that every double category is fibrant in this model structure. Indeed,this follows directly from the fact that it is right-induced from a model structure in whichevery object is fibrant.3.3.
Generating (trivial) cofibrations and cofibrant objects.
We now turn our at-tention to the cofibrations. As stated in Theorem 3.3, the Lack model structure on 2Catis cofibrantly generated. Since our model structure on DblCat is right-induced from copiesof it, it is also cofibrantly generated. The next result provides sets of generating (trivial)cofibrations for our model structure on DblCat.
Proposition 3.18.
Let I and J denote sets of generating cofibrations and generatingtrivial cofibrations, respectively, for the Lack model structure on . Then, the sets ofmorphisms in DblCat I = { H i, H i × V | i ∈ I } , and J = { H j, H j × V | j ∈ J } give sets of generating cofibrations and generating trivial cofibrations, respectively, for themodel structure on DblCat of Theorem 3.16.Proof.
Since the model structure on DblCat is right-induced from two copies of the Lackmodel structure on 2Cat along the adjunction H ⊔ L ⊣ ( H , V ), the sets of generatingcofibrations and of generating trivial cofibrations are given by the image under the leftadjoint H ⊔ L of the sets of generating cofibrations and generating trivial cofibrationsin 2Cat × i and i ′ be generating cofibrations of I in 2Cat. Then H i and L i = H i × V are cofibrations in DblCat. To see this apply H ⊔ L to the cofibrations ( i, id ∅ ) and (id ∅ , i ),respectively. Similarly, H i ′ and L i ′ = H i ′ × V are cofibrations in DblCat. Since coproducts of cofibrations are cofibrations, ( H ⊔ L )( i, i ′ ) = H i ⊔ L i ′ can be obtained from H i and L i ′ = H i ′ × V . This shows that I is a set of generating cofibrations of DblCat.Similarly, we can show that J is a set of generating trivial cofibrations of DblCat. (cid:3) In the Lack model structure on 2Cat, the cofibrant objects are precisely the 2-categorieswhose underlying categories are free, by [11, Theorem 4.8]. We get a similar characteriza-tion of the cofibrant objects in our model structure. Since double trivial fibrations are fullon horizontal morphisms and fully faithful on squares, the underlying horizontal categoryof a cofibrant double category is also free. To characterize these cofibrant double categoriescompletely, we also need a condition for vertical morphisms. This is expressed in termsof the underlying vertical category, which is not only required to be free, but, in addition,it cannot contain any composition of morphisms. This is intuitively coming from the factthat double trivial fibrations are only surjective on vertical morphisms instead of full.
Notation 3.19.
We write U : 2Cat → Cat for the functor that sends a 2-category to itsunderlying category.
Proposition 3.20.
A double category A is cofibrant if and only if its underlying horizontalcategory U H A is free and its underlying vertical category U V A is a disjoint union of copiesof and . We prove this proposition in Section 10, using a description of the cofibrations presentedtherein. In that same section, we also provide smaller sets of generating cofibrations andgenerating trivial cofibrations.4.
Quillen pairs between
DblCat , , and CatIn this paper, the model structure on DblCat was constructed in such a way as to becompatible with the Lack model structure on 2Cat. This section investigates the inter-actions between our model structure and Lack’s model structure by looking at differentQuillen pairs between them. Since our model structure on DblCat was right-induced alongthe adjunction H ⊔ L ⊣ ( H , V ), this adjunction is itself a Quillen pair. From it, we inducetwo Quillen pairs between DblCat and 2Cat: H ⊣ H , presented in Section 4.1, and L ⊣ V ,which is the content of Section 4.3.Since the derived unit of the adjunction H ⊣ H is an identity, this further implies thatthe functor H : 2Cat → DblCat is homotopically fully faithful. In Section 4.1, we also provethat the functor H and its left adjoint form a Quillen pair. These results imply that thefunctor H embeds the homotopy theory of 2Cat into that of DblCat in a reflective andcoreflective way, in the sense that the ( ∞ , ∞ , ∞ , H creates allhomotopy limits and colimits. Finally, our last result in Section 4.1 shows that the Lackmodel structure on 2Cat is right-induced from our model structure along H .In Section 4.2, we give a Quillen pair between Cat and DblCat, which horizontallyembeds the canonical homotopy theory of Cat into that of DblCat in a reflective way. Wealso show that the canonical model structure on Cat is right-induced from ours. We first give the following lemma, which allows us to directly induce the Quillen pairs H ⊣ H and L ⊣ V from the one given by H ⊔ L ⊣ ( H , V ). Lemma 4.1.
The following adjunctions × (id , ∅ )pr ⊥ × ( ∅ , id)pr ⊥ induced from the adjunction given by the initial object ∅ of ∅ ! ⊥ are Quillen pairs, where all copies of are endowed with the Lack model structure.Proof. It is clear that the projection functors pr i for i = 1 , (cid:3) Quillen pairs involving H . We present here the two Quillen pairs involving thefunctor H : 2Cat → DblCat and its right and left adjoints.
Proposition 4.2.
The adjunction H H ⊥ is a Quillen pair, where is endowed with the Lack model structure and DblCat isendowed with the model structure of Theorem 3.16. Moreover, the unit η A : A → H H A isan identity for all A ∈ .Proof.
By composing the Quillen pairs (id , ∅ ) ⊣ pr of Lemma 4.1 and H ⊔ L ⊣ ( H , V ) ofTheorem 3.16, we directly get the Quillen pair above. Moreover, we have H H = id . (cid:3) Remark . In particular, since every object is fibrant, the derived unit of the adjunction H ⊣ H is given by the components of the unit at cofibrant objects, and is therefore anidentity. This implies that the functor H is homotopically fully faithful.The functor H : 2Cat → DblCat also admits a left adjoint L . Indeed, this is given bythe Adjoint Functor Theorem, since H preserves all limits and the categories involved arelocally presentable. The next theorem shows that H is also a right Quillen functor. Theorem 4.4.
The adjunction
DblCat 2Cat L H ⊥ is a Quillen pair, where is endowed with the Lack model structure and DblCat isendowed with the model structure of Theorem 3.16. Moreover, the counit ǫ A : L H A → A is an isomorphism of -categories for all A ∈ .Proof.
We show that H is right Quillen, i.e., it preserves fibrations and trivial fibrations.Let F : A → B be a fibration in 2Cat; we prove that H F : H A → H B is a double fibrationin DblCat. Since H H F = F and F is a fibration, (df1-2) of Definition 3.7 are satisfied. Itremains to show (df3) of Definition 3.7. Let us consider a weakly horizontally invertiblesquare in H B B F CB F C . ≃ b ≃ d • • β ∼ = Note that its vertical boundaries must be trivial, since all vertical morphisms in H B areidentities. The square β is, in particular, vertically invertible by Lemma 2.17. Since F isa fibration in 2Cat, there exists an equivalence c : A ≃ −→ C such that F c = d , by (f1) ofDefinition 3.2. Now β can be rewritten as F A F CF A F C . ≃ b ≃ F c • • β ∼ = Then β is equivalently an invertible 2-cell β : b ⇒ F c in B . Since F is a fibration in 2Cat,there exist a morphism a : A → C in A and an invertible 2-cell α : a ⇒ c in A such that F α = β , by (f2) of Definition 3.2. In particular, since c is an equivalence, then so is a .This gives a vertically invertible square in H A A CA C ≃ a ≃ c • • α ∼ = such that F α = β ; furthermore, by Lemma 2.17, the square α is weakly horizontallyinvertible. This shows that H F is a double fibration. Now let F : A → B be a trivial fibration in 2Cat. We show that H F : H A → H B is a double trivial fibration in DblCat. Since H H F = F and F is a trivial fibration,(dt1-2) of Definition 3.8 are satisfied. Then (dt3) of Definition 3.8 follows from the factthat F is surjective on objects, since all vertical morphisms are identities. Finally, (dt4) ofDefinition 3.8 is a direct consequence of F being fully faithful on 2-cells, since all squaresin H A and H B are equivalently 2-cells in A and B , respectively. This shows that H F is adouble trivial fibration, and concludes the proof of L ⊣ H being a Quillen pair.Let A ∈ ǫ A : L H A → A is an isomorphismof 2-categories. Since H is fully faithful, this follows directly from evaluating at id A thefollowing isomorphism2Cat( A , A ) ∼ = DblCat( H A , H A ) ∼ = 2Cat( L H A , A ) , where the first isomorphism is induced by H and the second comes from the adjunction L ⊣ H . (cid:3) Remark . As we have seen in Remark 4.3, the functor H is homotopically fully faithful,and therefore the derived counit of the adjunction L ⊣ H is levelwise a biequivalence. Remark . By Proposition 4.2 and Theorem 4.4, we can see that H : 2Cat → DblCatis both left and right Quillen, and so it preserves all cofibrations, fibrations, and weakequivalences. In particular, since H is homotopically fully faithful by Remark 4.3, thissays that the homotopy theory of 2Cat is reflectively and coreflectively embedded in thatof DblCat via the functor H .In fact, more is true: the Lack model structure on 2Cat is right-induced from our modelstructure on DblCat along the adjunction L ⊣ H , which implies that the functor H alsoreflects fibrations and weak equivalences. Theorem 4.7.
The Lack model structure on is right-induced from the adjunction
DblCat 2Cat , L H ⊥ where DblCat is endowed with the model structure of Theorem 3.16.Proof.
