Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences

We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the bicategory of coloured symmetric sequences. In order to do this, we establish general results on extending monoidal structures to Kleisli bicategories. Our approach uses monoidal double categories, which help us to attack the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.

Context, aim and motivation.Joyal's theory of species of structures [Joy81] provides an illuminating and powerful approach to enumerative combinatorics, as amply illustrated in [BLL98], and finds applications also in algebra [AM10].By definition, a species of structures F is simply a functor from the category B of finite sets and bijections to the category of sets and functions, mapping a finite set U of 'labels' to a set F rU s of 'F -structures' (e.g.binary rooted trees) labelled by elements of U .Importantly for applications, species of structures support a calculus of operations (which includes substitution, sum, product and differentiation) that has a combinatorial interpretation and provides a 'categorification' of the calculus of exponential power series widely used in combinatorics [Wil06].This point of view is supported by the introduction of the so-called analytic functor associated to a species of structures [Joy86], which is defined by the formula where, for n P N, rns " t1, . . ., nu and the fraction denotes the quotient of F rns ˆXn by the evident action of the n-th symmetric group S n .The passage from species of structures to analytic functors goes via symmetric sequences, which are defined as functors from S, the skeleton of B whose objects are finite cardinals, into the category of sets.Under the equivalence between species of structures and symmetric sequences, the substitution operation of species of structures corresponds to the substitution monoidal structure on symmetric sequences defined in [Kel05], which is of interest since monoids with respect to it are precisely symmetric operads [BV73,May72].
In [MM08], Maia and Méndez introduced a new operation on species of structures, baptized arithmetic product, and provided a combinatorial interpretation for it.Given species F and G, their arithmetic product F b G is defined by letting pF 1 b F 2 qrU s " ÿ pπ1,π2qPRrUs where RrU s denotes a certain set of partitions of U , called rectangles.Independently of the work of Maia and Méndez, Dwyer and Hess rediscovered this operation1 in the context of symmetric sequences [DH14] in order to extend the Boardman-Vogt tensor product of symmetric operads [BV73] to operadic bimodules.For symmetric sequences, the arithmetic product is defined from the product of natural numbers on S (which is functorial, even if it is not the cartesian product) by Day convolution [Day70], via the coend formula Srm, m 1 ¨m2 s ˆF1 rm 1 s ˆF2 rm 2 s. (1) The connection between the arithmetic product for species in [MM08] and symmetric sequences in [DH14] seems to have been first noted by Bremner and Dotsenko in [BD20].
In [DH14], the authors also observed that the arithmetic product of symmetric sequences interacts with the substitution monoidal structure in an interesting way, in that there is a natural transformation with components pG 1 ˝F1 q b pG 2 ˝F2 q Ñ pG 1 b G 2 q ˝pF 1 b F 2 q (2) which are not necessarily invertible.Dwyer and Hess conjectured that this transformation underlies what is usually called a duoidal or 2-monoidal structure, in which two monoidal structures interact by means of an interchange law [AM10].The conjecture was settled positively by the second-named author and López Franco, who also showed that this duoidal structure is normal, in the sense that the units of the two monoidal structures essentially coincide.This was done as part of their general study of commutative operations [GL16], which involves introducing a general notion of commuting tensor product of ˝-monoids in a normal duoidal category pV, b, ˝q.When this notion is instantiated at the normal duoidal category of symmetric sequences, it re-finds the Boardman-Vogt tensor product P b BV Q of symmetric operads P and Q.In particular, to express that the operations of P and Q commute with each other in P b BV Q one uses a diagram of the form which involves both the arithmetic product and the substitution monoidal structures.Furthermore, [FV20] uses this duoidal structure to establish an enrichment of symmetric operads in symmetric cooperads.The aim of this paper is to generalise the definition of the arithmetic product, and the key results concerning it, from symmetric sequences to coloured symmetric sequences: this corresponds to the passage from symmetric operads to coloured symmetric operads, i.e. from the single-object to the many-object case.We will show that such generalisation not only is possible, but actually determines a new kind of low-dimensional categorical structure.
The motivation for this work is manifold.First, it is part of a wider research programme aimed at understanding the structure of the bicategory of coloured symmetric sequences and related bicategories, with applications to logic and theoretical computer science, cf.[CW05,Hyl10].In particular, it provides the basis to extend the Garner-López Franco theory of commutativity, to re-find the Boardman-Vogt tensor product of coloured symmetric operads (cf.(3)), and to develop a corresponding tensor product of bimodules between them, generalising the results of Dwyer and Hess, a project that we leave for future work.The results presented here are also useful to extend the study of enrichment in [Vas19] to relate coloured (co)operads and their (co)modules.Finally, we hope that our results may eventually be of interest in combinatorics, since the arithmetic product of coloured symmetric sequences defined here induces a corresponding operation on coloured species of structures [FGHW08], which extends the arithmetic product of Maia and Méndez to variants of Joyal's species of structures that are particular instances of coloured species of structures.
Main results.Whereas the original arithmetic product is extra structure on the substitution monoidal category of symmetric sequences, our generalised arithmetic product for coloured symmetric sequences will be extra structure on the bicategory Sym of coloured symmetric sequences of [FGHW08].Recall that, for sets X and Y , an pX, Y q-coloured symmetric sequence is a functor M : SY op ˆX Ñ Set, where SY is the free symmetric strict monoidal category on Y .Such a functor M assigns a set M p y, xq to each y " py 1 , . . ., y n q P SY and x P X, the elements of which can be thought of as operations f : y 1 , . . ., y n Ñ x, with inputs of sorts y 1 , . . ., y n and output of sort x, typically pictured as corollas.Taking X " Y " 1 recovers the notion of ordinary symmetric sequence since S1 " S; and as shown in [Fio05,FGHW08], the calculus of symmetric sequences can be extended to their coloured counterparts.In particular, the substitution monoidal structure can be generalised to a composition operation, which is the composition of the bicategory Sym whose objects are sets, and whose maps from X to Y are the symmetric pX, Y qcoloured sequences [FGHW08,FGHW18].The monads in this bicategory are then symmetric coloured operads [BD98].
Given an pX 1 , Y 1 q-coloured symmetric sequence M and a pX 2 , Y 2 q-coloured symmetric sequence M 2 , we will define their arithmetic product as the pX 1 ˆX2 , Y 1 ˆY2 q-coloured symmetric sequence M 1 b M 2 given by pM 1 b M 2 qp y, px 1 , x 2 qq " ż y 1 , y 2 PSY SpY 1 ˆY2 qr y, y 1 b y 2 s ˆM1 p y 1 , x 1 q ˆM2 p y 2 , x 2 q, (4) is an operation determined by lexicographic ordering of pairs.As expected, when X 1 " Y 1 " X 2 " Y 2 " 1, we obtain the arithmetic product in (1) of Dwyer and Hess.
Our main result, Theorem 10.10, asserts that the said arithmetic product determines a normal oplax monoidal structure on the bicategory Sym of coloured symmetric sequences.The notion of a normal oplax monoidal bicategory appears to be novel and is introduced here in Definition 4.9 as the natural 'many-object' generalisation of the normal duoidal structure in (2).The key challenge to be overcome to obtain our main result is the verification of the axioms for a normal oplax monoidal bicategory, which are of the same daunting complexity as those for a monoidal bicategory [GPS95,Gur13].As such, attempting a direct verification seems hopelessly complicated and unlikely to result in any insight.Instead, we solve the problem developing ideas of 2-dimensional monad theory [BKP89], obtaining some general results that are of independent interest.
Our approach exploits crucially the notion of a (pseudo) double category, which adeptly handles the bookwork around dealing with structures involving two kinds of morphisms.More specifically, in a double category one has objects, two kinds of 1-cells (called horizontal and vertical), and squares (which help to relate horizontal and vertical 1-cells).Of key importance for our development are the double category of profunctors Prof, which has categories as objects, profunctors [Bén73,Bén00] (also known as distributors or bimodules) as horizontal 1-cells and functors as vertical 1-cells; and the double category of matrices Mat, which is the full double subcategory of Prof spanned by sets (viewed as discrete categories).Double categories are important for us because they provide an efficient way of constructing threedimensional structures such as monoidal bicategories and, as we shall see, their oplax variants.The basic insight, as explained in [GG09,Shu10,HS19], is that, in order to obtain a monoidal structure on a bicategory E, it is sufficient to represent E as the horizontal bicategory of a double category and then construct a monoidal structure on this double category.Since the coherence data and axioms for a monoidal double category are of the same character as those of a monoidal category, rather than a monoidal bicategory, this significantly reduces the volume and complexity of the checks required to establish the structure.
The relevance of this to our situation is that the bicategory of coloured symmetric sequences Sym can be represented as the horizontal bicategory of a double category Sym in which vertical 1-cells are functions between sets.In fact, building on [CS10,FGHW18], this Sym can be seen as a full sub-double category (on the discrete objects) of the double category CatSym of categorical symmetric sequences; which can, in turn, be constructed as the Kleisli double category of a double monad on the double category Prof of profunctors.The double monad in question maps a category X to its symmetric strict monoidal completion SX, extending the corresponding 2-monad on the 2-category of categories.
Given the above, the desired normal oplax monoidal structure on the bicategory of coloured symmetric sequences can be obtained as follows.Firstly (Theorem 4.11), we extend the results of [GG09,Shu10,HS19] to establish that such a structure can be obtained from a normal oplax monoidal structure on Sym, or more generally, CatSym.Secondly, to obtain this, we prove and apply a result (Theorem 9.4) which isolates sufficient conditions on a double monad T on a double category C under which a monoidal structure on C will extend to an oplax monoidal structure on the Kleisli double category KlpT q.Pleasingly, this condition on T turns out to be a natural one, namely a suitably adapted form of the pseudocommutativity of [HP02].This is satisfied by the 2-monad S used in our application, and indeed, the operation (5) featured in the definition of the arithmetic product is part of this pseudo-commutative structure on S : Cat Ñ Cat.
