The Gray monoidal product of double categories

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double category whose 0-cells are the double functors $\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.


Introduction
The category 2-Cat of 2-categories and 2-functors carries different monoidal structures. The simplest one is given by the Cartesian product ×. It is symmetric and closed. For any 2-category A, the internal hom functor A, − sends a 2-category B to the 2-category of 2-functors A → B, 2-natural transformations, and modifications. This is, however, often too restrictive. For example, important examples of 2-categories which are intuitively monoidal, fail to be monoids for that [3,20,22]. A well established generalization is the so-called Gray monoidal product ⊗ of [19]. It is also symmetric and closed and for any 2-category A the corresponding internal hom functor (Again, the term monoidal functor is used in the same lax sense as in Section XI.2 of [21]. That is, a monoidal functor consists of coherent natural transformations which control the difference between the monoidal structure of the target category, and the image of the monoidal structure of the source category. ) We also give an explicit description of monoids in (DblCat, ⊗) which generalize the strict monoidal double categories of [7]; that is, the monoids in (DblCat, ×). They turn out to give rise to monoids in the category of double categories and double pseudo functors in the sense of [23,Definition 6.1], with respect to the Cartesian product. They do not seem to allow, however, for an interpretation as monoidal double categories in the sense of [24,Definition 2.9], [18,Section 3.1], [12,Definition 31] and [16,Section 5.5]; orwhat is the same-as one object locally cubical bicategories in [13]. That is to say, they do not seem to give rise to pseudo monoids in the monoidal bicategory of double categories, their homomorphisms in [12,16,18,24] and say, vertical transformations with the Cartesian product.
The author's personal motivation to study the Gray monoidal product of double categories arose from the work on [4]. In that paper, lifting of so-called multimonoidal structures (in the sense of [1]) to Eilenberg-Moore objects of multimonoidal monads is described in terms of a strict monoidal double functor between strict monoidal double categories (that is, monoids in DblCat with respect to the Cartesian product). While this is applicable to multimonoidal monads on nice enough categories as in [2], a generalization from Cat to more interesting Gray monoids (that is, monoids in 2-Cat with respect to the Gray tensor product) requires a notion of monoids in DblCat with respect to a suitable Gray monoidal product ⊗.
There should be further interesting applications. Gray categories-that is, categories enriched in (2-Cat, ⊗)-solve the coherence problem of tricategories: in [14] any tricategory is proven to be triequivalent to a Gray category. The Gray monoidal product ⊗ of DblCat is hoped to play some role in the coherence of triple categories.
One object groupoidal Gray categories are algebraic models of homotopy 3-types; an interesting application would be the appearance of (DblCat, ⊗) in some homotopy types.
2-Cat with the Gray tensor product is a monoidal model category. One can ask if the model structures on the category of small double categories in [11] are monoidal with respect to the Gray monoidal product in the current paper.

Existence
In this section we construct an adjunction − ⊗ D ⟦D, −⟧ of endofunctors on the category DblCat of double categories, for any double category D. Our line of reasoning is similar to [6,Proposition 3.10]. The occurring double functor ⊗ : DblCat × DblCat → DblCat is our candidate Gray monoidal product on DblCat. Mac Lane's coherence conditions are checked in Sect. 3.

The Category of Double Categories
We begin with introducing the category DblCat of double categories and double functors, and recording some of its basic properties. So a double category consists of 0-cells, also called objects, (interpreted as the objects of the category of objects), vertical 1-cells (which are the morphisms of the category of objects), horizontal 1-cells (the objects of the category of morphisms) and 2-cells (the morphisms of the category of morphisms). They can be composed vertically (in the category of objects and the category of morphisms, respectively) and horizontally (via the composition functor of the double category). As usual in the literature (see e.g. [15]), we denote 2-cells as squares surrounded by the appropriate horizontal and vertical source and target 1-cells. We denote by 1 both horizontal and vertical identity 1-cells; and also identity 2-cells for the horizontal or vertical composition. Usually we neither make notational difference between the compositions of horizontal and vertical 1-cells; both are denoted by a dot (if not a diagram is rather drawn).
By [11,Theorem 4.1] and its proof, DblCat is locally finitely presentable-so in particular cocomplete-and complete. Its terminal object 1 is the double category of a single object and only identity higher cells.
Consider the double category G which is freely generated by a single 2-cell. In more detail, G has four objects, we denote them by X , Y , V and Z . There are identity horizontal and vertical identity 1-cells for each object as well as non-identity horizontal and vertical 1-cells There are vertical identity 2-cells at each horizontal 1-cell, horizontal identity 2-cells at each vertical 1-cell, and a single non-identity 2-cell The functor DblCat(G, −) : DblCat → Set sends a double category A to the set of double functors G → A, which can be identified with the set of 2-cells in A. A double functor F is sent to its 2-cell part, which is an isomorphism in Set if and only if F is bijective on the 2-cells.
Since this includes bijectivity also on the identity 2-cells of various kinds, it is equivalent to F being bijective on all kinds of cells; that is, its being an isomorphism in DblCat. By the so obtained conservativity of the functor DblCat(G, −) : DblCat → Set we conclude that G is a strong generator of the finitely complete category DblCat with coproducts, see [5,Proposition 4.5.10].