We show that a 2-functor F : A → B is a biequivalence (resp. Lack fibration) if andonly if H F : H A → H B is a double biequivalence (resp. double fibration).Since H is right Quillen by Theorem 4.4, it preserves fibrations. Moreover, since allobjects in 2Cat are fibrant, by Ken Brown’s Lemma (see [9, Lemma 1.1.12]), we havethat H also preserves weak equivalences. This shows that if F is a biequivalence (resp. Lackfibration), then H F is a double biequivalence (resp. double fibration).Conversely, if H F is a double biequivalence (resp. double fibration), then H H F = F isa biequivalence (resp. Lack fibration) by definition of the model structure on DblCat.Since model structures are uniquely determined by their classes of weak equivalencesand fibrations, this shows that the Lack model structure on 2Cat is right-induced along H from the model structure on DblCat of Theorem 3.16. (cid:3) We saw that the derived unit (resp. counit) of the adjunction H ⊣ H (resp. L ⊣ H )is levelwise a biequivalence. However, these adjunctions are not expected to be Quillenequivalences, since the homotopy theory of double categories should be richer than that of2-categories. This is indeed the case, as shown in the following remarks. Remark . The components of the (derived) counit of the adjunction H ⊣ H are notdouble biequivalences. To see this, consider the double category V . The component ofthe counit at V is given by the inclusion ǫ V : H H ( V ) ∼ = ⊔ → V which is not a double biequivalence, as it does not satisfy (db3) of Definition 3.5. Since H V ∼ = ⊔ is cofibrant in 2Cat, this is also the component of the derived counit at V . Remark . The components of the (derived) unit of the adjunction L ⊣ H are not doublebiequivalences. Since the unique map ∅ → is a generating cofibration in 2Cat [11, § ∅ → V is a generating cofibration in DblCat, sothat V is cofibrant in DblCat. But we have that η V : V → H L ( V ) ∼ = is not a double biequivalence, where the isomorphism comes from the fact that the leftadjoint L collapses the vertical structure and thus L V ∼ = .4.2. Quillen pairs to
Cat . The category Cat of categories and functors also admits amodel structure, called the canonical model structure , in which the weak equivalences arethe equivalences of categories and the fibrations are the isofibrations. As shown by Lackin [11], with this model structure, the homotopy theory of Cat is reflectively embedded inthe homotopy theory of 2Cat. Combining this result with the one of Theorem 4.4, we getthat the homotopy theory of Cat is also reflectively embedded in that of DblCat.
Notation 4.10.
We write D : Cat → P : 2Cat → Cat for its left adjoint. In particular, the functor only P sends a 2-category A to the category P A with the same objects as A and with hom-sets P A ( A, B ) = π A ( A, B ),where π : Cat → Set is the functor sending a category to its set of connected components.
Remark . In [11, Theorem 8.2], Lack shows that the adjunction P ⊣ D is a Quillen pairbetween the Lack model structure on 2Cat and the canonical model structure on Cat, whosederived counit is levelwise a weak equivalence, but the unit is not. Composing this Quillenpair with the one of Theorem 4.4, we get a Quillen adjunction P L ⊣ H D between the modelstructure on DblCat of Theorem 3.16 and the canonical model structure on Cat whosederived counit is levelwise an equivalence of categories. In particular, H D : Cat → DblCatis homotopically fully faithful.The above remark guarantees that the functors D : Cat → H D : Cat → DblCatpreserve fibrations and weak equivalences, since all objects are fibrant. Furthermore, the following results imply that these two functors create fibrations and weak equiva-lences, since the canonical model structure on Cat is right-induced from the ones on 2Catand DblCat.
Proposition 4.12.
The canonical model structure on
Cat is right-induced from the ad-junction , PD ⊥ where is endowed with the Lack model structure.Proof. Let F : C → D be a functor in Cat. It suffices to show that F is an equivalence(resp. isofibration) if and only if DF is a biequivalence (resp. Lack fibration). Indeed,both statements can easily seen to be true, due to the fact that for any category A , the2-category D A only has trivial 2-cells, and thus a morphism in D A is an equivalenceprecisely if it is an isomorphism in A . (cid:3) Corollary 4.13.
The canonical model structure on
Cat is right-induced from the adjunc-tion
DblCat Cat , P L H D ⊥ where DblCat is endowed with the model structure of Theorem 3.16.Proof.
This follows directly from Proposition 4.12 and Theorem 4.7. (cid:3)
The Quillen pair L ⊣ V . Since we induced the model structure on DblCat fromthe adjunction H ⊔ L ⊣ ( H , V ), we directly get that the adjunction L ⊣ V forms a Quillenpair. The (derived) unit and counit of this adjunction, however, are not levelwise weakequivalences. Proposition 4.14.
The adjunction L V⊥ is a Quillen pair, where is endowed with the Lack model structure and DblCat isendowed with the model structure of Theorem 3.16.Proof.
By composing the Quillen pairs ( ∅ , id) ⊣ pr of Lemma 4.1 and H ⊔ L ⊣ ( H , V ) ofTheorem 3.16, we directly get the Quillen pair above. (cid:3) Remark . The components of the (derived) unit of the adjunction L ⊣ V are notbiequivalences. From [11, § ∅ → is a cofibration, so that is cofibrantin 2Cat. But we have that η : → V L ( ) ∼ = H [ V , V ] ∼ = ⊔ ⊔ is not a biequivalence. Remark . The components of the (derived) counit of the adjunction L ⊣ V are notdouble biequivalences. For example, if we consider the double category A generated by0 10 ′ ′ , α β • • then one can check that the double functor ǫ A : L V A = H H [ V , A ] × V → A given by evaluation is not full on squares, hence, it is not a biequivalence. By [11, The-orem 4.8], the 2-category V A is cofibrant in 2Cat since its underlying category is free.Therefore, the double functor ǫ A is also the component of the derived counit at A .5. 2Cat -enrichment of the model structure on DblCatThe aim of this section is to provide a 2Cat-enrichment on DblCat which is compatiblewith the model structure introduced in Theorem 3.16. Recall that a model category M is said to be enriched over a closed monoidal category N that is also a model category, ifit is a tensored and cotensored N -category and satisfies the pushout-product axiom (see[16, §
5] for more details). In particular, N is said to be a monoidal model category , if themodel structure is enriched over itself.In [11], it is shown that the Lack model structure is not monoidal with respect tothe cartesian product. However, it is established that such a compatibility exists whenconsidering instead the closed symmetric monoidal structure on 2Cat given by the Graytensor product. We first recall this result in Section 5.1.Similarly, the category of double categories admits two closed symmetric monoidal struc-tures, given by the cartesian product, and by an analogue of the Gray tensor product de-fined by B¨ohm in [1]. We show in Section 5.2 that the category DblCat is not a monoidalmodel category with respect to either of these monoidal structures.Nevertheless, the Gray tensor product on DblCat is not entirely unrelated to our modelstructure. By restricting this tensor product in one of the variables to 2Cat along H , weobtain in Section 5.3 a 2Cat-enrichment on DblCat compatible with our model structure.5.1. The Lack model structure is monoidal.
Let us recall how the Gray tensor producton 2Cat is defined, and state its compatibility with the Lack model structure.
Definition 5.1.
The
Gray tensor product ⊗ : 2Cat × → A , B and C , there is an isomorphism2Cat( A ⊗ B , C ) ∼ = 2Cat( A , Ps[ B , C ])natural in A , B and C , where Ps[ B , C ] denotes the 2-category of 2-functors B → C , pseudonatural transformations, and modifications.
Theorem 5.2 ([11, Theorem 7.5]) . The category endowed with the Lack model struc-ture is a monoidal model category with respect to the closed monoidal structure given bythe Gray tensor product.
The model structure on
DblCat is not monoidal.
We now turn to double cate-gories. As shown in the remark below, a similar argument to Lack’s [11, Example 7.2] alsoapplies in the case of DblCat, to see that the model structure on DblCat is not monoidalwith respect to the cartesian product.
Remark . Since the inclusion 2-functor i : ⊔ → is a generating cofibration in 2Catby [11, § H i : ⊔ → H is a generating cofibration in DblCat, dueto Proposition 3.18. However, the pushout-product H i (cid:3) H i with respect to the cartesianproduct is the double functor from the non-commutative square of horizontal morphisms tothe commutative square of horizontal morphisms, as in [11, Example 7.2]. As we will see inCorollary 10.11, cofibrations in DblCat are in particular faithful on horizontal morphisms,and therefore the pushout-product H i (cid:3) H i cannot be a cofibration in DblCat.As we mentioned before, a Gray tensor product for DblCat is introduced by B¨ohm in [1].In the same vein as in the 2-categorical case, the corresponding hom-double categories makeuse of the notions of pseudo horizontal and vertical transformations, and modificationsbetween them. Definition 5.4.
Let
F, G : A → B be double functors. A pseudo horizontal naturaltransformation h : F ⇒ G consists of(i) a horizontal morphism h A : F A → GA in B , for each object A ∈ A ,(ii) a square h u : ( F u h A h A ′ Gu ) in B , for each vertical morphism u : A A ′ in A , and(iii) a vertically invertible square h a : ( e F A Ga ◦ h A h B ◦ F a e GB ) in B , for each horizontal mor-phism a : A → B in A , expressing a pseudo naturality condition for horizontalmorphisms.These assignments of squares are functorial with respect to compositions of horizontal andvertical morphisms, and these data satisfy a naturality condition with respect to squares.Similarly, one can define a transposed notion of pseudo vertical natural transfor-mation between double functors.A modification in a square of pseudo horizontal and vertical transformations is definedsimilarly to Definition 2.3, with the horizontal and vertical coherence conditions taking thepseudo data of the transformations into account.See [5, § § Definition 5.5.
Let A and B be double categories. We define the double category [ A , B ] ps whose(i) objects are the double functors A → B ,(ii) horizontal morphisms are the pseudo horizontal transformations,(iii) vertical morphisms are the pseudo vertical transformations, and(iv) squares are the modifications. Proposition 5.6 ([1, § . There is a symmetric monoidal structure on
DblCat given bythe Gray tensor product ⊗ Gr : DblCat × DblCat → DblCat . Moreover, this monoidal structure is closed: for all double categories A , B , and C , there isan isomorphism DblCat( A ⊗ Gr B , C ) ∼ = DblCat( A , [ B , C ] ps ) , natural in A , B and C . Since this Gray tensor product deals with the horizontal and vertical directions in anequal manner, while our model structure does not, it is not surprising that these twostructures are not compatible.