Thus, using this result, the monoidal structure on Prof given by the cartesian product of sets extends to give the arithmetic product oplax monoidal structure on the double category CatSym of categorical symmetric sequences, which in turn induces the desired oplax monoidal structure on the double category Sym of coloured symmetric sequences.This general method leads exactly to the formula in (4), which is a natural generalisation of that in (1).
While our approach offers a clear pathway to prove our main results, and others besides, we still have to overcome significant technical challenges, dealing with coherence conditions at the double categorical level, keeping track of strictness and weakness of the structures involved.Roughly speaking, vertical structure tends to be stricter than horizontal one, but the two are closely related under the assumption that the double categories under consideration are fibrant, in the sense of [Shu08].This allows us to induce a lot of the horizontal, weaker, structure that we need for applications from vertical, stricter, one, that is already known, thereby keeping some control of the complexity of our calculations.
While it undoubtedly requires more groundwork to set up the abstract approach that we take, the end result is a modular framework which is easily applicable to other, related situations.For example, although we shall not do so here, it is entirely straightforward to adapt our results from the setting appropriate for studying symmetric coloured operads to the setting appropriate for many-sorted algebraic theories: it is simply a matter of replacing the double monad S for symmetric strict monoidal categories with a corresponding double monad F for categories with strictly associative finite products, and verifying that everything still carries through.In this setting, we obtain an oplax monoidal structure on the appropriate Kleisli bicategory-which is essentially the bicategory of sifted-cocontinuous functors between presheaf categories-which extends the duoidal structure on the functor category rF p1q op , Sets used in [GL16] to study the commuting tensor product of single-sorted algebraic theories.
For expository convenience, we outlined our results above for Set-valued coloured symmetric sequences, but in fact our development will be carried out in a more general enriched context.For any symmetric monoidal closed cocomplete V there is an analogue of the 2-monad S leading to a bicategory of V-enriched symmetric sequences.However, to obtain an oplax monoidal structure on this bicategory, we will restrict to the case where the tensor product of V is in fact cartesian product.The reason for this restriction, which was also made in [DH14,GL16], is that the 2-monad S is only pseudo-commutative when V is cartesian, since the structure maps in (5) for this pseudo-commutativity involve a 'duplication' of objects that is not available in the general symmetric monoidal closed setting.
Outline of the paper.Sections 2 and 3 recall the notions of double category, double functor, horizontal and vertical transformation, and modification.We pay particular attention to companions, leading to the notion of a special vertical transformation, and establish a few useful lemmas about them.Sections 4 and 5 focus on monoidal double categories, monoidal double functors, monoidal horizontal and vertical transformations and monoidal modifications.In particular, we show that, for monoidal double categories C and D, monoidal double functors between them are the objects of a functor double category (Proposition 5.6).Section 6 considers monoids in monoidal double categories.We use this notion in Sections 7 and 8 to define double monads and monoidal double monads and obtain results on them in a homogeneous manner.To do this, we show that the double category of monoidal double endofunctors on a monoidal double category admits a monoidal structure, given by composition (Proposition 8.1).In Section 9, we consider Kleisli double categories and establish sufficient conditions for a double monad on a monoidal double category to determine a monoidal structure on the Kleisli double category.We apply these results to coloured symmetric sequences in Section 10, leading up to our main results on the existence of oplax monoidal structures on the relevant double category (Theorem 10.9) and bicategory (Theorem 10.10).
Gambino acknowledges that this material is based upon work supported by the US Air Force Office for Scientific Research under award number FA9550-21-1-0007 and by EPSRC via grant EP/V002325/2.Garner acknowledges the support of Australian Research Council grants FT160100393 and DP190102432.Vasilakopoulou acknowledges the support of the General Secretariat of Research and Innovation (GSRI) and the Hellenic Foundation for Research and Innovation (HFRI).

Double categories
By double category, we will mean a (horizontally) weak double category, also known as a pseudo double category; relevant material can be found in [GP99, GP04, Gar06a, HS19].
Definition 2.1 (Double category).A double category C consists of the following data: ‚ a category C 0 , whose objects are 0-cells, and whose arrows are vertical 1-cells f : X Ñ X 1 ; ‚ a category C 1 , whose objects are horizontal 1-cells M : X ÝÞÑ Y , and whose arrows are 2morphisms ‚ two functors s, t : C 1 Ñ C 0 called source and target respectively; ‚ composition and identity functors ˝: These data are required to satisfy coherence axioms analogous to those for a bicategory; see [GP99, §7.1].
In the last item of the preceding definition, we use the notion of a globular 2-morphism in a double category C; this is a 2-morphism φ for which spφq and tpφq are identities: Remark 2.2.In our definition of double category, vertical 1-cells compose categorically, i.e., strictly associatively, while horizontal 1-cells compose bicategorically, i.e., with associativity only up to coherent isomorphism.Compared to the original definition in [GP99, GP04], we have chosen to interchange the vertical and horizontal directions so as to match up with later work such as [Gar06a,HS19].The reader should bear this reversal of sense in mind when comparing the definitions and constructions that follow with those of [GP99,GP04].
Given a double category C, we write HpCq for the its horizontal bicategory, comprising the 0-cells, horizontal 1-cells and globular 2-morphisms; and we write VpCq for its vertical 2-category, whose objects and morphisms are the 0-cells and vertical 1-cells of C, and where a 2-cell Notice that vertical composition of 2-cells in VpCq is given by pasting in the horizontal direction in C, and vice versa.
Example 2.3 (Bicategories and monoidal categories as double categories).Any bicategory can be seen as a double category with only identity vertical arrows.In particular, any monoidal category pV, ˝, Jq can be regarded as a double category with a single object and only the identity vertical arrow.
Example 2.4 (The double category of matrices).Fix a monoidal category pV, ˝, Jq with small coproducts, such that the tensor product preserves coproducts in each variable; in particular, this holds whenever the monoidal structure is closed.Given sets X and Y , an pX, Y q-matrix M : X ÝÞÑ Y is a family of objects `M py, xq P V : x P X, y P Y ˘.The double category of V-matrices Mat V has objects and vertical 1-cells being sets and functions, respectively; horizontal 1-cells M : X ÝÞÑ Y being pX, Y q-matrices; and 2-morphisms φ : M ñ N as in (6) being families of V-arrows `φyx : M py, xq Ñ N pgy, f xq : x P X, y P Y ˘.
Horizontal composition N ˝M : X ÝÞÑ Y ÝÞÑ Z is given by pN ˝M qpz, xq " ÿ yPY N pz, yq ˝M py, xq, while the horizontal identities id X : X ÝÞÑ X are defined by id X px, x 1 q " J if x " x 1 and id X px, x 1 q " 0 if x ‰ x 1 .The horizontal bicategory of this double category is the usual bicategory of enriched matrices; see [BCSW83].
The choice of a different notation for Example 2.5 below is needed in view of Example 4.6.
Example 2.5 (The double category of profunctors).Fix a braided monoidal cocomplete category pV, b, Iq in which the tensor product preserves colimits in each variable.Below, we use freely the standard notions of V-category, V-functor and V-natural transformation, for which we invite readers to refer to [Kel82].
Recall that, for small V-categories X and Y , a pY, Xq-profunctor M : We may now define the double category Prof V of V-profunctors as follows.The objects and vertical 1-cells of Prof V are small V-categories and V-functors; the horizontal 1-cells are V-profunctors, and squares as in (6) (writing now F and G instead of f and g) are V-natural transformations φ : The horizontal identity id X : X ÝÞÑ X is defined by id X px 1 , xq " Xrx 1 , xs2 .Note that Mat V can be regarded as a sub-double category of Prof V by considering sets as discrete V-categories.The horizontal bicategory of Prof V is the familiar bicategory of profunctors, while the vertical 2-category is the 2-category of small V-categories.
These examples illustrate the fact, pointed out in Remark 2.2, that in our double categorical structures, the vertical structure is strict, and relatively straightforward, while the horizontal structure is weaker and requires careful consideration of coherence issues.The management of these coherence issues is simplified when the horizontal and vertical structures can be universally related in the following way.
Definition 2.6.Let C be a double category.A companion for a vertical 1-cell f : Although companions are defined algebraically, they also have a universal characterisation.
Lemma 2.7.Let f : X Ñ X 1 be vertical 1-cell of a double category C. Giving a companion p p f , p 1 , p 2 q for f is equivalent to giving either of the following: f and 2-morphism p 1 as in (9) such that, for every horizontal 1-cell M : X 1 ÝÞÑ Y 1 , the composite 2-morphism to the left below is cartesian with respect to ps, tq : C 1 Ñ C 0 ˆC0 ; or ‚ A horizontal 1-cell p f and 2-morphism p 2 as in (9) such that, for every horizontal 1-cell M : W ÝÞÑ X, the composite 2-morphism to the right below is opcartesian with respect to ps, tq : By virtue of this result, companions of a vertical 1-cell are unique up to a unique globular 2-isomorphism, so that by the usual abuse of notation we may refer simply to the companion.In what follows we will often require the existence of certain companions, but in many examples of interest we have all companions and also all conjoints-the dual notion to companion, which associates to a vertical 1-cell f : X Ñ X 1 a horizontal 1-cell q f : X 1 ÝÞÑ X along with unit and counit 2-morphisms.In this case, we may speak of a framed bicategory in the sense of [Shu08] or fibrant double category in other references.By the above lemma and an appropriate dual lemma for conjoints, a double category is a framed bicategory if and only if ps, tq : C 1 Ñ C 0 ˆC0 is a Grothendieck fibration, or equivalently, a Grothendieck opfibration.
Example 2.8.The double category Prof V of V-profunctors is a fibrant double category.The companion and conjoint of a V-functor F : X Ñ X 1 are the V-profunctors p F : X ÝÞÑ X 1 and q F : X 1 ÝÞÑ X given by p F px 1 , xq " X 1 rx 1 , F xs and q F px, x 1 q " X 1 rF x, x 1 s.
It follows a fortiori that the double category Mat V of matrices is also fibrant, where for a function f : X Ñ X 1 its companion and conjoint p f : X ÝÞÑ X 1 and q f : X 1 ÝÞÑ X are the V-matrices given by: Note that, in these examples, we have that p F % q F and p f % q f in the horizontal bicategory.In fact, it is always true that the companion and conjoint of a vertical 1-cell are adjoint in this way.