The Double Categories of Double Functors
Using similar constructions to those in [ These ingredients are subject to the following axioms.
(i) Vertical functoriality, saying that for the identity vertical 1-cell 1 on any object A in A, x 1 is equal to the vertical identity 2-cell on the left; and for any composable vertical 1-cells f and g in A, the equality on the right holds: (ii) Horizontal functoriality, saying that for the identity horizontal 1-cell 1 on any object A, x 1 is equal to the same vertical identity 2-cell on the left; and for any composable horizontal 1-cells h and k in A, the equality on the right holds: (iii) Naturality, saying that for any 2-cell ω in A,  These ingredients are subject to the following axioms.
(i) Horizontal functoriality, saying that for the identity horizontal 1-cell 1 on any object A in A, y 1 is equal to the horizontal identity 2-cell on the left; and for any composable horizontal 1-cells h and k in A, the equality on the right holds: FA satisfying the following axioms.
(i) For any horizontal 1-cell h in A, (ii) For any vertical 1-cell f in A, The identity horizontal pseudo transformation has the components while the composite of some horizontal pseudo transformations F The components of the horizontal composite of modifications on the left of are the horizontal composites of their components on the right. Symmetrically, the components of the vertical composite of modifications on the left of are the vertical composites of their components on the right. Throughout, we identify any double category A with the isomorphic double category ⟦1, A⟧.

The Functor ⟦−, −⟧ : DblCat op × DblCat → DblCat
In this section we interpret the map, sending a pair of double categories A and B to the double category ⟦A, B⟧ of Sect. 2.2, as the object map of a functor in the title. So we need to construct its morphism map, sending a pair of double functors F : A → A and G : B → B to a double functor ⟦F, G⟧ : ⟦A, B⟧ → ⟦A , B ⟧.
The horizontal 1-cell part sends a horizontal pseudo transformation H x H to the horizontal pseudo transformation with the components for any horizontal 1-cell h and vertical 1-cell f in A . Finally, the 2-cell part sends a modification in the first diagram to the modification with components in the second diagram:

The Extranatural Transformation l
In this section we construct an extranatural transformation • The horizontal pseudo transformation with components Symmetrically, the vertical 1-cell part sends a vertical pseudo transformation y from G to G to the vertical pseudo transformation from ⟦1, G⟧ to ⟦1, G ⟧ with the following components, for any horizontal pseudo transformation p and vertical pseudo transformation q between double functors H, H : D → A.
• The vertical pseudo transformation with components • The horizontally invertible modification with components The 2-cell part sends a modification in the first diagram to the modification with components in the second diagram, for any 0-cell A in D: to the identity double functor, where 1 A : 1 → ⟦A, A⟧ is the double functor sending the single object of 1 to the identity double functor 1 A : A → A.

The Extranatural Transformation r
In this section we construct another extranatural transformation • The vertically invertible modification with the components Symmetrically, the vertical 1-cell part sends a vertical pseudo transformation y from F to F to the vertical pseudo transformation ⟦F, 1⟧ to ⟦F , 1⟧ with the following components, for any horizontal pseudo transformation p and vertical pseudo transformation q between double functors H, H : B → D. • The 2-cell part sends a modification in the first diagram to the modification with components in the second diagram, for any 0-cell A in A: for any double category D, which obeys the idempotent-like property that is the identity natural transformation. This f is natural in its lower indices by the naturality of r. It is natural in the upper index as well (here the upper index no longer refers to extranaturality).
In order to see that, both naturality and extranaturality of r are needed. The extranatural transformation r in this section and l in Sect. 2.4 together render commutative the following diagram, for any double categories A, B and D.
for any horizontal 1-cell n and vertical 1-cell v in G. They satisfy the naturality condition (2.5) Symmetrically, the left and right vertical 1-cells are vertical pseudo transformations labelled by the vertical 1-cells w ∈ {l, r } of G; with components denoted by for any horizontal 1-cell n and vertical 1-cell v in G. They satisfy the naturality condition (2.6) Finally, the 2-cell itself is a modification with components denoted by They satisfy the horizontal compatibility conditions (2.7) for all horizontal 1-cells h ∈ {t, b} in G; and the vertical compatibility conditions From all that we can read off that the functor DblCat(G, ⟦G, −⟧) : DblCat → Set is represented by the following double category.
All further cells are generated by their compositions modulo the associativity and unitality conditions, the middle four interchange law, and the identities (2.5), (2.6), (2.7) and (2.8).