Notation 5.7.
Let I : A → B and J : A ′ → B ′ be double functors in DblCat. We write I (cid:3) Gr J for their pushout-product I (cid:3) Gr J : A ⊗ Gr B ′ a A ⊗ Gr A ′ B ⊗ Gr A ′ → B ⊗ Gr B ′ with respect to the Gray tensor product ⊗ Gr on DblCat. Remark . The model structure defined in Theorem 3.16 is not compatible with the Graytensor product ⊗ Gr . To see this, recall that i : ∅ → is a generating cofibration in 2Cat andtherefore L i : ∅ → V is a generating cofibration in DblCat, by Proposition 3.18. Howeverthe pushout-product L i (cid:3) Gr L i : δ ( V ⊗ Gr V ) → V ⊗ Gr V is not a cofibration, where V ⊗ Gr V is00 ′ ′ ′ • •• • ∼ = and δ ( V ⊗ Gr V ) is its sub-double category without the horizontally invertible square(just four vertical morphisms sharing some boundaries and no other relation). The factthat L i (cid:3) Gr L i is not a cofibration is a consequence of (dt3) of Definition 3.8 requiring onlysurjectivity on vertical morphisms, instead of fullness. -enrichment of the model structure on DblCat . By restricting the Graytensor product on DblCat along H in one of the variables, we get rid of the issue con-cerning the vertical structure that obstructs the compatibility with the model structureof Theorem 3.16. With this variation, we show that DblCat is a tensored and cotensored2Cat-category, and that the corresponding enrichment is now compatible with our modelstructure. Definition 5.9.
We define the tensoring functor ⊗ : 2Cat × DblCat → DblCat to be thecomposite 2Cat × DblCat DblCat × DblCat DblCat. H × id ⊗ Gr Proposition 5.10.
The category
DblCat is enriched, tensored, and cotensored over ,with(i) hom- -categories given by H [ A , B ] ps , for all A , B ∈ DblCat ,(ii) tensors given by
C ⊗ A , where ⊗ is the tensoring functor of Definition 5.9, for all A ∈ DblCat and
C ∈ , and(iii) cotensors given by [ H C , B ] ps , for all B ∈ DblCat and
C ∈ .Proof.
This follows directly from the definition of ⊗ , and the universal properties of thetensor ⊗ Gr and of the adjunction H ⊣ H . (cid:3) We now present the main result of this section.
Theorem 5.11.
The model structure on
DblCat of Theorem 3.16 is a -enriched modelstructure, where the enrichment is given by [ − , − ] ps . The rest of this section is devoted to the proof of this theorem. With that goal, we firstprove several auxiliary lemmas.
Notation 5.12.
Let i : A → B and j : A ′ → B ′ be 2-functors in 2Cat, and let I : A → B be a double functor in DblCat. We denote by i (cid:3) j the pushout-product i (cid:3) j : A ⊗ B ′ a A⊗ A ′ B ⊗ A ′ → B ⊗ B ′ with respect to the Gray tensor product ⊗ on 2Cat, and we denote by i (cid:3) I the pushout-product i (cid:3) I : A ⊗ B a A⊗ A B ⊗ A → B ⊗ B with respect to the tensoring functor ⊗ : 2Cat × DblCat → DblCat. In particular, we havethat i (cid:3) I = H i (cid:3) Gr I . Lemma 5.13.
Let A and B be -categories. There is an isomorphism of double categories A ⊗ H B ∼ = H ( A ⊗ B ) , natural in A and B . Proof.
This follows from the universal properties of ⊗ and ⊗ , and the isomorphism H [ H B , C ] ps ∼ = Ps[ B , H C ], natural in B ∈ C ∈ DblCat. This isomorphism holds,since there are no non trivial vertical morphisms in H B , and therefore pseudo horizontalnatural transformations out of H B are canonically the same as pseudo natural transforma-tions out of B . (cid:3) Remark . In particular, the natural isomorphism H [ H ( − ) , − ] ps ∼ = Ps[ − , H ( − )] impliesthat the adjunction H ⊣ H is enriched with respect to the 2Cat-enrichments H [ − , − ] ps andPs[ − , − ] of DblCat and 2Cat, respectively. Lemma 5.15.
Let A be a -category. There is an isomorphism of double categories A ⊗ V ∼ = H A × V , natural in A .Proof. This follows from the universal property of ⊗ and of × , and the fact that we have H [ V , B ] ps = H [ V , B ], for all B ∈ DblCat. This equality holds, since there are no non triv-ial horizontal morphisms in V , and therefore pseudo horizontal natural transformationsout of V correspond to (strict) horizontal natural transformations out of V . (cid:3) Lemma 5.16.
Let i : A → B and j : A ′ → B ′ be -functors in . Then the followingstatements hold.(i) There exists an isomorphism i (cid:3) H j ∼ = H ( i (cid:3) j ) in the arrow category.(ii) There exists an isomorphism i (cid:3) ( H j × V ) ∼ = H ( i (cid:3) j ) × V in the arrow category.Proof. Since H is a left adjoint, it preserves pushouts. Moreover, by Lemma 5.13, wehave that it is compatible with the tensors ⊗ and ⊗ . Therefore, i (cid:3) H j ∼ = H ( i (cid:3) j ). ByLemma 5.15, by associativity of ⊗ Gr , and by (i), we get that i (cid:3) ( H j × V ) ∼ = i (cid:3) ( j ⊗ V ) ∼ = ( i (cid:3) H j ) ⊗ Gr V ∼ = ( i (cid:3) j ) ⊗ V ∼ = H ( i (cid:3) j ) × V . (cid:3) We are now ready to prove Theorem 5.11.
Proof of Theorem 5.11.
Recall from Proposition 3.18 that a set I of generating cofibrationsand a set J of generating trivial cofibrations for the model structure on DblCat are given bymorphisms of the form H j and H j × V , where j is a generating cofibration or a generatingtrivial cofibration in 2Cat, respectively.We show that the pushout-product of a generating cofibration in I with any (trivial)cofibration in 2Cat is a (trivial) cofibration in DblCat, and that the pushout-product ofa generating trivial cofibration in J with any cofibration in 2Cat is a trivial cofibrationin DblCat.Given cofibrations i and j in 2Cat, we know by Lemma 5.16 that i (cid:3) H j ∼ = H ( i (cid:3) j ) and i (cid:3) ( H j × V ) ∼ = H ( i (cid:3) j ) × V = L ( i (cid:3) j ) , and by Theorem 5.2 that i (cid:3) j is also a cofibration in 2Cat, which is trivial when either i or j is. Since H and L preserve (trivial) cofibrations, then H ( i (cid:3) j ) and L ( i (cid:3) j ) arecofibrations in DblCat, which are trivial if either i or j is. Taking j to be a generating cofibration or generating trivial cofibration in 2Cat, weget the desired results. More precisely, for all cofibrations i in 2Cat and all generatingcofibrations I ∈ I , we have that i (cid:3) I is a cofibration in DblCat, which is trivial if i istrivial. Similarly, for all cofibrations i of 2Cat and all generating trivial cofibrations J ∈ J ,we have that i (cid:3) J is a trivial cofibration in DblCat. (cid:3) Comparison with other model structures on
DblCatIn [3], Fiore, Paoli, and Pronk construct several model structures on the category DblCatof double categories. We show in this section that our model structure on DblCat is notrelated to these model structures in the following sense: the identity adjunction on DblCatis not a Quillen pair between the model structure of Theorem 3.16 and any of the modelstructures of [3]. This is not surprising, since our model structure was constructed in sucha way that the functor H : 2Cat → DblCat embeds the homotopy theory of 2Cat intothat of DblCat, while there seems to be no such relation between their model structureson DblCat and the Lack model structure on 2Cat, e.g. see end of Section 9 in [3].We start by recalling the categorical model structures on DblCat constructed in [3].Since our primary interest is to compare them to our model structure, we only describe theweak equivalences; the curious reader is encouraged to visit their paper for further details.The first model structure we recall is induced from the canonical model structure on Catby means of the vertical nerve . Definition 6.1 ([3, Definition 5.1]) . The vertical nerve of double categories is the functor N v : DblCat → Cat ∆ op sending a double category A to the simplicial category N v A such that ( N v A ) is the categoryof objects and horizontal morphisms of A , ( N v A ) is the category of vertical morphismsand squares of A , and ( N v A ) n = ( N v A ) × ( N v A ) . . . × ( N v A ) ( N v A ) , for n ≥ Proposition 6.2 ([3, Theorem 7.17]) . There is a model structure on
DblCat in which adouble functor F is a weak equivalence if and only if N v F is levelwise an equivalence ofcategories. The next model structure on DblCat requires a different perspective. For a 2-category A whose underlying 1-category U A admits limits and colimits, there exists a model structureon U A in which the weak equivalences are precisely the equivalences of the 2-category A ;see [13]. When applying this construction to the 2-category DblCat h of double categories,double functors, and horizontal natural transformations, we obtain the following modelstructure on DblCat. Proposition 6.3.
There is a model structure on
DblCat , called the trivial model structure ,in which a double functor F : A → B is a weak equivalence if and only if it is an equivalencein the -category DblCat h , i.e., there exist a double functor G : B → A and two horizontalnatural isomorphisms id ∼ = GF and F G ∼ = id . The last model structure is of a more algebraic flavor. Let T be a 2-monad on a2-category A . In [13], Lack gives a construction of a model structure on the categoryof T -algebras, in which the weak equivalences are the morphisms of T -algebras whose un-derlying morphism in A is an equivalence. In particular, double categories can be seen asthe algebras over a 2-monad on the 2-category Cat(Graph) whose objects are the categoryobjects in graphs; see [3, § Proposition 6.4.