The universality of companions in Lemma 2.7 immediately implies the following omnibus proposition.
Proposition 2.9.Let C be a double category.
(i) The vertical identity 1-cell 1 X : X Ñ X has the horizontal identity id X : X ÝÞÑ X as a companion.
(ii) If the vertical 1-cells f : X Ñ X 1 and g : X 1 Ñ X 2 have the companions p f and p g, then g ˝f has the companion p g ˝p f .(iii) If the vertical 1-cells f : X Ñ X 1 and g : Y Ñ Y 1 admit companions, then pasting with the companion 2-morphisms for f and g gives a bijection between 2-morphisms φ as in (6) and globular 2-morphisms If f and g are invertible in C 0 , then under this correspondence φ is invertible in C 1 if and only if p φ is.(iv) If VpCq 1 denotes the locally full sub-2-category of VpCq with the same objects, and as morphisms just those the vertical 1-cells which admit companions, then taking companions underlies an identity-on-objects homomorphism of bicategories VpCq 1 Ñ HpCq.(v) If f : X Ñ X 1 is a vertical 1-isomorphism in C and both f and f -1 admit companions, then p f is an equivalence in HpCq with pseudoinverse x f -1 ; (vi) If φ : f ñ g is an invertible 2-cell in VpCq, and f and g admit companions, then p φ is an invertible globular 2-morphism p f ñ p g in C.

Maps of double categories
In this section, we recall the various kinds of maps existing between double categories, starting with the notions of double functor and oplax double functor.In most of the paper we will work with double functors, which preserve composition and identities up to isomorphism.However, we will also need oplax double functors, which preserve composition and identities only up to a non-invertible 2-cell, in one important situation, namely when we define oplax monoidal structure in Definition 4.1.Definition 3.1 (Oplax double functor, double functor).Let C and D be double categories.An oplax double functor F : C Ñ D consists of the following data: ‚ two ordinary functors F 0 : C 0 Ñ D 0 , F 1 : C 1 Ñ D 1 , denoted by the same letter F below, such that sF 1 " F 0 s and tF 1 " F 0 t, as displayed in: ‚ two natural transformations with components These data are required to satisfy coherence conditions similar to those for an oplax functor between bicategories, one regarding associativity and two for unitality; see [GP99, §7.2].
A (pseudo) double functor F : C Ñ D is an oplax double functor for which the 2-cells ξ M,N and ξ X are invertible.
Lemma 3.2.Let F : C Ñ D be an oplax (pseudo) double functor.Then F induces an oplax (pseudo) functor of bicategories HpF q : HpCq Ñ HpDq.This assignment extends to an ordinary functor from the category of double categories and oplax (pseudo) double functors to the category of bicategories and oplax (pseudo) functors.
Proof.This follows from the definition, and is an oplax analogue of [HS19, Theorem 4.1].Recall that functors of double categories and bicategories compose strictly associatively.
Lemma 3.3.Let F : C Ñ D be a double functor and f : X Ñ X 1 a vertical 1-cell of C. Assume that f admits a companion p f with structure cells p 1 , p 2 .Then the vertical 1-cell F f : F X Ñ F X 1 of D admits the companion F p f via the structure cells In future, we will tend to suppress the unit coherence cells ξ appearing above, and simply write F p 1 and F p 2 for these pasting composites.
Since a double category has two kinds of 1-cell, vertical and horizontal, there are two kinds of natural transformations between double functors, vertical and horizontal, depending on the directions of their components.For our applications, it is the horizontal transformations, recalled in Definition 3.4, which will be most important, since these induce structure on the horizontal bicategory.However, the vertical transformations, recalled in Definition 3.5, are simpler to construct and work with, and so fundamental to our development will be the possibility of turning a vertical natural transformation into a horizontal one in the presence of well-behaved companions.The precise conditions needed are isolated in the the notion of a special vertical transformation (Definition 3.6), and are justified in Proposition 3.10, where we show that the special vertical transformations are exactly the vertical 1-cells of the functor double category (Proposition 3.9) which admit companions.
The following definition can be found, for example, in [Gar06b, §2.4]; or in [GP99, §7.4] under the name 'strong vertical transformation' (recalling the reversal of sense of Remark 2.2).Definition 3.4 (Horizontal transformation).Let F, G : C Ñ D be oplax double functors.A horizontal transformation β : F G consists of These data are required to satisfy, firstly, the two axioms expressing that β p-q : C 0 Ñ D 1 is a functor; then the axiom expressing that the 2-morphisms β M are components of a natural transformation; and finally, the axioms expressing compatibility of β with the double structure of F and G.
The following definition can be found e.g. in [Gar06b, §2.3] or [HS19, Definition 2.8] under the name 'tight transformation'.Definition 3.5 (Vertical transformation).Let F, F 1 : C Ñ D be oplax double functors.A vertical transformation σ : F ñ F 1 consists of the following data: in D for each horizontal 1-cell M : X Y in C.These data are required to satisfy, firstly, the axiom expressing that we have two ordinary natural transformations F 0 ñ F 1 0 and F 1 ñ F 1 1 ; then the axiom expressing compatibility with horizontal composition; and finally, the axiom expressing compatibility with horizontal identities.
It is easy to see that a horizontal transformation β : F F 1 between double functors induces a pseudonatural transformation Hpβq : HpF q ñ HpF 1 q between the associated homomorphisms of bicategories.On the other hand, from a vertical transformation σ : F ñ F 1 , there is no direct way of inducing anything HpF q ñ HpF 1 q.However, there is an indirect way of doing so, if we can first turn the vertical transformation σ : F ñ F 1 into a horizontal one p σ : F F 1 .The following definition isolates the properties required of σ for this to be possible.Definition 3.6 (Special vertical transformation).Let σ : F ñ F 1 be a vertical transformation between oplax double functors C Ñ D. We say that σ is special if for every X P C, the vertical 1-cell component σ X : F X Ñ F 1 X has a companion p σ X : F X ÝÞÑ F 1 X in D, and the companion transposes of the 2-morphism components (16) of σ are invertible.
A special vertical transformation was called a transformation with loosely strong companions in [HS19, Definition 4.10], characterised precisely by the following proposition; when considered in the setting of fibrant double categories, it was called a horizontally strong transformation in [CS10,Definition A.4].
Proposition 3.7.Let σ : F ñ F 1 be a special vertical transformation between oplax double functors.The companion 1-cells p σ X : F X F 1 X are the horizontal 1-cell components of a horizontal transformation p σ : F F 1 , whose 2-morphism components p σ f are the companion transposes of the equalities of vertical 1-cells σ X 1 ˝F f " F 1 f ˝σX as in: and whose globular coherence 2-morphisms are given by (20).In particular, σ induces a pseudonatural transformation Hpσq : HpF q ñ HpF 1 q between the induced oplax functors of horizontal bicategories.
Proof.The horizontal transformation axioms are a straightforward diagram chase using the universal property of companions.
We could now proceed to verify by hand further desirable properties of the construction σ Þ Ñ p σ (for example, its functoriality); however, this turns out to be unneccessary, as we can in fact characterise p σ as a companion for σ in a suitable functor double category, and then apply results such as Proposition 2.9.We first define these functor double categories.Definition 3.8 (Modification).Let β : F G and β 1 : F 1 G 1 be horizontal transformations and let σ : F ñ F 1 and τ : G ñ G 1 be vertical transformations between oplax double functors C Ñ D. A modification X in D for every object X P C, subject to the naturality axiom: ; and the following axiom expressing compatibility with β, β 1 , σ, τ : whose component 2-morphisms are those witnessing that each p σ X is a companion of σ X ; the modification axioms of Definition 3.8 are now easily verified, and it is clear that these modifications satisfy the companion axioms since they do so componentwise.
Suppose conversely that σ has a companion in the functor double category, namely a horizontal transformation p σ with the modifications witnessing this given as in (21).The components of these modifications witness that each horizontal 1-cell p σ X is a companion for the vertical 1-cell σ X .Furthermore, the second modification axiom for p 1 ensures that the invertible coherence 2-morphism p σ M is the companion transpose of the 2-morphism component σ M ; in particular, this says that σ is special as required.
As a sample application of the utility of this result, let us use it to give an efficient proof of: Proposition 3.11.Let σ : F ñ F 1 : C Ñ D be an invertible vertical transformation between oplax double functors.If the 1-cell components of σ and σ -1 have companions, then they induce a horizontal equivalence p σ : F F 1 and so a pseudonatural equivalence p σ : HpF q ñ HpF 1 q between oplax functors of bicategories.
Proof.Since σ is invertible and its components have companions, it is special; likewise, σ -1 is special.So by Proposition 3.10, both σ and σ -1 admit companions in DblCat oplax rC, Ds.It follows by Proposition 2.9(v) that p σ is an equivalence in HpDblCat oplax rC, Dsq as desired.
We conclude this section with a miscellaneous technical lemma concerning components of a vertical transformation, which will be used in Sections 6, 9 and 10.Lemma 3.12.Let σ : F ñ F 1 be a vertical transformation and f : X Ñ X 1 be a vertical 1-cell in C. If f has a companion p f , then the component σ p f is the transpose of the naturality vertical identity as in Proof.It suffices to show these two 2-morphisms have the same companion transposes, which follows by the calculation (in which we again suppress unit coherence 2-morphisms for F and F 1 ): id using naturality of σ; the companion axiom (10); and axiom (19) for a vertical transformation.

Monoidal double categories
The aim of this section is to introduce the notions of monoidal double category and oplax monoidal double category, and to prove some useful facts about them.Both notions describe double categories endowed with a monoidal product: the key difference is that in the former case, this tensor product is a double functor, while in the latter case, it is merely an oplax double functor as in Definition 3.1.In particular, it should be emphasised that 'oplax' only modifies the functoriality of the tensor product, rather than the nature of the associativity and unit constraints for this tensor, which for us will always be invertible.