Representability of the Functors DblCat(A, ⟦B, −⟧) : DblCat → Set
In this section we investigate the functors in the title, for any double categories A and B. Our main tool is the representability result of Sect. 2.6 and the following fact.
Consider a functor U : C → C between locally presentable categories. Bourke and Gurski's [6, Lemma 3.9] says that the functor C (X , U(−)) : C → Set is representable for all objects X -that is, U possesses a left adjoint -if and only if C (G i , U(−)) : C → Set is representable for all G i in a strong generator of C . The 0-cell part of the iso double functor (2.3) yields a bijection DblCat(B, ⟦G, A⟧) ∼ = DblCat(G, ⟦B, A⟧) for any double categories A and B; which is natural in A by the extranaturality and naturality of r. Therefore also the functor DblCat(G, ⟦B, −⟧) : DblCat → Set is representable for any double category B. Applying again Bourke and Gurski's lemma, we obtain the representability of DblCat(A, ⟦B, −⟧) for any double categories A and B.
In other words, the functor ⟦B, −⟧ : DblCat → DblCat possesses a left adjoint for any double category B which we denote by − ⊗ B. By the functoriality in B, it gives raise to a double functor ⊗ : DblCat × DblCat → DblCat. For any double categories A and B, an explicit description of A ⊗ B can be given analogously to Sect. 2.6. However, our verification of the coherence of ⊗ in Sect. 3 completely avoids using a presentation in terms of generators and relations (which we find an appealing feature). Therefore we do not work out the details of such a presentation.

Coherence
This section is devoted to the proof that the double functor ⊗ : DblCat × DblCat → DblCat of Sect. 2.7 renders DblCat a symmetric monoidal category (which is then closed with the internal hom functors ⟦B, −⟧, for all double categories B).
Taking mates under the adjunctions B ⊗ − ⟦B, −⟧ of Sect. 2 and using the Yoneda Lemma, we relate In what follows, the unit of the adjunction − ⊗ B ⟦B, −⟧ will be denoted by η B A : A → ⟦B, A ⊗ B⟧, and the counit will be denoted by B A : ⟦B, A⟧ ⊗ B → A, for all double categories A and B.

The Associativity Natural Isomorphism
For any double category C, consider the natural transformation  Since all of the occurring maps but the top row are known to be bijections, we conclude that so is the top row. Whence by Yoneda's lemma a C is a natural isomorphism. Using the adjunction isomorphisms in the first and last steps, we obtain a natural isomorphism By Yoneda's lemma again, it determines a natural isomorphism α A,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) which is our candidate associativity natural isomorphism.

The Pentagon Condition
By Yoneda's lemma, Mac Lane's pentagon condition on the natural isomorphism α of Sect. 3.1 is equivalent to the commutativity of the exterior of the diagram of Fig. 1; hence also to the commutativity of the diagram of for any double categories B, C, P and K.

The Unitality Natural Isomorphisms
For any double categories A and K there are natural isomorphisms where f K 0 denotes the 0-cell part of the iso double functor (2.3). By Yoneda's lemma, they induce respective natural isomorphisms and λ with the components at any double category A (where 1 A : 1 → ⟦A, A⟧ is the double functor which sends the single object of 1 to the identity double functor 1 A : A → A). They are our candidate unitality natural isomorphisms.

The Triangle Conditions
By Yoneda's lemma, Mac Lane's triangle condition on the natural isomorphisms α of Sect. 3.1 and λ, of Sect. 3.3 is equivalent to the commutativity of the exterior of Fig. 3  whose left-bottom path is (2.2); that is, the canonical (usually omitted) isomorphism.

The Symmetry
The natural isomorphism constructed from the 0-cell part of f in (2.3) induces a natural isomorphism ϕ : ⊗ → ⊗.flip with the components at any double categories A and B. It is our candidate symmetry.