There is a model structure on
DblCat , called the algebra model struc-ture , in which a double functor F is a weak equivalence if and only if its underlying mor-phism in the -category Cat(Graph) is an equivalence.Remark . In [3, Corollary 8.29 and Theorems 8.52 and 9.1], Fiore, Paoli, and Pronkshow that the model structures on DblCat of Propositions 6.2 to 6.4 coincide with modelstructures given by Grothendieck topologies, when double categories are seen as internalcategories to Cat. Then, it follows from [3, Propositions 8.24 and 8.38] that a weak equiv-alence in the algebra model structure is in particular a weak equivalence in the modelstructure induced by the vertical nerve N v . Remark . At this point, we must mention that [3] defines one other model structureon DblCat, which is not equivalent to any of the above. However, this is constructed fromthe Thomason model structure on Cat, and is therefore not expected to have any relationto our model structure, which is closely related to the canonical model structure on Cat,as shown in Corollary 4.13.We now proceed to compare these three model structures on DblCat to the one defined inTheorem 3.16. Our strategy will be to find a trivial cofibration in our model structure thatis not a weak equivalence in any of the other model structures. Let E adj be the 2-categorycontaining two objects 0 and 1 and an adjoint equivalence between them. By [12, § j : → E adj at 0 is a generating trivial cofibration in the Lackmodel structure on 2Cat. By Proposition 3.18, the double functor H j : → H E adj is agenerating trivial cofibration in the model structure of Theorem 3.16. Lemma 6.7.
The double functor H j : → H E adj is not a weak equivalence in any of themodel structures of Propositions 6.2 to 6.4.Proof. We first prove that H j is not a weak equivalence in the model structure on DblCatof Proposition 6.2 induced by the vertical nerve. For this, we need to show that N v ( H j ) : N v ( ) = ∆ → N v ( H E adj )is not a levelwise equivalence of categories. Indeed, the category N v ( H E adj ) is the freecategory generated by { ⇄ } which is not equivalent to .By Remark 6.5, a weak equivalence in the algebra model structure on DblCat of Propo-sition 6.4 is in particular a weak equivalence in the model structure induced by the verticalnerve. Therefore, H j is not a weak equivalence in the algebra model structure either.Finally, we show H j is not a weak equivalence in the trivial model structure on DblCatof Proposition 6.3. If H j was an equivalence in the 2-category DblCat h , then its weak inverse would be given by the unique double functor ! : H E adj → and we would have ahorizontal natural isomorphism id H E adj ∼ = H j !, where H j ! is constant at 0. But sucha horizontal natural isomorphism does not exist since 1 is not isomorphic to 0 in H E adj .Therefore H j is not an equivalence. (cid:3) Proposition 6.8.
The identity adjunction on
DblCat is not a Quillen pair between themodel structure of Theorem 3.16 and any of the model structures of Propositions 6.2 to 6.4.Proof.
We consider the identity functor id : DblCat → DblCat from the model structure ofTheorem 3.16 to any of the other model structures of Propositions 6.2 to 6.4, and we showthat it is neither left nor right Quillen.Since H j is a trivial cofibration in the model structure of Theorem 3.16, but is not aweak equivalence in any of the other model structures by Lemma 6.7, we see that id doesnot preserve trivial cofibrations; therefore, it is not left Quillen. Moreover, every object isfibrant in the model structure of Theorem 3.16, so that if id was right Quillen, it wouldpreserve all weak equivalences by Ken Brown’s Lemma (see [9, Lemma 1.1.12]). However,it does not preserve the weak equivalence H j , and thus it is not right Quillen. (cid:3) Model structure for weak double categories
In this section, we turn our attention to weak double categories, and show that ourresults for double categories still hold in this weaker setting. In particular, in Section 7.1we establish the existence of a model structure on the category wkDblCat s of weak doublecategories and strict double functors, which is right-induced from two copies of the Lackmodel structure of [12] for Bicat s , the category of bicategories and strict functors.As in the strict context, the horizontal embedding of bicategories into weak doublecategories is both left and right Quillen, and homotopically fully faithful; thus the homotopytheory of bicategories can be read off of the homotopy of weak double categories. Moreover,the Lack model structure on Bicat s is right-induced from this model structure on weakdouble categories along the horizontal embedding.Finally, in Section 7.2, we show the interplay between the model structures on thecategories considered so far. More precisely, we prove that the inclusion of double categoriesinto weak double categories is the right part of a Quillen equivalence, which gives rise tothe following commutative square of right Quillen functors2Cat Bicat s DblCat wkDblCat s . I ≃ Q I ≃ Q H H w Constructing the model structure for wkDblCat s . Let us first define weak doublecategories.
Definition 7.1. A weak double category B consists of objects, horizontal morphisms,vertical morphisms, and squares, together with horizontal and vertical compositions and identities such that composition of vertical morphisms is strictly associative and unital, asfor double categories, but composition of horizontal morphisms is associative and unitalup to a vertically invertible square with trivial boundaries. Notation 7.2.
We write wkDblCat s for the category of weak double categories and strictdouble functors between them. Remark . A bicategory B can be seen as a horizontal weak double category H w B withonly trivial vertical morphisms. This induces a functor H w : Bicat s → wkDblCat s . Remark . As in the case of 2Cat and DblCat, we have the following adjunctionswkDblCat s Bicat s , H w H w ⊥ wkDblCat s Bicat s , L w V w ⊥ where the functors involved are defined as the functors H , H , L , and V of Section 2 onobjects, morphisms, and cells. Note that the functor V w can be described as H w [ V w , − ],where V w is the weak double category having two objects 0 and 1, a vertical morphism u : 0 1, and non-trivial unitors for the horizontal identities.Just as in the case of 2Cat, Lack defines a model structure on Bicat s , whose weakequivalences and fibrations are also given by the biequivalences and the Lack fibrations;see [12]. Similarly, in wkDblCat s , we define double biequivalences and double fibrationsas in Definitions 3.5 and 3.7. Then, analogues of Propositions 3.10 and 3.11 hold in thisweak context, so that these double biequivalences and double fibrations correspond to theweak equivalences and fibrations in the right-induced model structure on wkDblCat s along( H w , V w ); this is the content of the following result. Theorem 7.5.
Consider the adjunction
Bicat s × Bicat s wkDblCat s , H w ⊔ L w ( H w , V w ) ⊥ where Bicat s is endowed with the Lack model structure. Then the right-induced modelstructure on wkDblCat s exists. In particular, a strict double functor is a weak equiva-lence (resp. fibration) in this model structure if and only if it is a double biequivalence(resp. double fibration).Proof. The proof is given by applying Theorem 3.14. Indeed, the model structure on Bicat s is combinatorial and all objects are fibrant (see [12]). As for the path objects, a slightalteration of the path object construction for DblCat produces the desired constructionfor the weak case. These path objects will have the same objects, horizontal morphisms,vertical morphisms, and squares as their double categorical counterpart, and only thehorizontal composition is modified in the obvious way; for details, see Section 8. (cid:3) Remark . Note that every weak double category is fibrant in this model structure.
Remark . One can show that the adjunction H w ⊣ H w gives a Quillen pair betweenBicat s and wkDblCat s whose unit is an identity, as shown in Proposition 4.2 for the strictcase. Therefore, H w is also homotopically fully faithful. Similarly, the adjunction L w ⊣ V w also gives a Quillen pair between Bicat s and wkDblCat s , as in Proposition 4.14.More interestingly, we also obtain an adjunction L w ⊣ H w analogous to the one inTheorem 4.4, which allows us to exhibit the model structure on Bicat s as right-inducedfrom our model structure on wkDblCat s . Theorem 7.8.
The adjunction wkDblCat s Bicat s L w H w ⊥ is a Quillen pair, where Bicat s is endowed with the Lack model structure and DblCat isendowed with the model structure of Theorem 7.5. Moreover, the Lack model structure on
Bicat s is right-induced along this adjunction.Proof. The proof proceeds as in Theorems 4.4 and 4.7. (cid:3)
Remark . By Remark 7.7 and Theorem 7.8, we can see that H w : Bicat s → wkDblCat s is both left and right Quillen, i.e., it preserves all cofibrations, fibrations, and weak equiv-alences. In particular, since H w is homotopically fully faithful by Remark 7.7, this saysthat the homotopy theory of Bicat s is reflectively and coreflectively embedded in that ofwkDblCat s via the functor H w .A characterization of the cofibrant objects, similar to that of Proposition 3.20, also holds.In the weak setting, we do not have an underlying horizontal category, since horizontalcomposition is not well defined when discarding the squares. Instead, the underlyinghorizontal structure is given by a compositional graph ; these are directed graphs, with achosen identity for each object, and a chosen composite for each composable pair of edges,but with neither the associativity law nor the unitality laws assumed to hold. Proposition 7.10.
A weak double category is cofibrant if and only if it is a retract of aweak double category whose(i) underlying horizontal compositional graph is free, and(ii) underlying vertical category is a disjoint union of copies of and .Remark . By [12, Lemma 8], a bicategory is cofibrant if and only if it is a retractof a bicategory whose underlying compositional graph is free. Therefore, the functor H w : wkDblCat s → Bicat s preserves cofibrant objects.7.2. Quillen equivalence between
DblCat and wkDblCat s . Now that we have defineda model structure for weak double categories, we study its relation with our model structureon DblCat. First, we recall Lack’s results, relating the model structures for 2-categoriesand bicategories.