In what follows, we will be concerned with the the question of extending a monoidal structure on a double category to an oplax monoidal structure on an associated Kleisli double category.Since we need both notions, we here define them simultaneously.‚ monoidal structures pb 0 , I 0 q and pb 1 , I 1 q on the categories C 0 and C 1 ; ‚ strict monoidality of s, t : C 1 Ñ C 0 .For example, the associativity constraint for C 1 has components and subject to axioms that make b into an oplax double functor; ‚ globular 2-morphisms and subject to axioms that make I into an oplax double functor; ‚ two axioms ensuring that the associativity constraint α is a vertical transformation between oplax double functors; ‚ four axioms ensuring that the unit constraints λ and ρ are vertical transformations between oplax double functors.The above axioms are written explicitly in Appendix A.1.We have a monoidal double category when the tensor and unit are specified by double functors, rather than oplax double functors; said another way, when each of the 2-morphisms in ( 23) and ( 24) is invertible.
What we call here an oplax monoidal double category is what is called simply a monoidal double category in [GP04, §5.5]; it is equally well a pseudomonoid in the cartesian monoidal 2-category of double categories, oplax double functors and vertical transformations.
Remark 4.2.We defined a monoidal double category to be an oplax monoidal double category satisfying some additional properties; but these additional properties in fact allow us to simplify the structure further, as explained in [HS19, Page 8].Indeed, in a monoidal double category, the monoidal unit I 1 of C 1 is always canonically isomorphic to id I0 via ι; and it does no harm to assume that, in fact, I 1 is id I0 and ι is the identity 2-morphism-which in turn forces δ " ℓ id I 0 " r id I 0 for the globular isomorphisms (7) for horizontal identities in C. As such, if in specifying a monoidal double category we follow these conventions, then we need only provide the invertible structure 2-morphisms τ and η satisfying the appropriate coherence axioms.By contrast, in an oplax monoidal double category, none of the data are redundant: indeed, δ and ι as in (24) now specify a comonad structure on I 1 in HpCq, see (80).
Moreover, notice that just as a double category is an internal pseudocategory in the 2-category of small categories, functors and natural transformations, an oplax monoidal double category is an internal pseudocategory in the 2-category of monoidal categories, lax monoidal functors and monoidal transformations for which the source and target functors are strict monoidal.
As mentioned in the introduction, the notion of oplax monoidal structure will be exploited in future work in order to provide a general notion of commuting tensor product, generalising the theory of [GL16], which will in particular recover the Boardman-Vogt tensor product of symmetric coloured operads and its extension to operadic bimodules in [DH14].For these applications, it will be important that the oplax monoidal structure is normal in the sense of the following definition.Definition 4.3 (Normal oplax monoidal double category).An oplax monoidal double category C is said to be normal if: (i) I : 1 Ñ C is a (pseudo) double functor; (ii) for all objects X 1 , X 2 P C, the following restricted oplax double functors are (pseudo) double functors: Said another way, both 2-morphisms δ and ι in (24) are invertible; while in (23), η is invertible, and each τ for which M 1 " N 1 " id X1 or M 2 " N 2 " id X2 is also invertible.
We give now some examples of monoidal double categories and oplax monoidal double categories.The example of profunctors in Example 4.7 will be fundamental for our application in Section 10.
Example 4.4 (Duoidal categories).Recall from Example 2.3 that a monoidal category pV, ˝, Jq is the same thing as a double category with a single object and only identity vertical arrow.To equip this double category with an oplax monoidal structure in the sense of Definition 4.1 is the same thing as equipping V with additional structure making it into a duoidal category [AM10].Explicitly, this amounts to providing a second monoidal structure pb, Iq on V, along with maps satisfying appropriate axioms.Here, the interchange law ξ corresponds to the square τ in (23).This oplax monoidal structure is a genuine monoidal structure whenever all of ξ, µ, γ and ν are invertible.In this case, by the Eckmann-Hilton argument, the identity functor underlies a monoidal isomorphism pV, b, Iq Ñ pV, ˝, Jq, and the two isomorphic monoidal structures are each braided.Loosely, then, we may say that in this situation, V 'is' a braided monoidal category.In particular, if we merely start with a braided monoidal category pV, b, Iq, then it becomes duoidal on taking ˝" b, J " I, ν " id, µ " r I , γ " r -1 I , and ξ the canonical constraint built from associativity and braiding maps.Returning to the general situation, the oplax monoidal structure on V qua double category is normal if, and only if, the duoidal structure on V is normal meaning that ν, γ and µ are all invertible.Indeed, to say that the unit double functor of the oplax monoidal structure is pseudo is precisely to say that γ and ν are invertible, which by the duoidal axioms implies the invertibility of µ also.So the duoidal structure on V is normal precisely when its oplax monoidal structure qua double category satisfies Definition 4.3(i).What is less obvious is that, in this one-object case, Definition 4.3(ii) is an automatic consequence of Definition 4.3(i); but (ii) in this case amounts to the invertibility of ξ when X 1 " Y 1 " J or X 2 " Y 2 " J, and this follow from the oplax monoidality of the unit constraints for b.
Remark 4.5.Looking at the previous example, the reader may wonder why we impose Definition 4.3(ii) at all, given that Definition 4.3(i) by itself gives a faithful "many-object" generalisation of the notion of normal duoidal category.The reason for imposing the extra condition is that it ensures companions in our double category are stable under tensoring by objects, which will be crucial when we come to show that normal oplax monoidal double categories give rise to normal oplax monoidal bicategories-see the proofs of Lemma 4.10 and Theorem 4.11 below.A second justification for the condition comes from work-in-progress, which generalises the theory of commuting tensor products developed in [GL16] to a "many-object" setting.In this generalisation, the normal duoidal categories used as an enrichment base in op.cit.will be replaced with normal oplax monoidal double categories in the above sense-and, again, Definition 4.3(ii) will be necessary in order to make any progress with the theory.
Example 4.6 (Oplax monoidal structure on Mat V ).Recall from Example 2.4 that, for a monoidal category pV, ˝, Jq in which the tensor product preserves coproducts in each variable, we have a double category Mat V of V-matrices.If V is further equipped with a second monoidal structure pb, Iq which also preserves coproducts in each variable, and data as above making it into a duoidal category, then Mat V acquires an oplax monoidal structure.The tensor product is given by the cartesian product of sets and functions on the vertical level, and for V-matrices The monoidal unit is I : 1 ÝÞÑ 1 with unique component I ˚,˚" I.The structure cells (23) and ( 24) are formed using the duoidal structure maps, with the most complex case being that the 2-morphism τ : pN 1 ˝M1 q b pN 2 ˝M2 q Ñ pN 1 b N 2 q ˝pM 1 b M 2 q has components: It is not hard to see that this oplax monoidal structure is normal precisely when V is normal as a duoidal category in the sense of the preceding example, and that it is genuinely monoidal just when the duoidal structure of V comes from a braided monoidal structure.
Example 4.7 (Monoidal structure on Prof V ).Let V be a braided monoidal category in which the tensor product preserves colimits in each variable and recall the double category Prof V of Example 2.5.This double category admits a monoidal structure extending that of Mat V in the braided case.On objects and vertical 1-cells, this is simply the monoidal structure of the 2-category Cat V .On horizontal 1-cells, given V-profunctors with a corresponding definition on 2-morphisms.The key structure isomorphism τ in ( 23) is formed using the braiding of the tensor product on V and the fact that it preserves colimits in each variable.
One can carry out the construction above also when V is merely a duoidal category, thereby extending Example 4.6, but we shall not need this level of generality for our application in Section 10.
One of the main results of [HS19], building on [Shu10,GG09], is that under suitable assumptions, the horizontal bicategory of a monoidal double category is a monoidal bicategory in the sense of [GPS95]; see Theorem 1.1 of op.cit.In view of our application in Section 10, we would like a generalisation of this result which endows the horizontal bicategory of an oplax monoidal double category with an oplax monoidal structure.
The first obstacle to be faced is the definition of oplax monoidal structure on a bicategory K.As before, 'oplax' refers to the strictness of the tensor product functor, and so we might attempt the following naive adaptation of the usual notion of monoidal bicategory.Firstly, we would require oplax functors of bicategories b : K ˆK Ñ K and I : 1 Ñ K; then pseudonatural equivalences giving the associativity and unit constraints; then invertible modifications witnessing the pseudo-coherence of the constraint cells; and, finally, the appropriate coherence axioms for these pseudo-coherences, as found, for example, in [McC99, §A.1].
The problem with this definition can be seen in ( 28) above.In the domain of π we have, among other things, the pseudonatural transformation α ˆ1 whiskered by the oplax functor b : K 2 Ñ K.However, such a composition does not yield another pseudonatural transformation, nor even a lax or oplax transformation.So π is not well-defined; and similar issues arise for µ, L and R in (29) and (30).
We will resolve this issue by imposing a further constraint on the components of the pseudonatural equivalences α, λ, ρ of (27) which we will term centrality, loosely inspired by the nomenclature of [PR97].Centrality of the components of a pseudonatural transformation γ will ensure that composites of the form γ b 1 :" b ˝pγ ˆ1q and 1 b γ :" b ˝p1 ˆγq are well-defined pseudonatural transformations; and, furthermore, that the same pseudonaturality holds for any iterated tensorings such as pp1 b γq b 1q b 1. Applied to the case of α, λ and ρ, this will ensure that the transformations appearing in (28) to (30), as well as all of those appearing in the corresponding coherence axioms, make sense.Definition 4.8.Let K be a bicategory endowed with an oplax functor pb, τ, ηq : K ˆK Ñ K, an oplax functor pI, δ, ιq : 1 Ñ K and pseudonatural equivalences as in (27).A 1-cell f : X Ñ Y of K is said to be central when for all maps g : X 1 Ñ X, h : Y Ñ Y 1 , k : U Ñ W and ℓ : V Ñ Z, the following composite oplax structure cells are invertible: Note that in (31), we consider three-fold tensor products bracketed to the left.We could equally have chosen to bracket to the right, but this would make no difference, since composing (31) with the components of α and their pseudoinverses would yield invertibility of the corresponding cells for the other bracketing.Now, by taking k " id I or ℓ " id I in (31), and composing with the components of λ or ρ and their pseudoinverses, we obtain the invertibility of oplax constraints of the following forms: Because of this, if γ : F ñ G : L Ñ K is a pseudonatural transformation with central components, then both γ b 1 and 1 b γ will also be pseudonatural; for example, in the case of γ b 1, the pseudonaturality with respect to f : X Ñ Y and g : C Ñ D is witnessed by the invertible 2-cell In a similar way, the general form of (31) implies that p1 b γq b 1 is also pseudonatural; note that this does not seem to follow from the pseudonaturality of 1 b γ and γ b 1.However, once we have pseudonaturality of p1 b γq b 1, we obtain a fortiori that of, say, p1 b γq b p1 b 1q and so by composing with the equivalence components of α, the pseudonaturality of pp1 b γq b 1q b 1.By following this pattern, we see that any tensoring of γ with identity pseudonatural transformations will again be pseudonatural.