The Hexagon Condition
By Yoneda's lemma, the hexagon condition on the natural isomorphisms α of Sect

Examples
Obviously, what we get for the closed Gray monoidal product ⊗ of double categories essentially depends on our choice of the internal hom double categories ⟦A, B⟧. Although our notions of (horizontal and vertical) pseudo transformations and of corresponding modification in Sect. 2.2 may look quite natural, admittedly no higher principle fixes their choice. Therefore there is no a'priori good resulting Gray monoidal product of double categories.
In this final section we support our construction by relating it to existing structures. Namely, we verify the monoidality of some well-known functors between our monoidal category (DblCat, ⊗) and some other monoidal categories which occur in the literature quite frequently.

Monoidal Functors Between Closed Monoidal Categories
In any closed monoidal category we may take the mate    for all objects A, B and C of the domain category.

The Closed Monoidal Category (DblCat, ⊗)
For the closed monoidal category (DblCat, ⊗) of Sects. 2 and 3, is equal to 1 A ; that is, the double functor sending the single object of 1 to the identity double functor 1 A : A → A for any double category A. The double functor is the canonical isomorphism for any double category A. The double functor of (4.1) is equal to that in Sect. 2.4.

Monoids in (DblCat, ⊗)
Monoidal 2-categories can be defined at different levels of generality. The most restrictive one in the literature is a monoid in the category of 2-categories and 2-functors with respect to the Cartesian monoidal structure. This is known as a strict monoidal 2-category. The most general one is a single object tricategory [14]; known as a monoidal bicategory. In between them are the so-called Gray monoids; these are again monoids in the category of 2-categories and 2-functors, but in this case with respect to the Gray monoidal structure [19]. Their importance stems from the coherence result of [14], proving that any monoidal bicategory is equivalent to a Gray monoid (as a tricategory). Analogously, a strict monoidal double category [7] is a monoid in the category of double categories and double functors with respect to the Cartesian monoidal structure. In [18,24] it was generalized to a pseudo monoid in the 2-category of (pseudo) double categories and pseudo double functors and, say, vertical transformations. However, no double category analogues of Gray monoids and of monoidal bicategories seem to be available in the literature. While the considerations in this paper do not promise any insight how to define most general monoidal (pseudo) double categories, monoids in (DblCat, ⊗) are natural candidates for the double category analogue of Gray monoid. In this section we give their explicit characterization, similar to the characterization of Gray monoids in [3,Lemma 4].
A monoid in (DblCat, ⊗) is equivalently a monoidal functor from the terminal double category 1 (with the trivial monoidal structure) to (DblCat, ⊗). It can be described in terms of the data in Sect. 4.1. Namely, a monoid structure on a double category A translates to double functors I : 1 → A and M : A → ⟦A, A⟧ which render commutative the diagrams of (4.2) and (4.3). Spelling out the details, this amounts to the following data.
• A distinguished 0-cell I . (iii) For any 0-cells X and Y , the following equalities of double functors hold.
(iv) For any 0-cell X , any horizontal 1-cells h, p and any vertical 1-cells v, q, the following equalities of 2-cells hold.
(v) For any 0-cell X , we denote by 1 X the horizontal identity 1-cell; and by 1 X the vertical identity 1-cell on X . For any horizontal 1-cell h and any vertical 1-cell v, the following equalities of 2-cells hold.
Consequently, monoids in (DblCat, ⊗) in this section determine monoids-with this multiplication -in the category whose objects are the double categories, whose morphisms are the double pseudo functors of [23, Definition 6.1], and which is monoidal via the Cartesian product. However, the multiplication of this monoid need not be a double functor or at least a pseudo double functor in the sense of [16, Section 2.1] and [24,Definition 2.7] (recall that a pseudo double functor in these references is defined to strictly preserve the composition in one direction; and up-to a coherent natural family of invertible 2-cells in the other direction). Consequently, there seems to be no evident way to interpret a monoid in (DblCat, ⊗) like above as a monoidal double category in the sense of [24, Definition 2.9], [18, Section 3.1], [12,Definition 31] and [16,Section 5.5]; or-what is the same-as a one object locally cubical bicategory in [13]. By this reason, there seems to be no easy way to regard a monoid in (DblCat, ⊗) in this section as a suitably degenerate intercategory [17,18] To any double category A, the double category Mnd(A) of monads in A was associated in [10]. This construction can be seen as the object map of the functor in the title, which sends a morphism; that is, a double functor F : It  , t), θ, τ ) acts as and on the vertical 1-cells it acts as in the horizontal 2-category of ⟦A, B⟧ (so that in particular p is a horizontal pseudo transformation and π is a modification). The double functor χ A,B should send it to a horizontal 1-cell in ⟦Mnd(A), Mnd(B)⟧; that is, the following horizontal pseudo transformation , t ), θ , τ ).