Remark . The full inclusion I : 2Cat → Bicat s has a left adjoint, which we denote by S : Bicat s → B to the 2-category S B that has(i) the same objects as B , and(ii) morphisms and 2-cells obtained as a quotient of the morphisms and 2-cells of B ,where the associators and unitors are universally made into identities.More precisely, for all composable morphisms a, b, c in B , the composites ( cb ) a and c ( ba )are identified in S B , and, for every morphism a : A → B in B , the composites id B a and a id A are identified with a in S B . Furthermore, the unit component η B : B → I S B isdescribed as the quotient strict functor corresponding to the above identifications. Theorem 7.13 ([12, Theorem 11]) . The adjunction
Bicat s S I ⊥ is a Quillen equivalence, where and Bicat s are endowed with the Lack model structure.Remark . As Lack explains in [14, § S is not the usual strictification,but rather the “free strictification”. Notably, it is such that, for all bicategories B , the unitcomponents η B : B → I S B are strict functors, as opposed to the usual strictificationprocess that produces only pseudo-functors.We now study the setting of double categories, and show that an analogous result to the2-categorical one is also true. Remark . The full inclusion I : DblCat → wkDblCat s has a left adjoint, which wedenote by S : wkDblCat s → DblCat. This functor sends a weak double category B to thedouble category S B that has(i) the same objects and vertical morphisms as B , and(ii) horizontal morphisms and squares obtained as a quotient of the horizontal mor-phisms and squares of B , where the associators and unitors are universally madeinto identities.More precisely, for all composable horizontal morphisms a, b, c in B , the composites ( cb ) a and c ( ba ) are identified in S B , and, for every horizontal morphism a : A → B in B , thecomposites id B a and a id A are identified with a in S B . Furthermore, the unit component η B : B → IS B is described as the quotient strict double functor corresponding to the aboveidentifications. Remark . From the definitions of the functors involved, we directly see that I H = H w I, S H w = H S, I V = V w I, and S V w = V S. Furthermore, if we denote by η : id ⇒ I S and η : id ⇒ IS the units of the adjunctions S ⊣ I and S ⊣ I respectively, then we have that η H w = H w η : H w ⇒ I S H w = H w IS and η V w = V w η : V w ⇒ I S V w = V w IS.
Theorem 7.17.
The adjunction
DblCat wkDblCat s SI ⊥ is a Quillen equivalence, where DblCat is endowed with the model structure of Theorem 3.16and wkDblCat s with the model structure of Theorem 7.5.Proof. Since the inclusion I : DblCat → wkDblCat s creates weak equivalences, it is enoughto show, by [9, Corollary 1.3.16], that for every cofibrant B ∈ wkDblCat s the unit com-ponent η B : B → IL B is a double biequivalence in wkDblCat s . Let B be a cofibrant weakdouble category. By construction, it suffices to show that both H w η B and V w η B are biequiv-alences in Bicat s .By Remark 7.11, the bicategory H w B is cofibrant in Bicat s . Therefore, the unit compo-nent η H w B : H w B → I S H w B is a biequivalence, since I ⊣ S is a Quillen equivalence byTheorem 7.13. As H w η = η H w by Remark 7.16, this shows that H w η B is a biequivalence.We now prove that V w η B is a biequivalence. Since V w η = η V w by Remark 7.16, itis enough to show that the unit component η V w B : V w B → I S V w B is a biequivalence.By [12, Proposition 2], we know that η V w B is bijective on objects, surjective on morphisms,and full on 2-cells; it remains to show that it is faithful on 2-cells. For this, recall that a2-cell in V w B consists of two squares in B of the form A BA B ac • • σ A ′ B ′ A ′ B ′ bd • • σ satisfying a certain vertical pasting equality. Note that the squares σ and σ are 2-cells in H w B , and that, by the first part of the proof, the unit component η H w B : H w B → I S H w B is faithful on 2-cells. Then the quotient strict double functor η V w B : V w B → I S V w B isalso faithful on 2-cells, since it is given by applying η H w B to the components σ and σ separately. This shows that V w η B is a biequivalence, and concludes the proof. (cid:3) Finally, considering the different embeddings between the categories involved in thispaper, we get a commutative square of homotopically fully faithful right Quillen functors.
Corollary 7.18.
We have a commutative square of right Quillen functors s DblCat wkDblCat s , I ≃ Q I ≃ Q H H w where the horizontal functors are Quillen equivalences. Part II. Technical results Path objects in double categories
This section is devoted to the proof of Proposition 3.15, regarding the existence of pathobjects for double categories. For the reader’s convenience, we recall the precise statementof Proposition 3.15 here.
Proposition 3.15.
For every double category A , there exists a double category P A togetherwith a factorization of the diagonal double functor A W −→ P A P −→ A × A such that W is a double biequivalence and P is a double fibration. We start by defining the path object P A for a double category A . Definition 8.1.
Let A be a double category. The path object P A is the double categorydefined by the following data.(i) An object ( A, a , a ) in P A consists of a triple ( A, A , A ) of objects in A togetherwith horizontal equivalences a : A ≃ −→ A and a : A ≃ −→ A in A .(ii) A horizontal morphism ( f, ϕ , ϕ ) : ( A, a , a ) → ( B, b , b ) in P A consists of threehorizontal morphisms in A f : A → B, f : A → B , and f : A → B , together with two vertically invertible squares ϕ and ϕ in A as depicted below. A B BA A B f ≃ b a ≃ f • • ϕ ∼ = A B BA A B f ≃ b a ≃ f • • ϕ ∼ = (iii) Composition of horizontal morphisms is defined as follows: given horizontal mor-phisms ( f, ϕ , ϕ ) : ( A, a , a ) → ( B, b , b ) and ( g, ψ , ψ ) : ( B, b , b ) → ( C, c , c ),their composite is given by the composites gf : A → C, g f : A → C , and g f : A → C , together with the vertically invertible squares in A obtained by the pastings A B C A B B f ≃ b f g • • ψ ∼ = CC ≃ c g • A B C f g • • ϕ ∼ = A ≃ a • e g e f A B C A B B f ≃ b f g • • ψ ∼ = CC ≃ c g • A B C . f g • • ϕ ∼ = A ≃ a • e g e f (iv) A vertical morphism ( u, µ , µ ) : ( A, a , a ) ( A ′ , a ′ , a ′ ) consists of three verticalmorphisms in A u : A A ′ , u : A A ′ , and u : A A ′ , together with weakly horizontally invertible squares µ and µ in A as depictedbelow. A AA ′ A ′ u ua ≃ a ′ ≃ • • µ ≃ A AA ′ A ′ u ua ≃ a ′ ≃ • • µ ≃ (v) Composition of vertical morphisms is given by the vertical composition in eachcomponent of vertical morphisms and squares in A .(vi) A square in P A ( A, a , a ) ( B, b , b )( A ′ , a ′ , a ′ ) ( B ′ , b ′ , b ′ ) ( f, ϕ , ϕ )( g, ψ , ψ )( u, µ , µ ) ( v, ν , ν ) • • ( α, α , α ) consists of three squares in A AA ′ BB ′ , u vfg • • α A A ′ B B ′ , u v f g • • α A A ′ B B ′ u v f g • • α satisfying the following pasting equality, for each i = 0 , A i B i BA i A B f i b i ≃ a i ≃ f • • ϕ i ∼ = A ′ A ′ B ′ a ′ i ≃ gu i u v • • • αµ i ∼ = = A ′ i B ′ i B ′ A ′ i A ′ B ′ g i b ′ i ≃ a ′ i ≃ g • • ψ i ∼ = A i B i B f i b i ≃ u u i v i • • • ν i ∼ = α i (vii) Horizontal and vertical compositions of squares are given by componentwise hori-zontal and vertical compositions of squares in A .This path object comes with two double functors A W −→ P A P −→ A × A . Definition 8.2.
The double functor W : A → P A sends(i) an object A ∈ A to the object ( A, id A , id A ) ∈ P A ,(ii) a horizontal morphism f in A to the horizontal morphism ( f, e f , e f ) in P A ,(iii) a vertical morphism u in A to the vertical morphism ( u, id u , id u ) in P A , and(iv) a square α : ( u fg v ) to the square ( α, α, α ) in P A . Definition 8.3.
The double functor P : P A → A × A sends(i) an object ( A, a , a ) ∈ P A to the object ( A , A ) ∈ A × A ,(ii) a horizontal morphism ( f, ϕ , ϕ ) in P A to the horizontal morphism ( f , f ) in A × A ,(iii) a vertical morphism ( u, µ , µ ) in P A to the vertical morphism ( u , u ) in A × A ,and(iv) a square ( α, α , α ) in P A to the square ( α , α ) in A × A .It follows directly from the definitions that the composite P W is the diagonal functor A → A × A . Furthermore, the functors W and P have the desired properties, as we showin the following lemmas. Lemma 8.4.
The double functor W : A → P A is a double biequivalence.Proof. We prove (db1-4) of Definition 3.5.We first prove (db1). Let (
A, a , a ) be an object in P A . Then W A = ( A, id A , id A )comes with a horizontal equivalence ( A, a , a ) ≃ −→ W A in P A given byid A : A ≃ −→ A, a : A ≃ −→ A, and a : A ≃ −→ A, and the weakly invertible squares e a and e a .We now prove (db2). Let ( f, ϕ , ϕ ) : W A → W B be a morphism in P A , i.e., ϕ and ϕ are vertically invertible squares in A of the form A BA B , f f • • ϕ ∼ = A BA B . f f • • ϕ ∼ = Then
W f = ( f, e f , e f ) : W A → W B comes with a vertically invertible square in P A W A W BW A W B ( f, ϕ , ϕ ) W f • • ∼ = given by the triple of vertically invertible squares ( e f , ϕ , ϕ ).We now prove (db3). Let ( u, µ , µ ) : ( A, a , a ) ( A ′ , a ′ , a ′ ) be a vertical morphismin P A . Then W u = ( u, id u , id u ) : W A W A ′ comes with a weakly horizontally invertiblesquare in P A ( A, a , a ) W A ( A ′ , a ′ , a ′ ) W A ′ ≃ (id A , e a , e a ) ≃ (id A ′ , e a ′ , e a ′ )( u, µ , µ ) W u • • ≃ given by the triple of weakly horizontally invertible squares (id u , µ , µ ).Finally, we prove (db4). Let ( α, α , α ) : ( W u
W fW g
W v ) be a square in P A . By therelations in Definition 8.1 (vi), we must have α = α = α . Then W α = ( α, α, α ) =( α, α , α ) and it is the unique square in A satisfying this equality. (cid:3) Lemma 8.5.