In particular, if we require the pseudonatural transformations α, λ and ρ to themselves have central components, then we see that every 2-cell pasting which appears in the axioms (28) to (30) will be a welldefined pseudonatural transformation, and likewise for the pastings appearing in the coherence axioms.Thus, we are justified in giving: Definition 4.9.Let K be a bicategory.An oplax monoidal structure on K consists of: ‚ an oplax functor of bicategories b : K ˆK Ñ K; ‚ an oplax homomorphism I : 1 Ñ K; ‚ pseudonatural equivalences α, λ, ρ as in (27), whose components are central; ‚ invertible modifications π, µ, L, R as in ( 28)-(30); satisfying the coherence axioms for a monoidal bicategory as found, for example, in [McC99, §A.1].Like in Definition 4.3, we say that the oplax monoidal structure on K is normal if I : 1 Ñ K is a homomorphism of bicategories, and b is pseudo in each variable, i.e. for each X, Y P K the oplax functors X bp-q : K Ñ K and p-q b Y : K Ñ K are homomorphisms of bicategories.
We now explain how a normal oplax monoidal double category C gives rise to a normal oplax monoidal bicategory.First of all, the oplax double functor b and the pseudo double functor I induce functors on the horizontal bicategory b : HpCq ˆHpCq Ñ HpCq, I : 1 Ñ HpCq (32) which we denote with the same symbol.Here, as per Lemma 3.2, b is an oplax functor of bicategories (which is pseudo in each variable) and I is a homomorphism of bicategories.If now we assume that the components of the invertible vertical transformations α, λ and ρ associated to the monoidal structure on C have companions, then they will induce pseudonatural equivalences between oplax functors of bicategories, according to Proposition 3.11 and since H is functorial.To proceed further, we need the components of p α, p λ and p ρ to be central.This will be a consequence of the following lemma.Note that normality of the oplax monoidal structure on C is important for the proof.It is not clear to us if the corresponding result without it would hold; however, since normality will be present in our applications, we have not pursued this point any further.
Lemma 4.10.Let C be a normal oplax monoidal double category in which the vertical 1-cells giving associativity, left and right unit constraints have companions.If f : X Ñ Y is any vertical isomorphism in C that has a companion, then p f : X Y is central in the horizontal bicategory HpCq with respect to the structure of (32) and (33).
Proof.Fix f : X Ñ Y with companion p f : X Y as in the statement.To check the conditions in Definition 4.8, we must show, for any horizontal 1-cells g : X 1 X, h : Y Y 1 , k : U W and ℓ : V Z, that the two composite 2-cells in (31) are invertible.For the one on the left-hand side, we must show invertibility of the globular 2-morphism in Because b is pseudo in each variable and f has companion p f , it follows that p1 b f q b 1 has companion pid b p f q b id.Thus, τ pτ b 1q in (34) is invertible if and only if its companion transpose pkbhqbℓ is invertible.We claim that this companion transpose is actually given by the following tensor product in C: Note that this is clearly invertible, since f is an isomorphism and so p 1 is invertible.To show that (35) is a transpose companion of (34) it suffices to use the explicit definition of transposition of a 2-morphism.Indeed, pasting p1 b p 2 q b 1 to the left of (35) and using the (right-hand side) axiom (10) and naturality of the components of τ , we obtain the 2-cell (34).It is possible to verify that the composite 2-cell on the right-hand side of (31) is invertible by a similar argument, but pasting with p1 b p ´1 1 q b 1 instead of p1 b p 2 q b 1.
Theorem 4.11.If C is a normal oplax monoidal double category in which the vertical 1-cells giving associativity, left and right unit constraints have companions, then the horizontal bicategory HpCq inherits a normal oplax monoidal structure with underlying data (32) and (33).
Proof.Since the pseudonatural transformations p α, p λ and p ρ of (33) have as their components the horizontal companions of vertical isomorphisms, we can apply Lemma 4.10 to see that these components are all central in the sense of Definition 4.8.We now need to provide the four invertible modifications of Definition 4.9 for HpCq to have the structure of a normal oplax monoidal bicategory.The components of (28)-( 30) are of the form Notice that the two sides in each case are companions of the corresponding sides of the pentagon axiom, the triangle axiom and two known equations for the ordinary monoidal category C 0 , due to Proposition 2.9 and Lemma 3.3.For example, since b is a pseudo double functor in each variable, each p-q b X and Y b p-q preserves companions thus p α b id X is canonically a companion of α b 1 X .As a result, we take π, µ, L, R to be the unique isomorphisms between companions of the same vertical 1-cells.It can then be verified that these invertible cells form a modification between pseudonatural transformations of oplax double functors by [Shu10, Lemma 4.8]3 .
Finally, the three equations that relate those π, µ, L, R can be checked in exactly the same way as in the proof of [Shu10, Theorem 5.1].In more detail, the domain and codomain of the pasted 2-cells involved in the equations are companions of the same isomorphism in C 0 , namely the unique pppX 1 b X 2 q b X 3 q b X 4 qbX 5 -X 1 bpX 2 bpX 3 bpX 4 bX 5 qqq as well as the associator pX 1 bX 2 qbX 3 -X 1 bpX 2 bX 3 q.Using a collection of technical lemmas [Shu10, Lemma 3.11, 3.14, 3.15, 3.19, 4.10] concerning the composition as well as the tensoring of the canonical isomorphisms between companions (the latter adjusted in the normal oplax monoidal case in a straightforward way), we deduce that there can only be a unique invertible 2-cell inside each one of the diagrams which is compatible with the companion data, hence the equations must hold.

Maps of monoidal double categories
For our development in Sections 7 and 8, we will need results concerning both double monads and pseudomonoidal double monads.It turns out that many of these results can be proved uniformly across the two cases, by exhibiting both kind of structure as monoids in suitable endofunctor double categories.This is much as ordinary monads and monoidal monads can be seen as monoids in appropriate endofunctor categories.In order to do this for the case of pseudomonoidal double monads, we need to construct a suitable double category of (lax) monoidal double functors and monoidal transformations.While the notions of lax monoidal double functor and monoidal horizontal transformation (recalled in Definition 5.1 and Definition 5.2 below) are as expected, it turns out that in our motivating examples, the vertical transformations which we need are not monoidal in the obvious way but only pseudomonoidal.While this may seem an innocuous change, it adds an additional layer of subtlety to our development, very much in analogy with what happens in the purely 2-categorical setting [HP02].
We begin with the notion of a lax monoidal double functor.If we view monoidal double categories as pseudomonoids in a 2-category of double categories, double functors and vertical transformations, then the lax monoidal functors are simply the lax morphisms of pseudomonoids.This definition can also be found in [HS19, Definition 2.14], though note that there, the (invertible) structure maps τ and η of a monoidal double category (Definition 4.1) are oriented in the opposite direction.‚ a vertical transformation F 2 : b ˝pF ˆF q ñ F ˝b, whose vertical 1-cell components we denote by and whose 2-morphism components we denote by ‚ a vertical transformation F 0 : I D ñ F ˝IC , whose vertical 1-cell component we denote by F 0 : I Ñ F I and whose 2-morphism component we denote by subject to axioms expressing that the vertical 1-cells F 2 X1,X2 : F X 1 bF X 2 Ñ F pX 1 bX 2 q and F 0 : I Ñ F I endow F 0 : C 0 Ñ D 0 with the structure of a lax monoidal functor, and that the 2-morphisms of (36) and (37) do the same for The reader will notice that we have not named the 2-morphism in (37).This is because its definition is forced: for indeed, since F 0 is a vertical transformation between double functors, the axiom (19) causes (37) to be equal to id F 0 followed by the unit structure isomorphism of F .
We now turn to monoidal transformations between lax monoidal double functors.We begin with the horizontal case, which is as expected, though we could not find it in the literature.G is a horizontal transformation endowed with cells which, firstly, make β p-q : C 0 Ñ D 1 into a lax monoidal functor; in other words, such that the naturality condition is satisfied, along with the usual associativity and unitality conditions, identifying the two evident 2morphisms pβ X1 b β X2 q b β X3 Ñ β X1bpX2bX3q , the two 2-morphisms β X1 b id I Ñ β X1bI and the two 2-morphisms id I b β X2 Ñ β IbX1 .We moreover require the equality of the pastings: expressing that the natural transformation giving the globular cell components of β is a monoidal natural transformation.(Note that the 'nullary' axiom corresponding to this 'binary' axiom holds automatically and need not be stated explicitly.) We now consider monoidality of vertical transformations.Given the view of monoidal double categories and lax monoidal double functors as pseudomonoids and lax pseudomonoid maps, the obvious thing to consider would be the corresponding transformations of pseudomonoids, and this would yield the notion of monoidal vertical transformation considered in [HS19, Definition 2.15].However, we will need something slightly more general for our applications (cf.Remark 5.4), which we will term a pseudomonoidal vertical transformation.The difference can be appreciated by noting that monoidality of a vertical transformation in the sense of loc.cit.makes the underlying 2-natural transformation on the vertical 2-category into a Catenriched monoidal transformation, while for our transformations, this underlying 2-natural transformation is only a monoidal pseudonatural transformation in the sense of [DS97, Definition 3].