The Closed Monoidal Category (DblCat, ×)
Recall that for any double category A, the internal hom functor A, − : DblCat → DblCat of the closed monoidal category in the title sends an object; that is, a double category B to the following double subcategory A, B of ⟦A, B⟧.
• The 0-cells are still the double functors A → B.
• The horizontal 1-cells are the horizontal transformations of [15]. That is, those horizontal pseudo transformations x (see Sect. 2.2) whose 2-cell parts x h are vertical identity 2-cells for all horizontal 1-cells h in A. • Symmetrically, the vertical 1-cells are the vertical transformations of [15]. That is, those vertical pseudo transformations y (see Sect. 2.2) whose 2-cell parts y f are horizontal identity 2-cells for all vertical 1-cells f in A. • Finally, the 2-cells are the modifications of [15] (this is the same notion as in Sect. 2.2).
The functor A, − : DblCat → DblCat sends a morphism; that is, a double functor H : B → C to the restriction A, H : A, B → A, C of the double functor ⟦A, H⟧ : ⟦A, B⟧ → ⟦A, C⟧ in Sect. 2.3.
For any double category A, the double functor is 1 A , sending the single object of 1 to the identity double functor 1 A : A → A; and the double functor For any double categories A, B and C, the double functor l C × : A, B → C, A , C, B of (4.1) is constituted by the following maps.
• It sends a 0-cell; that is, a double functor F : A → B to the double functor C, F : C, A → C, B .
• It sends a horizontal 1-cell; that is, a horizontal transformation F x G to the horizontal transformation C, F → C, G whose component at any vertical transformation on the left-between double functors C → A-is the modification on the right: • Symmetrically, it sends a vertical 1-cell; that is, a vertical transformation on the leftbetween double functors A → B-to the vertical transformation whose component at any horizontal transformation H p H' -between double functors C → A-is the modification on the right: • Finally, it sends a modification on the left to the modification whose component at a double functor H : C → A is the modification on the right: In particular, a strict monoidal double category [7]-which is a monoid in (DblCat, ×)gives rise to a monoid in (DblCat, ⊗)-described in Sect. 4.3.

The Closed Monoidal Category (2-Cat, ⊗)
In this section we regard the category 2-Cat of 2-categories and 2-functors as a closed monoidal category via the Gray monoidal product ⊗ in [19]. Recall that, for any 2-category A, the internal hom functor • It sends a 2-cell; that is, a modification ω to the modification whose component at any 2-functor F : C → A is ω F− .

Monoidality of the Functors (DblCat, ⊗) → (2-Cat, ⊗) Sending Double Categories to Their Horizontal-or Vertical-2-Categories
Consider the functor H : DblCat → 2-Cat which sends a double category A to its so-called horizontal 2-category. The 0-cells of HA are the 0-cells of A, the 1-cells of HA are the horizontal 1-cells of A and the 2-cells of HA are those 2-cells of A which are surrounded by identity vertical 1-cells (and arbitrary horizontal 1-cells). Compositions in HA are inherited from A. The functor H sends a morphism; that is, a double functor F : A → B to the 2-functor HF : HA → HB which acts on the various cells as F does. The horizontal 2-category of the terminal double category 1 is the terminal 2-category 1. So we may choose the nullary part H 0 of the candidate monoidal structure on H to be the identity 2-functor 1 → 1. As the 2-functor χ A,B : H⟦A, B⟧ → [HA, HB] for any double categories A and B, encoding the binary part, we propose the following.
• A 0-cell; that is, a double functor F : A → B is sent to the 2-functor HF : HA → HB.
• A 1-cell; that is, a horizontal pseudo transformation x : F → G is sent to the pseudo natural transformation HF → HG whose component at any 1-cell of HA-that is, horizontal 1-cell h : A → C of A-is the 2-cell x h : x C .Fh → Gh.x A of HB. • A 2-cell; that is, a modification of the form Applying the so defined functor Sqr to the terminal 2-category 1, we obtain the terminal double category 1. So as the nullary part of the candidate monoidal structure on Sqr, we may choose the identity double functor 1 → 1. For any 2-categories A and B, for the double

Appendix A. Diagrams
Large size diagrams, used in the earlier sections, are collected in this appendix (see Figs. 1, 2, 3, 4 and 5).