The double functor P : P A → A × A is a double fibration.Proof. We prove (df1-3) of Definition 3.7.We first prove (df1). Let (
C, c , c ) be an object in P A with P ( C, c , c ) = ( C , C ),and let ( f , f ) : ( B , B ) ≃ −→ ( C , C ) be a horizontal equivalence in A × A . We define theobject ( C, c f , c f ) in P A through the horizontal equivalences B C C , ≃ f ≃ c B C C , ≃ f ≃ c and then the horizontal equivalence (id C , e c f , e c f ) : ( C, c f , c f ) ≃ −→ ( C, c , c ) in P A given by the tuple of horizontal equivalences (id C , f , f ) is sent via P to ( f , f ). We now prove (df2). Let ( g, ψ , ψ ) : ( A, a , a ) → ( C, c , c ) be a horizontal morphismin P A , and let ( β , β ) be a vertically invertible square in A × A as below, A C A C f g • • β ∼ = A C A C f g • • β ∼ = where ( g , g ) = P ( g, ψ , ψ ). We define the morphism ( g, ϕ , ϕ ) : ( A, a , a ) → ( C, c , c )through the horizontal morphisms g : A → C, f : A → C , and f : A → C , and the vertically invertible squares ϕ and ϕ given by the following pastings. A C CA A C g c ≃ a ≃ g • • ψ ∼ = A C C f c ≃ • • • e c β ∼ = A C CA A C g c ≃ a ≃ g • • ψ ∼ = A C C f c ≃ • • • e c β ∼ = Then the vertically invertible square in P A ( A, a , a ) ( C, c , c )( A, ϕ , ϕ ) ( C, ψ , ψ ) ( g, ϕ , ϕ )( g, ψ , ψ ) • • ∼ = given by ( e g , β , β ) is sent via P to ( β , β ).Finally, we prove (df3). Let ( u, ν , ν ) : ( C, c , c ) ( C ′ , c ′ , c ′ ) be a vertical morphismin P A , and let ( β , β ) be a weakly horizontally invertible square in A × A as below, B C B ′ C ′ v u f ≃ g ≃ • • β ≃ B C B ′ C ′ v u f ≃ g ≃ • • β ≃ where ( u , u ) = P ( u, ν , ν ). We set a = c f , a = c f , a ′ = c ′ g , and a ′ = c ′ g .We define the morphism ( u, µ , µ ) : ( C, a , a ) → ( C ′ , a ′ , a ′ ) in P A through the verticalmorphisms u : C C ′ , v : B B ′ , and v : B B ′ and the weakly invertible squares µ and µ given by the following pastings. B C B ′ C ′ CC ′ v u uf ≃ g ≃ c ≃ c ′ ≃ • • • β ≃ ν ≃ B C B ′ C ′ CC ′ v u uf ≃ g ≃ c ≃ c ′ ≃ • • • β ≃ ν ≃ Then the weakly horizontally invertible square in P A ( C, a , a ) ( C, c , c )( C ′ , a ′ , a ′ ) ( C ′ , c ′ , c ′ ) ≃ (id C , e a , e a ) ≃ (id C ′ , e a ′ , e a ′ )( u, µ , µ ) ( u, ν , ν ) • • ≃ given by (id u , β , β ) is sent via P to ( β , β ). (cid:3) The above constructions provide a proof of Proposition 3.15, as we now summarize.
Proof of Proposition 3.15.
Let A be a double category, and let P A be the path objectconstructed in Definition 8.1. Consider the functors A W −→ P A P −→ A × A as in Definitions 8.2 and 8.3; these provide a factorization of the diagonal functor. Moreover,Lemmas 8.4 and 8.5 show that W is a double biequivalence and P is a double fibration. (cid:3) Characterization of weak equivalences and fibrations
This section provides proofs of Propositions 3.10 and 3.11, which claim that the weakequivalences and fibrations of the right-induced model structure on DblCat of Theorem 3.16are precisely the double biequivalences of Definition 3.5 and the double fibrations of Defi-nition 3.7.We first focus on Proposition 3.10, dealing with weak equivalences. In order to charac-terize the functors F such that ( H , V ) F is a weak equivalence, we express what it meansfor H F and V F to be biequivalences in 2Cat; this is done by translating (b1-3) of Defini-tion 3.1 for these 2-functors. Remark . Let F : A → B be a double functor. Then H F : H A → H B is a biequivalencein 2Cat if and only if it satisfies (db1-2) of Definition 3.5, and the following condition:(hb3) for every square in B of the form F A F CF A F C , F aF c • • β there exists a unique square α : ( e A ac e C ) in A such that F α = β . Remark . Let F : A → B be a double functor. Then V F : V A → V B is a biequivalencein 2Cat if and only if it satisfies (db3) of Definition 3.5, and the following conditions:(vb2) for every square β : ( F u bd F u ′ ) in B , there exist a square α : ( u ac u ′ ) in A and twovertically invertible squares in B such that the following pasting equality holds, F A F CF A F CF A ′ F C ′ bF aF u F u ′ F c • •• • ∼ = F α = F A F CF A ′ F C ′ F A ′ F C ′ bdF u F u ′ F c • •• • β ∼ = (vb3) for all squares τ and τ as in the pasting equality below left, there exist uniquesquares σ : ( e A aa ′ e C ) and σ : ( e A ′ cc ′ e C ′ ) in A satisfying the pasting equality belowright, and such that F σ = τ and F σ = τ . F A F CF A F CF A ′ F C ′ F aF a ′ F u F u ′ F c ′ • •• • τ F α ′ = F A F CF A ′ F C ′ F A ′ F C ′ F aF cF u F u ′ F c ′ • •• • F ατ A CA CA ′ C ′ aa ′ u u ′ c ′ • •• • σ α ′ = A CA ′ C ′ A ′ C ′ acu u ′ c ′ • •• • ασ The reader may have noticed that condition (db4) in Definition 3.7 has not been men-tioned so far. The following lemma explains how the additional conditions (hb3) and(vb2-3) introduced in Remarks 9.1 and 9.2 relate to (db4).
Lemma 9.3.
Suppose F : A → B is a double functor satisfying (hb3) of Remark 9.1,and (vb2-3) of Remark 9.2. Then, for every square in B of the form F A F CF A ′ F C ′ , F aF cF u F u ′ • • β there exists a unique square α : ( u ac u ′ ) in A such that F α = β .Proof. Suppose β : ( F u
F aF c
F u ′ ) is a square in B as above. By (vb2) of Remark 9.2, thereexists a square α : ( u ac u ′ ) in A and two vertically invertible squares ψ , ψ in B such thatthe following pasting equality holds. F A F CF A F CF A ′ F C ′ F aF aF u F u ′ F c • •• • ∼ = ψ F α = F A F CF A ′ F C ′ F A ′ F C ′ F aF cF u F u ′ F c • •• • β ∼ = ψ By (hb3) of Remark 9.1, there exist unique squares ϕ and ϕ in A A CA C aa • • ϕ A ′ C ′ A ′ C ′ cc • • ϕ such that F ϕ = ψ and F ϕ = ψ . Moreover, the squares ϕ and ϕ are verticallyinvertible by the unicity condition in (hb3) of Remark 9.1. Therefore, the square α givenby the following vertical pasting A CA ′ C ′ acu u ′ • • α = A CA CA ′ C ′ A ′ C ′ aau u ′ cc • •• •• • ∼ = ϕ α ∼ = ϕ − is such that F α = β . This settles the matter of the existence of the square α . Now supposethere are two squares α and α ′ in A A CA C acu u ′ • • α A ′ C ′ A ′ C ′ acu u ′ • • α ′ such that F α = β = F α ′ . Take τ = e F a and τ = e F c in (vb3) of Remark 9.2. This givesunique squares σ and σ in A such that the following pasting equality holds A CA CA ′ C ′ aau u ′ c • •• • σ α ′ = A CA ′ C ′ A ′ C ′ acu u ′ c • •• • ασ and F σ = e F a and
F σ = e F c . By unicity in (hb3) of Remark 9.1, we must have σ = e a and σ = e c . Replacing σ and σ by e a and e c in the pasting diagram above impliesthat α = α ′ . This proves unicity. (cid:3) We can now use the above results to prove Proposition 3.10, giving the characterizationof the weak equivalences.
Proof of Proposition 3.10.