Definition 5.3 (Pseudomonoidal vertical transformation).Let F , F 1 : C Ñ D be lax monoidal double functors.A pseudomonoidal vertical transformation σ : F ñ F 1 is a vertical transformation equipped with squares which are invertible in the vertical 2-category VpDq and satisfying the following five coherence axioms: Note that the final four of these axioms only involve structure in the vertical 2-category VpDq; and in fact, they correspond exactly to the axioms for a monoidal pseudonatural transformation from [DS97, Definition 3].More explicitly, the second axiom expresses that the 2-cells σ 2 X1,X2 are components of a modification, while the third through fifth axioms are precisely the three coherence axioms of loc.cit.
If σ 0 and the components of σ 2 are identity 2-cells, then σ becomes a monoidal vertical transformation in the sense of [HS19, Definition 2.15].In that case, σ 0 : F 0 ñ F 1 0 and σ 1 : F 1 ñ F 1 1 are monoidal transformations in the usual sense between lax monoidal functors.
Remark 5.4.The notion of a monoidal (rather than pseudomonoidal) vertical transformation is insufficiently general for the situation we are interested in: the monoidality of the free symmetric monoidal category double monad on the double category of small categories, functors and profunctors, as considered in Section 10.The underlying double functor of this double monad can be equipped with lax monoidal structure, with respect to which the monad unit is a monoidal vertical transformation; however, the monad multiplication is not a monoidal as a vertical transformation, but only pseudomonoidal.This can be seen as a consequence of the fact that the free symmetric monoidal category monad is not commutative, but only pseudocommutative in the sense of [HP02].
We now describe the final piece of structure needed for a double category of monoidal double functors.
Definition 5.5 (Monoidal modification).Let β, β 1 be monoidal horizontal transformations and let σ, τ be pseudomonoidal vertical transformations, as displayed on the boundary of: A monoidal modification γ filling this boundary is a modification of the displayed shape satisfying the axioms: We now provide an analogue of Proposition 3.9 in the monoidal setting, by constructing a double category of monoidal double functors between two monoidal double categories C and D. It would be routine to construct a double category of lax monoidal double functors, monoidal vertical transformations, monoidal horizontal transformations, and monoidal modifications; however, because we wish to involve pseudomonoidal vertical transformations, a little more care is needed in checking the details.
Proposition 5.6 (Functor double categories, monoidal case).Let C, D be monoidal double categories.There is a double category MonDblCatrC, Ds of lax monoidal (pseudo) double functors, pseudomonoidal vertical transformations, monoidal horizontal transformations, and monoidal modifications.
Note that in Proposition 3.9, we considered oplax double functors; here we consider only (pseudo) double functors, but endowed with lax monoidal structure.While it certainly would be possible to consider "lax monoidal oplax double functors", this is not needed for our applications.
Proof.We first show that lax monoidal double functors and pseudomonoidal vertical transformations form a category.Given pseudomonoidal vertical transformations σ : F ñ F 1 and τ : F 1 ñ F 2 , we endow the composite vertical transformation τ ¨σ : F ñ F 2 with pseudomonoidal structure via the pasting composites: It is now routine to verify the pseudomonoidal vertical transformation axioms for τ ¨σ, and to check that this composition law is associative and unital, so yielding the desired category.
We next show that monoidal horizontal transformations and monoidal modifications form a category; for which it suffices to verify that, given a pair of composable monoidal modifications, their composite qua modification, as in Definition 5.5, is again monoidal.This is straightforward.
We now provide the horizontal composition law for MonDblCatrC, Ds.Given monoidal horizontal transformations β : F G and γ : G H, we endow the composite horizontal transformation γ ˝β with monoidal structure via the pastings: Direct verification yields the horizontal transformation axioms.To make the assignment β, γ Þ Ñ γ ˝β into a functor, it now suffices to observe that that the horizontal composition of two monoidal modifications qua modification is again monoidal; this is again a matter of direct verification.Finally, the globular constraints a, ℓ, r of MonDblCatrC, Ds are inherited from DblCatrC, Ds, and it is simply a matter of checking that these are indeed monoidal modifications.
The next result builds on Proposition 3.10.
Proposition 5.7.A pseudomonoidal vertical transformation σ : F ñ F 1 has a companion as a vertical 1-cell of MonDblCatrC, Ds if and only if the underlying vertical transformation of σ has a companion as a vertical 1-cell of DblCatrC, Ds, i.e. it is special.
Proof.The 'only if' direction is trivial: if σ has a companion in MonDblCatrC, Ds, then applying the forgetful double functor MonDblCatrC, Ds Ñ DblCatrC, Ds shows it has a companion in DblCatrC, Ds.For the 'if' direction, given a pseudomonoidal transformation σ as in Definition 5.3, the additional necessary data for the induced horizontal transformation p σ : F F 1 as described in the proof of Proposition 3.10 to be monoidal are the cells of (38).We obtain these as companion transposes of the structure data (39) of the pseudomonoidal vertical transformation σ, as in: where the top-left isomorphism arises due to the double functor b preserving companions.That this makes p σ into a monoidal horizontal transformation can now be checked by lengthy, but straightforward, calculations.Similarly, it is straightforward to verify that with respect to this structure, the companion 2-morphisms p 1 and p 2 in DblCatrC, Ds are monoidal, and so lift to MonDblCatrC, Ds as required.

Monoids in monoidal double categories
In this section, we consider horizontal and vertical monoids in a monoidal double category.When instantiated in the monoidal double categories DblCatrC, Cs and MonDblCatrC, Cs of Propositions 3.9 and 5.6, these will give us the notions of horizontal and vertical double monad, and of monoidal horizontal and vertical double monad respectively, to be considered in Sections 7 and 8.
We begin with the notion of a horizontal monoid in a monoidal double category.This is analogous to a pseudomonoid in a monoidal bicategory, in that the associativity and unit axioms do not hold on the nose, but rather up to invertible squares.The next result shows how we may induce horizontal monoids from vertical ones, and will be applied in Theorem 7.4, relating horizontal and vertical double monads, and Theorem 8.4, relating monoidal horizontal and vertical double monads.Theorem 6.4.Let C be a monoidal double category and pA, m, eq be a vertical monoid in C, such that m and e have companions.The companion transposes x m id of the monoid identities endow pA, p m, p eq with the structure of a horizontal pseudomonoid.
Proof.The displayed 2-morphisms are constructed using transpose operations like (12) from the vertical associativity and unitality monoid axioms for A. They are vertically invertible since α, ρ and λ are, so that a -1 may be constructed as companion transposes of the identities m ˝p1 b mq ˝α-1 " m ˝pm b 1q, and similarly for l -1 and r -1 .The coherence axioms of Definition 6.1 for a horizontal pseudomonoid can now be checked by computing appropriate transposes of the required diagrams and making use of Lemma 3.12.

Double monads
For an ordinary category C, the category of endofunctors of C has a monoidal structure given by composition, and a monoid therein is precisely a monad on C. In the case of double categories, we can do something similar by exploiting our work in Sections 3 and 6, so leading to a notion of double monad: or rather, two notions of double monad, horizontal and vertical.
To begin with, observe that Proposition 3.9 states in particular that for any double category C, there is a double category DblCatrC, Cs of double endofunctors, vertical transformations, horizontal transformations and modifications (Definitions 3.1, 3.4, 3.5 and 3.8).In fact, as is well-known, this double category is monoidal: Proposition 7.1 (Composition monoidal structure).Let C be a double category.The double category DblCatrC, Cs admits a monoidal structure given by composition.
Proof.We only sketch the proof; for a full treatment see, for example, [Gar06a, Proposition 39].
Given double endofunctors F 1 , F 2 : C Ñ C, we define F 1 b F 2 to be the double endofunctor F 2 F 1 with underlying ordinary functors pF 2 q 0 ˝pF 1 q 0 : C 0 Ñ C 0 and pF 2 q 1 ˝pF 1 q 1 : C 1 Ñ C 1 , and with coherence data obtained by vertically pasting those for F and G.Given vertical transformations σ 1 : F 1 ñ F 1 1 and σ 2 : F 2 ñ F 1 2 , we define σ 1 b σ 2 to be the vertical transformation σ 2 σ 1 : F 2 F 1 ñ F 1 2 F 1 1 with underlying ordinary natural transformations given by the horizontal composites pσ 2 q 0 ˚pσ 1 q 0 and pσ 2 q 0 ˚pσ 1 q 0 .With the identity double functor as unit, this yields a strict monoidal structure on the category of double endofunctors and vertical transformations.
Next, given horizontal transformations and remaining data obtained in an analogous way; whereas for modifications γ 1 , γ 2 as in 4 In the provided reference, the alternate choice pβ 2 β 1 q X " pβ 2 q G 1 X ˝F2 pβ 1 q X is used; this results in a different but equivalent monoidal structure.
By looking at horizontal and vertical monoids (as introduced in Definition 6.1 and Definition 6.3) in the endofunctor double category, we obtaine notions of horizontal and vertical double monad.These notions differ by the direction of the transformations for the the multiplication and unit and by their strictness: a horizontal monad induces a pseudomonad on the horizontal bicategory, while a vertical monad induces a 2-monad on the vertical 2-category.We shall relate these notions in Theorem 7.4.Definition 7.2 (Horizontal double monad).Let C be a double category.A horizontal double monad on C is a horizontal monoid in the monoidal double category DblCatrC, Cs.Explicitly, it consists of: for each object X, vertical 1-cell f : X Ñ X 1 and horizontal 1-cell M : X ÝÞÑ Y ; ‚ a horizontal transformation e : 1 T , with components e X : X ÝÞÑ T X, for each object X, vertical 1-cell f : X Ñ X 1 and horizontal 1-cell M : X ÝÞÑ Y ; ‚ invertible modifications a, l and r with respective components at X P C given by: These data are subject to the axioms of Definition 6.1, noting carefully the order-reversal stemming from the fact that A horizontal double monad is exactly the structure we need to define a horizontal Kleisli double category, as we shall do in Theorem 9.1 below.However, horizontal double monads involves non-trivial coherence axioms for associativity and unit; it is therefore useful in practice to have some ways of constructing them from simpler kinds of data.For this purpose, we recall from [GP04, §7] the following definition: Definition 7.3 (Vertical double monad).Let C be a double category.A vertical double monad on C is a vertical monoid in DblCatrC, Cs.Explicitly, it consists of the following data: ‚ a double endofunctor T : C Ñ C; ‚ a vertical transformation m : T T ñ T , with components m X : T T X Ñ T X and for each object X and horizontal 1-cell M : X ÝÞÑ Y ; ‚ a vertical transformation e : 1 ñ T , with components e X : X Ñ T X and for each object X and horizontal 1-cell M : X ÝÞÑ Y .These data are required to satisfy associativity and unitality conditions, as in Definition 6.3.
The notion of a vertical monad is stricter than that of a horizontal monad and thus easier to exhibit in examples.Once we have a vertical monad, the following result allows us to enhance it to a horizontal one.
Theorem 7.4.Let C be a double category and T : C Ñ C be a vertical double monad.Assume that its multiplication m : T T ñ T and unit e : 1 C ñ T are special vertical transformations.Then T induces a horizontal double monad pT, p m, p eq on C.
Proof.If we consider T as a vertical monoid in DblCatrC, Cs, Theorem 6.4 ensures that it induces a horizontal monoid therein (namely a horizontal double monad) whenever the unit e and multiplication m have companions as vertical transformations.By Proposition 3.10, this will happen if and only if they are special.
While a direct proof of Theorem 7.4 would certainly be possible, the more abstract approach we take has the advantage of being equally applicable to the case of monoidal double monads (Theorem 8.4), for which a direct approach seems less practicable.It is to this that we now turn.

Monoidal double monads
In this section, we retread the material of the previous section in the context of monoidal double categories, leading to the notions of a monoidal horizontal and monoidal vertical double monad, and results relating the two.In Section 9, we will exploit these notions in order to impose monoidal structure on the Kleisli double category of a horizontal double monad.
As a first step, we show that when C is a monoidal double category, we can extend the composition monoidal structure on the endofunctor double category DblCatrC, Cs as recalled in Proposition 7.1, to a monoidal structure on the monoidal endofunctor double category MonDblCatrC, Cs of Proposition 5.6.Proof.We must lift each of the pieces of data exhibited in the proof of Proposition 7.1 to the monoidal context.We first lift the strict monoidal structure on the category of 0-cells and vertical 1-cells.If F 1 , F 2 : C Ñ C are lax monoidal double endofunctors of C, then their composite F 2 F 1 bears lax monoidal structure with vertical 1-cell components and 2-morphism components given similarly by and σ 2 : F 2 ñ F 1 2 are pseudomonoidal vertical transformations, then the composite vertical transformation σ 2 σ : F 2 F 1 ñ F 1 2 F 1 1 bears pseudomonoidal structure witnessed by the 2-morphisms where the empty squares are horizontal identities existing due to naturality of σ 2 and F 12 2 .It is direct to check that these pseudomonoidal constraint cells are stable under vertical composition, so that we have a functorial tensor product on the category of lax monoidal functors and pseudomonoidal vertical transformations.Taking this tensor product together with the (strict) monoidal 1 C as unit, we obtain a strict monoidal category: indeed, for any lax monoidal double endofunctors F 3 , F 2 , F 1 of C, pF 3 F 2 qF 1 " F 3 pF 2 F 1 q as lax monoidal double functors, since both have coherence vertical 1-cells given by using the fact that double functors strictly preserve vertical composition; and correspondingly for the coherence 2-morphisms.
We now show that the category of horizontal monoidal transformations and monoidal modifications is monoidal.If β 1 : F 1 G 1 and β 2 : F 2 G 2 are two monoidal horizontal transformations, then the horizontal transformation β 2 β 1 : F 2 F 1 G 2 G 1 given as in ( 44) is monoidal, via the structure 2-morphisms Moreover, given monoidal modifications γ 1 , γ 2 as in (45), their composite γ 2 γ 1 : β 2 β 1 ⇛ β 1 2 β 1 1 given by ( 46) can be verified to satisfy the axioms (41) that render it monoidal, using, among other things, the monoidality of γ 1 and γ 2 .The functoriality of this tensor product is now inherited from DblCatrC, Cs 1 , given that monoidality is a mere condition on a modification.It is moreover easy to check that the associativity and unitality modifications in DblCatrC, Cs 1 become monoidal on lifting them to MonDblCatrC, Cs 1 , so providing the last pieces of data for the desired monoidal structure.
It remains to lift τ and η from DblCatrC, Cs to MonDblCatrC, Cs: and this is again simply a matter of checking that the modifications obtained from DblCatrC, Cs do indeed become monoidal modifications.
Using this result, and paralleling the developments of Section 7, we can now give succinct definitions of the notions of monoidal horizontal and vertical double monad.Definition 8.2 (Monoidal horizontal double monad).Let C be a monoidal double category.A monoidal horizontal double monad on C is a horizontal monoid in the monoidal double category MonDblCatrC, Cs.Explicitly, it is a horizontal double monad pT, m, eq on C in the sense of Definition 7.2 such that: satisfying the axioms of Definition 5.1; ‚ the horizontal transformation m : T T T is monoidal, i.e., it comes equipped with structure 2-morphisms: satisfying the axioms of Definition 5.2; ‚ the horizontal transformation e : 1 T is monoidal, i.e. comes with structure 2-morphisms satisfying the axioms of Definition 5.2; ‚ the modifications a, l, r of (51) are monoidal as in Definition 5.5.Definition 8.3 (Pseudomonoidal vertical double monad).Let C be a monoidal double category.A pseudomonoidal vertical double monad on C a vertical monoid in MonDblCatrC, Cs.Explicitly, it is a vertical double monad pT, m, eq on C in the sense of Definition 7.3, such that: ‚ the double functor T : C Ñ C is lax monoidal, as in Definition 5.1; ‚ the vertical transformation m : T T ñ T is pseudomonoidal, i.e., it comes equipped with 2morphisms satisfying the axioms of Definition 5.3; ‚ the vertical transformation e : 1 ñ T is pseudomonoidal , i.e., it comes equipped with 2-morphisms satisfying the axioms of Definition 5.3; ‚ the pseudomonoidal structures of the composites m˝T m and m˝mT : T T T ñ T are equal, while the pseudomonoidal structures of m ˝T e and m ˝eT : T ñ T are both trivial.
For a horizontal monad that arises from a vertical one via Theorem 7.4, we are naturally interested in conditions on the vertical monad such that the induced horizontal monad is monoidal.Thankfully, the conditions under which a pseudomonoidal vertical double monad induces a monoidal horizontal double monad are the same as for the non-monoidal case of Theorem 7.4, as the next theorem shows.
Theorem 8.4.Let C be a monoidal double category and pT, m, eq be a pseudomonoidal vertical double monad on C. Assume that the underlying vertical transformations m : T T ñ T and e : 1 C ñ T are special.Then pT, m, eq induces a monoidal horizontal double monad pT, p m, p eq on C.
Proof.If we consider T as a vertical monoid in MonDblCatrC, Cs, Theorem 6.4 ensures that it induces a horizontal pseudomonoid therein (namely a monoidal horizontal monad) when m and e have companions as pseudomonoidal vertical transformations.By Proposition 5.7, this is true if and only if m and e are special (Definition 3.6).For example, the unit of the induced monoidal horizontal double monad structure on T is the horizontal transformation p e : 1 T which becomes monoidal with structure cells that bijectively correspond, under transpose operations, to those of (57).
While a direct proof of Theorem 8.4 should be possible, the construction of all the data for a monoidal horizontal double monad from that of a pseudomonoidal vertical double monad using companions, let alone the verification of the coherence axioms, would be a daunting task.It is at this point that the advantage of our abstract view becomes clear; as an added bonus, the proofs of Theorems 7.4 and 8.4 become essentially the same.

Monoidal Kleisli double categories
In this section, we first introduce the (horizontal) Kleisli double category KlpT q for a horizontal double monad T (Definition 7.2) on a double category C, with a particularly important case being where T is induced from a vertical double monad (Definition 7.3) as in Theorem 7.4.
We next consider what happens when the double category C is monoidal and the double monad T is also monoidal.We would naturally expect the monoidal structure of C to extend to KlpT q, just as happens with an ordinary monoidal monad on an ordinary monoidal category.However, because the monoidal constraint data for a horizontal double monad does not point exclusively in the horizontal direction, things are slightly more subtle.To even obtain monoidal structure we must assume certain companions exist, and even then, this structure is only oplax monoidal in general (Theorem 9.4).Again, the situation where T is induced from a vertical double monad will be important, and in this special case, we describe sufficient conditions for this oplax monoidal structure on KlpT q to be normal oplax (Proposition 9.7) or (pseudo) monoidal (Corollary 9.8).
We begin with the construction of the horizontal Kleisli double category of a horizontal double monad.This construction is essentially contained in [CS10]; there, the authors start from a vertical double monad pT, m, eq, and define from it a horizontal Kleisli double category (Definition 4.1 of op.cit.) which in general is only a so-called virtual double category.These are weaker structures than double categories, in which horizontal 1-cells do not compose, but instead are formed into a structure of "multi-2-morphisms"; however, [CS10, Theorem A.8] shows that, when the vertical transformations e : 1 ñ T and m : T T ñ T are special, this virtual double category is in fact a double category.In this case, the horizontal Kleisli double category of loc.cit.can be obtained as follows: first apply Theorem 7.4 to form the horizontal double monad pT, p m, p eq associated to pT, m, eq; and then apply the following result.