Suppose that F : A → B is a double functor such that both H F and V F are biequivalences in 2Cat. By Remarks 9.1 and 9.2, we directly have (db1-3) ofDefinition 3.5. Moreover, by Lemma 9.3, we also have (db4) of Definition 3.5. This showsthat F is a double biequivalence.Now suppose that F : A → B is a double biequivalence. We want to show that both H F and V F are biequivalences in 2Cat. To show that H F is a biequivalence, it suffices toshow that (hb3) of Remark 9.1 is satisfied; this follows directly from taking u and u ′ to bevertical identities in (db4) of Definition 3.5.It remains to show that V F is a biequivalence; we do so by proving (vb2-3) of Remark 9.2.To prove (vb2), let β be a square in B of the form F A F CF A ′ F C ′ . bdF u F u ′ • • β By (db2) of Definition 3.5, there exist horizontal morphisms a : A → C and c : A ′ → C ′ in A and vertically invertible squares ϕ and ϕ in B as depicted below. F A F CF A F C bF a • • ∼ = ϕ F A ′ F C ′ F A ′ F C ′ dF c • • ∼ = ϕ By (db4) of Definition 3.5, there exists a unique square α : ( u ac u ′ ) in A such that F A F CF A ′ F C ′ F aF cF u F u ′ • • F α = F A F CF A F CF A ′ F C ′ F A ′ F C ′ , F abF u F u ′ dF c • •• •• • ∼ = ϕ − β ∼ = ϕ which gives (vb2). Finally, we prove (vb3). Suppose we have the following pasting equalityin B . F A F CF A F CF A ′ F C ′ F aF a ′ F u F u ′ F c ′ • •• • τ F α ′ = F A F CF A ′ F C ′ F A ′ F C ′ F aF cF u F u ′ F c ′ • •• • F ατ Applying (db4) of Definition 3.5 to τ and τ gives unique squares σ : ( e A aa ′ e C ) and σ : ( e A ′ cc ′ e C ′ ) in A such that F σ = τ and F σ = τ . Moreover, by unicity in (db4) ofDefinition 3.5, we have that A CA CA ′ C ′ aa ′ u u ′ c ′ • •• • σ α ′ = A CA ′ C ′ A ′ C ′ , acu u ′ c ′ • •• • ασ since applying F to each vertical composite yields the same squares in B . This proves (vb3),and thus concludes our proof. (cid:3) Now we turn our attention to Proposition 3.11, dealing with fibrations. Our treatmentis analogous to that of weak equivalences: in order to characterize the functors F such that( H , V ) F is a fibration, we express what it means for H F and V F to be Lack fibrationsin 2Cat; this is done by translating (f1-2) of Definition 3.2 for these 2-functors. Remark . Let F : A → B be a double functor. Then H F : H A → H B is a fibrationin 2Cat if and only if it satisfies (df1-2) of Definition 3.7. Remark . Let F : A → B be a double functor. Then V F : V A → V B is a fibrationin 2Cat if and only if it satisfies (df3) of Definition 3.7, and the following condition:(vf2) for every square α ′ : ( u a ′ c ′ u ′ ) in A and every square β : ( F u bd F u ′ ) in B , togetherwith vertically invertible squares τ and τ in B as in the pasting equality belowleft, there exists a square α : ( u ac u ′ ), together with vertically invertible squares σ and σ in A as in the pasting equality below right, such that F α = β , F σ = τ ,and F σ = τ . F A F CF A F CF A ′ F C ′ bF a ′ F u F u ′ F c ′ • •• • τ ∼ = F α ′ = F A F CF A ′ F C ′ F A ′ F C ′ bdF u F u ′ F c ′ • •• • βτ ∼ = A CA CA ′ C ′ aa ′ u u ′ c ′ • •• • σ ∼ = α ′ = A CA ′ C ′ A ′ C ′ acu u ′ c ′ • •• • ασ ∼ = We can now use the above remarks to provide a proof of Proposition 3.11, giving thecharacterization of the fibrations.
Proof of Proposition 3.11.
It is clear that if a double functor F : A → B is such thatboth H F and V F are Lack fibrations in 2Cat, then it is a double fibration, by Remarks 9.4and 9.5.Suppose now that F : A → B is a double fibration. By Remark 9.4, we directly getthat H F is a Lack fibration in 2Cat. To show that V F is is also a Lack fibration, it suffices to show that (vf3) of Remark 9.5 is satisfied. Let α ′ : ( u a ′ c ′ u ′ ) be a square in A and β : ( F u bd F u ′ ) be a square in B , together with vertically invertible squares τ and τ in B such that the following pasting equality holds. F A F CF A F CF A ′ F C ′ bF a ′ F u F u ′ F c ′ • •• • τ ∼ = F α ′ = F A F CF A ′ F C ′ F A ′ F C ′ bdF u F u ′ F c ′ • •• • βτ ∼ = By (df2) of Definition 3.7, there exist vertically invertible squares σ and σ in A A CA C aa ′ • • σ ∼ = A ′ C ′ A ′ C ′ cc ′ • • σ ∼ = such that F σ = τ and F σ = τ . Then the square α given by the vertical composite A CA ′ C ′ acu u ′ • • α = A CA CA ′ C ′ A ′ C ′ aa ′ u u ′ c ′ c • •• •• • ∼ = σ α ′ ∼ = σ − is such that F α = β , which proves (vf3). (cid:3) Generating (trivial) cofibrations and cofibrant objects
In this section, we take a closer look at the (trivial) cofibrations and cofibrant objectsin our model structure on DblCat. In Proposition 3.18, we specified sets of generatingcofibrations and generating trivial cofibrations for this model structure. In fact, thereexist smaller sets of generating cofibrations and generating trivial cofibrations, determineddirectly by their left lifting properties, as we show in Proposition 10.2.
A closer study of the lifting properties further reveals that cofibrations can be char-acterized through their underlying horizontal and vertical (1-)functors; this is done inProposition 10.5. Finally, in Proposition 10.12, we use this characterization to describe thecofibrant double categories in our model structure.Let us first describe new sets of generating cofibrations and generating trivial cofibra-tions.
Notation 10.1.
Let S be the double category containing a square, δ S be its boundary,and S be the double category containing two squares with same boundaries. S = 0 10 ′ ′ ; α • • δ S = 0 10 ′ ′ ; • • S = 0 10 ′ ′ α α • • We fix notation for the following double functors, which form a set of generating cofibrationsfor our model structure on DblCat: • the unique map I : ∅ → , • the inclusion I : ⊔ → H , • the unique map I : ∅ → V , • the inclusion I : δ S → S , and • the double functor I : S → S sending both squares in S to the non-trivial squareof S .We also fix notation for the following double functors, which form a set of generating trivialcofibrations for our model structure on DblCat: • the inclusion J : → H E adj , where E adj is the 2-category containing an adjointequivalence, • the inclusion J : H → H C inv , and • the inclusion J : V → H E adj × V ; note that H E adj × V is the double categorycontaining a weakly horizontally invertible square. Proposition 10.2.
In the model structure on
DblCat of Theorem 3.16, a set of generatingcofibrations is given by I ′ = { I : ∅ → , I : ⊔ → H , I : ∅ → V , I : δ S → S , I : S → S } and a set of generating trivial cofibrations is given by J ′ = { J : → H E adj , J : H → H C inv , J : V → H E adj × V } . Proof.
It is a routine exercise to check that a double functor is a double trivial fibrationas defined in Definition 3.8 if and only if it has the right-lifting property with respectto the cofibrations in I ′ , and that a double functor is a double fibration as defined inDefinition 3.7 if and only if it has the right-lifting property with respect to the trivialcofibrations of J ′ . This shows that I ′ and J ′ are sets of generating cofibrations andgenerating trivial cofibration for DblCat, respectively. (cid:3) Our next goal is to provide a characterization of the cofibrations in DblCat. In [11,Lemma 4.1], Lack shows that a 2-functor is a cofibration in 2Cat if and only if its underlyingfunctor has the left lifting property with respect to all surjective on objects and full functors.A similar result applies to our model structure; indeed, we show that a double functor isa cofibration in DblCat if and only if its underlying horizontal functor and its underlyingvertical functor satisfy respective lifting properties.
Remark . The functor U H : DblCat → Cat, which sends a double category to itsunderlying category of objects and horizontal morphisms, has a right adjoint. It is givenby the functor R h : Cat → DblCat that sends a category C to the double category with thesame objects as C , horizontal morphisms given by the morphisms of C , a unique verticalmorphism between every pair of objects and a unique square ! : (! fg !) for every pair ofmorphisms f, g in C . Remark . The functor U V : DblCat → Cat, which sends a double category to itsunderlying category of objects and vertical morphisms, has a right adjoint. It is given bythe functor R v : Cat → DblCat that sends a category C to the double category with thesame objects of C , a unique horizontal morphism between every pair of objects, verticalmorphisms given by the morphisms of C , and a unique square ! : ( u !! v ) for every pair ofmorphisms u, v in C . Proposition 10.5.
A double functor F : A → B is a cofibration in DblCat if and only if(i) the functor U H F : U H A → U H B has the left lifting property with respect to allsurjective on objects and full functors, and(ii) the functor U V F : U V A → U V B has the left lifting property with respect to allsurjective on objects and surjective on morphisms functors.Proof. Suppose first that F : A → B is a cofibration in DblCat, i.e., it has the left liftingproperty with respect to all double trivial fibrations. In order to show (i), let P : X → Y be a surjective on objects and full functor. By the adjunction U H ⊣ R h , saying that U H F has the left lifting property with respect to P is equivalent to saying that F has the leftlifting property with respect to R h P . We now prove this latter statement.Note that the double functor R h P : R h X → R h Y is surjective on morphisms and fullon horizontal morphisms, since P is. Moreover, by construction of R h , there is exactlyone vertical morphism and one square for each boundary in both its source and target;therefore R h P is surjective on vertical morphisms and fully faithful on squares. Hence R h P is a double trivial fibration, and F has the left lifting property with respect to R h P sinceit is a cofibration in DblCat.Similarly, one can show that (ii) holds, by considering the adjunction U V ⊣ R v andreplacing fullness by surjectivity on morphisms.Now suppose that F : A → B satisfies (i) and (ii). Given a double trivial fibration P : X → Y and a commutative square as below, we want to find a lift L : B → X . AB XY
GF H PL
Using (ii), since U V P is surjective on objects and surjective on morphisms, we have alift L v in the following diagram. U V A U V B U V X U V Y U V GU V F U V H U V PL v Now, using (i), since U H P is surjective on objects and full, we can choose a lift L h in thefollowing diagram U H A U H B U H X U H Y U H GU H F U H H U H PL h such that L h coincides with L v on objects. This comes from the fact that, by fullnessof U H P , we can first choose an assignment on objects and then choose a compatible as-signment on morphisms. Then, since P : X → Y is fully faithful on squares, the assignmenton objects, horizontal morphisms, and vertical morphisms given by L h and L v uniquelyextend to a double functor L : B → Y , which gives the desired lift. (cid:3) Remark . Note that the functor H : DblCat → Proposition 10.7.
There is a cofibrantly generated weak factorization system ( G , R ) on Cat , where R is the set of surjective on objects and full functors. A set of generatingmorphisms for G can be chosen to be { i : ∅ → , i : ⊔ → } . Corollary 10.8.
A functor F : A → B is in G if and only if(i) the functor F is injective on objects and faithful, and (ii) there exist functors I : B → C and R : C → B such that RI = id B , where thecategory C is obtained from the image of F by freely adjoining objects and thenfreely adjoining morphisms between specified objects.Moreover, a functor ∅ → A is in G if and only if A is a retract of a free category C . Inparticular, the category A is itself free. Proposition 10.9.
There is a cofibrantly generated weak factorization system ( H , S ) on Cat , where S is the set of surjective on objects and surjective on morphisms functors.A set of generating morphisms for H can be chosen to be { i : ∅ → , i : ∅ → } . Corollary 10.10.
A functor F : A → B is in H if and only if(i) the functor F is injective on objects and faithful, and(ii) there exist functors I : B → C and R : C → B such that RI = id B , where the category C is obtained from the image of F by freely adjoining objects and morphisms.Moreover, a functor ∅ → A is in H if and only if A is a retract of a category C that is adisjoint union of copies of and . In particular, the category A is itself a disjoint unionof copies of and . From these characterizations, we get the following result.
Corollary 10.11.
If a double functor is a cofibration in
DblCat , then it is injective onobjects, faithful on horizontal morphisms, and faithful on vertical morphisms.Proof.
This follows directly from Proposition 10.5 and Corollaries 10.8 and 10.10. (cid:3)
Finally, we use the above results to obtain a characterization of the cofibrant doublecategories in terms of their underlying horizontal and vertical categories.
Proposition 10.12.
A double category A is cofibrant if and only if its underlying horizontalcategory U H A is free and its underlying vertical category U V A is a disjoint union of copiesof and .Proof. By Proposition 10.5, a double category A is cofibrant if and only if(i) the functor ∅ → U H A has the left lifting property with respect to all surjective onobjects and full functors, and(ii) the functor ∅ → U V A has the left lifting property with respect to all surjective onobjects and surjective on morphisms functors.By Corollary 10.8, (i) is equivalent to the category U H A being free, and, by Corollary 10.10,(ii) is equivalent to the category U V A being a disjoint union of copies of and . (cid:3) Part III. The Whitehead Theorem
A Whitehead Theorem for double categories
In this section, we show that a Whitehead Theorem for double biequivalences is availablein some cases. As we will see, this continues to highlight the close connection between ourmodel structure and Lack’s model structure on 2Cat.Recall the statement of the Whitehead Theorem for biequivalences between 2-categories:a 2-functor F : A → B is a biequivalence if and only if there exists a pseudo functor G : B → A together with two pseudo natural equivalences id ≃ GF and F G ≃ id. Thisis a long-established result in the literature; a proof can be found, for example, in [10,Theorem 7.4.1].Under certain conditions, we can show an analogous characterization of double biequiv-alences using pseudo horizontal natural equivalences; this is done in Theorem 11.14, whichwe interpret as a Whitehead Theorem for double categories. In particular, this holds fordouble biequivalences of the form H F : H A → H B , for any 2-functor F : A → B , and sothis result recovers the 2-categorical version.Let us first introduce the notions of pseudo double functors and pseudo horizontal naturalequivalences, which are needed to state the theorem.
Definition 11.1. 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 [5, 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 also notions of pseudo horizontal natural transformations between(normal) pseudo double functors, and modifications between them (with trivial verticalboundaries). These are defined analogously to Definition 5.4 and satisfy similar coherenceconditions to the ones in [5, § Definition 11.3.
Let
F, G : A → B be (normal) pseudo double functors. A pseudo hor-izontal natural equivalence ϕ : F ⇒ G is an equivalence in the 2-category of (normal)pseudo double functors A → B , pseudo natural horizontal transformations, and modifica-tions with trivial vertical boundaries. Remark . Equivalently, a pseudo horizontal natural equivalence is a pseudo horizontalnatural transformation h : F ⇒ G such that the horizontal morphisms h A : F A ≃ −→ GA are horizontal equivalences, for all A ∈ A , and the squares h u : ( F u h A h A ′ Gu ) are weaklyhorizontally invertible, for all vertical morphisms u : A A ′ in A ; see [15].We now introduce a notion of horizontal biequivalence for a double functor which admitsa pseudo weak inverse. Definition 11.5.
A double functor F : A → B is a horizontal biequivalence if there exista pseudo double functor G : B → A and pseudo horizontal natural equivalences η : id ⇒ GF and ǫ : F G ⇒ id. 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 pseudo horizontal natural adjoint equivalence( η : id ⇒ GF, η ′ : GF ⇒ id , λ : id ∼ = η ′ η, κ : ηη ′ ∼ = id) , where λ and κ satisfy the triangle identities,(iii) a pseudo horizontal natural adjoint equivalence( ǫ : F G ⇒ id , ǫ ′ : id ⇒ 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 [7, Theorem 3.2], saying that a biequivalence can alwaysbe promoted to a biadjoint biequivalence.Our next goal is to show one direction of the characterization provided in the WhiteheadTheorem; namely, that a horizontal biequivalence is in particular a double biequivalence.In order to prove this result, we need the following lemma. Lemma 11.7.
The data of Remark 11.6 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 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 11.8. 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 3.5. Let ( F, G, η, ǫ ) bethe data of a horizontal adjoint biequivalence as in Remark 11.6.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 11.7.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. 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 11.7; 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 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 a converse statement forProposition 11.8. Such a result will not hold in the same generality, and we will haveto require either a condition saying that all vertical morphisms are trivial, or anothercondition, which we now introduce. The formulation of this second condition is inspiredby the definition of horizontal invariance for a double category given by Grandis in [5,Theorem and Definition 4.1.7]. This latter definition is needed to prove a version of theWhitehead theorem for a stricter notion of equivalence of double categories; see [5, Theorem4.4.5].
Definition 11.9.
A double category A is weakly horizontally invariant if, for allhorizontal equivalences a : A ≃ −→ C and c : A ′ ≃ −→ C ′ in A and every vertical morphism u ′ : C C ′ in A , there exist a vertical morphism u : A A ′ and 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 [5, § Lemma 11.11.
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 3.5,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 3.5, 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 11.8, under the additional assumption that ourdomain double category is weakly horizontally invariant.
Proposition 11.12.
Let F : A → B be a double biequivalence, where the double category A is weakly horizontally invariant. Then F is a horizontal biequivalence.Proof. We simultaneously define the pseudo double functor G : B → A and the pseudohorizontal natural transformation ǫ : F G ⇒ id. G and ǫ on objects. Let B ∈ B be an object. By (db1) of Definition 3.5, 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 3.5, 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 3.5, 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 3.5, 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 ≃ • • 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 11.11, 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 Lemma 2.17,and thus so is ǫ v . We write ǫ ′ v for its unique weak inverse with respect to the horizontaladjoint equivalences ( ǫ B , ǫ ′ B , µ B , ν B ) and ( ǫ B ′ , ǫ ′ B ′ , µ B ′ , ν B ′ ), as given by Lemma 2.16.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 3.5, 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. 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 3.5, 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 pseudo horizontalnatural equivalence ǫ : F G ⇒ id. 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 pseudo horizontal naturalequivalence ǫ ′ : id ⇒ F G , where ǫ ′ b is defined in a similar manner as ǫ b was. In particular, ǫ : F G ⇒ id and ǫ ′ : id ⇒ F G are adjoint equivalences, where the invertible modificationsare given by µ : id ∼ = ǫ ′ ǫ and ν : ǫǫ ′ ∼ = id.It remains to define the pseudo horizontal natural equivalence η : id ⇒ GF . For thispurpose, we use the pseudo horizontal natural equivalence ǫ ′ : id ⇒ 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 3.5, 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 11.11. 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 3.5, 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. 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 Lemma 2.17,and thus so is ψ u . By (db4) of Definition 3.5, there exists a unique weakly horizontallyinvertible 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 ⇒ F G is a pseudo horizontal natural transformation, then η A , η a , and η u assemble into a pseudo horizontal natural transformation η : id ⇒ GF . Notethat η is a pseudo horizontal natural equivalence, because η A are horizontal equivalencesand η u are weakly horizontally invertible squares. Moreover, ρ : ǫ ′ F ∼ = F η gives the data ofan invertible modification. (cid:3)
Remark . A careful study of the proof of Proposition 11.12 reveals that, if all thevertical morphisms in the double category B are identities, then the result holds withoutrequiring that A be weakly horizontally invariant, since this condition is only needed fordefining the pseudo functor G : B → A on vertical morphisms. Put together, Propositions 11.8 and 11.12 and Remark 11.13 give the following character-ization of the double biequivalences, when the source double category is weakly horizontallyinvariant or the target double category is horizontal.
Theorem 11.14 (Whitehead Theorem for double categories) . Let A and B be doublecategories such that A is weakly horizontally invariant or B has only trivial vertical mor-phisms. Then a double functor F : A → B is a double biequivalence if and only if thereexists a pseudo double functor G : B → A together with horizontal pseudo natural equiva-lences id ≃ GF and F G ≃ id . In particular, we can restrict our results to double functors arising from 2-functors,and recover the well-known statement of the aforementioned Whitehead Theorem for2-categories.
Corollary 11.15 (Whitehead Theorem for 2-categories) . Let F : A → B be a -functor.Then F is a biequivalence if and only if there exists a pseudo functor G : B → A togetherwith two pseudo natural equivalences id ≃ GF and F G ≃ id .Proof. This can be obtained as a direct application of Theorem 11.14 to the double functor H F : H A → H B . It follows from the fact that pseudo double functors and pseudo horizontalnatural equivalences between double categories in the image of H are equivalently pseudofunctors and pseudo natural transformations between their preimages. (cid:3) References [1] Gabriella B¨ohm. The gray monoidal product of double categories.
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UPHESS BMI FSV, Ecole Polytechnique F´ed´erale de Lausanne, Station 8, 1015 Lausanne,Switzerland
E-mail address : [email protected] Department of Mathematics, Cornell University, Ithaca NY, 14853, USA
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