Theorem 9.1.Let C be a double category and pT, m, eq be a horizontal double monad on it.There is a double category KlpT q, called the horizontal Kleisli double category of C, wherein: ‚ 2-morphisms as to the left below, are the 2-morphisms of C as to the right: Proof.Vertical composition in KlpT q is the same as in C; horizontal composition of Kleisli 1-cells M : X ù Y and N : Y ù Z is given by while horizontal pasting of Kleisli 2-morphisms φ and ψ is given by and the horizontal identity 2-morphism on f : X Ñ X 1 is It is easy to see that horizontal composition of 2-morphisms is vertically functorial, and so it remains to give the coherence constraints.Kleisli 1-cells M : X ù Y and N : Y ù Z and P : Z ù W , the associativity constraint is given by the following pasting, in which the horizontal 1-cell at the top is pP ˝Kl N q ˝Kl M and the one at the bottom is P ˝Kl pN ˝Kl M q: The unit constraints are as follows, where the horizontal top 1-cells are id Kl Y ˝Kl M and M ˝Kl id Kl X : Above, the 2-cells labelled a, l, r are as in (51) and the components of m, e are as in (49, 50).The coherence axioms follow by the usual argument for a Kleisli bicategory, cf.[CS10,FGHW18].
We have not ascribed any kind of universal property to the construction of the Kleisli double category, and for our purposes we do not need to; however, if we were to do so, then, following [Str72], we would express it in terms of universal opalgebra structure on the canonical embedding of C into KlpT q: Definition 9.2.Let C be a double category and pT, m, eq be a horizontal double monad on it.The canonical embedding F T : C Ñ KlpT q is the double functor which is the identity on objects and vertical 1-cells; sends a horizontal 1-cell M : X ÝÞÑ Y to e Y ˝M : X ù Y , and correspondingly for 2-morphisms between horizontal 1-cells.
Applying Lemma 3.3 to this canonical embedding, we immediately obtain the following result concerning companions in Kleisli double categories (cf.[CS10, Proposition 7.5]): Proposition 9.3.Let C be a double category, T a horizontal double monad on C, and f : X Ñ X 1 a vertical 1-cell of C. If f has a companion as a vertical 1-cell of C, then it has a companion also as a vertical 1-cell of KlpT q.
We now consider the Kleisli double category when C is a monoidal double category and T is a monoidal horizontal double monad (Definition 8.2).As discussed above, it does not seem to be true in general that the monoidal structure of C will extend to KlpT q; however, under mild assumptions which are satisfied in our applications, we do obtain at least an oplax monoidal structure (Definition 4.1) on KlpT q: Theorem 9.4.Let C be a monoidal double category and T a monoidal horizontal double monad on C. If the vertical 1-cell T 0 : I Ñ T I and each vertical 1-cell T 2 X1,X2 : T X 1 b T X 2 Ñ T pX 1 b X 2 q has a companion, then the monoidal structure of C induces an oplax monoidal structure on KlpT q.
Proof.The monoidal structure on the category of objects and vertical 1-cells KlpT q 0 " C 0 is inherited from C. For the monoidal structure on the category KlpT q 1 of horizontal 1-cells and 2-morphisms, we define the tensor product of M 1 : X 1 ù Y 1 and M 2 : X 2 ù Y 2 and the monoidal unit J to be: while the binary tensor product of 2-morphisms is given by: where the right-hand 2-morphism is a companion transpose of the equality T pg vertical 1-cells expressing naturality of T 2 .The associativity constraint is given by the following pasting, where the horizontal composite at the top is pM 1 b M 2 q b M 3 and the one at the bottom is Here, the 2-cell p˚q is the transpose of the equality pT αqT 2 pT 2 b 1q " T 2 p1 b T 2 qα of vertical 1-cells expressing the associativity axiom for the vertical part of the monoidal double functor T ; note that p˚q is invertible (as all other 2-cells in the above composite) by Proposition 2.9(iii) and (vi).The unit constraints are formed similarly as follows, where the top horizontal 1-cells are J b M and M b J, respectively: (61) Here, the 2-cells p˚q are the transposes of the unit axioms for the vertical part of the monoidal double functor T , and are invertible by the same reasoning as before.
The oplax monoidal structure map to the left of (23) has the form: where ˝Kl is defined as in (59); we obtain it as the pasting composite where the top left isomorphism is the monoidal interchange (23) applied twice, the 2-morphism labelled x m 2 is a companion transpose of the structure 2-morphism m 2 of the monoidal horizontal transformation m as in (54), and the 2-morphism x T 2 N1,N2 is a companion transpose of the component T 2 N1,N2 of the lax monoidal structure on the double functor T as in (36).
The globular 2-morphism η to the right of ( 23) is obtained as follows, where the horizontal 1-cell at the top is id Kl X1 b id Kl X2 and the one at the bottom is id Kl X1bX2 : where id Kl is defined as in (60).Here, the 2-morphism filling the square is a companion transpose of the structure 2-morphism e 2 of the monoidal horizontal transformation e as in (55).Finally, the globular structure 2-morphisms δ and ι of (24) are defined to be δ " obtained as the companion transpose of the structure 2-morphism m 0 from (54) (using that T x T 0 is a companion of T T 0 by Lemma 3.3); and the companion transpose of the structure 2-morphism e 0 from (55) respectively.
With some effort, one may show that with these structure cells, the horizontal double Kleisli category KlpT q is an oplax monoidal double category in the sense of Definition 4.1.We do not provide the details here, but in Appendix A.2 we give some sample verifications, along with a number of technical lemmas used repeatedly in the calculations.
It is very natural to ask when the oplax monoidal structure of the preceding definition is in fact a genuine (pseudo) monoidal structure, or at least a normal oplax monoidal structure.For our purposes, we will only answer this question in the case of primary interest, where our monoidal horizontal monad is induced from a pseudomonoidal vertical monad (Definition 8.3).To start with, putting together Theorem 8.4 and Theorem 9.4 gives us: Corollary 9.5.Let C be a monoidal double category and T be a pseudomonoidal vertical double monad.If it is true that: (i) the multiplication and unit of T are special; and (ii) all vertical 1-cells T 2 X1,X2 : T X 1 b T X 2 Ñ T pX 1 b X 2 q and T 0 : I Ñ T I have companions, then the Kleisli double category KlpT q of the induced monoidal horizontal monad pT, p m, p eq admits an oplax monoidal structure found as in Theorem 9.4.Remark 9.6.Notice that in the situation of the above corollary, it is only the interchange 2-morphisms τ of the oplax monoidal structure which may not be invertible.Indeed, each of the other structure 2-morphisms η, δ, ι, as displayed in ( 63) and (64), must be invertible; for example, in this case η is the transpose of the 2-morphism (58) which is, in turn, the transpose of the VpCq-invertible 2-morphism e 2 (57) of the pseudomonoidal vertical transformation e, and as such, is invertible by Proposition 2.9(vi).
As explained above, we will now investigate when, in the situation of Corollary 9.5, the oplax monoidal structure on KlpT q is in fact normal in the sense of Definition 4.3.To this end, motivated by the theory of pseudo-commutative monads [HP02], we define for a pseudomonoidal vertical monad pT, m, eq as in Definition 8.3 a vertical double transformation κ : p-q b T p?q ñ T ˝p-b ?q called the strength; this is given as the vertical composite T 2 ˝pe b 1q with components :" In an analogous way, we can also define the costrength as a vertical transformation T p-qbp?q ñ T ˝p-b?q.
It turns out that requiring these two vertical transformations to be special, as in Definition 3.6, is sufficient to make the oplax monoidal structure on KlpT q normal: Proposition 9.7.Let C be a monoidal double category and T be a pseudomonoidal vertical double monad.
If it is true that: (i) the multiplication and unit of T are special; (ii) all vertical 1-cells T 2 X1,X2 : T X 1 b T X 2 Ñ T pX 1 b X 2 q and T 0 : I Ñ T I have companions; and (iii) the strength and costrength of T are special vertical transformations, then the oplax monoidal double structure on KlpT q found as in Corollary 9.5 is normal.
Proof.We need to verify that each oplax double functor X 1 b p-q : KlpT q Ñ KlpT q and p-q b X 2 : KlpT q Ñ KlpT q is in fact a (pseudo) double functor.By Remark 9.6, we already know that the square η in (63) is invertible, which expresses the invertibility of the identity constraints for these double functors.As for the binary functoriality constraints, it suffices by symmetry to consider the case of X 1 b p-q.To say that its binary constraints are invertible is to say that the 2-morphism in (62) is invertible when X " X 1 " Y 1 " Z 1 and M 1 " N 1 " p e X1 .The 2-morphism in question is given by: Clearly the first and final rows of this diagram are invertible.On the second row, the 2-morphism x m 2 corresponds under transpose to the VpCq-invertible cell m 2 of the pseudomonoidal vertical transformation m in (56), and as such is invertible by Proposition 2.9(vi).Thus, we will be done if we can also prove the invertibility of the third row of the diagram.Now, since the strength (65) is by assumption a special vertical transformation, it is in particular true that the companion transpose of the component κ p e,N2 is invertible.This transpose is equally well the composite: where the 2-morphism p˚q on the top row is the transpose of the 2-cell e p eX 1 b 1 N2 .But by Lemma 3.12, this p˚q is itself the transpose of a vertical identity, and as such, is invertible by Proposition 2.9(vi).It follows that the composite 2-morphism comprising the bottom row of this diagram is invertible, which now implies the invertibility of the third row of (66) as desired.
Although this will not be the case in our applications, we note in particular the following sufficient conditions for the induced oplax monoidal structure on KlpT q to be not just normal oplax, but in fact a genuine (pseudo) monoidal structure.
Corollary 9.8.Let C be a monoidal double category and T be a pseudomonoidal vertical double monad.If it is true that: (i) the multiplication and unit of T are special; (ii) all vertical 1-cells T 2 X1,X2 : T X 1 b T X 2 Ñ T pX 1 b X 2 q and T 0 : I Ñ T I have companions; and (iii) the monoidality constraint T 2 of T is a special vertical transformation, then the oplax monoidal double structure on KlpT q found as in Corollary 9.5 is genuinely monoidal.
Proof.η, δ and ι are already known to be invertible, and, arguing as before, the assumption that T 2 is special ensures that every component (62) of the oplax monoidal interchange τ is invertible.
In the situation of Proposition 9.7, the fact that KlpT q is a normal oplax monoidal double category implies that its horizontal bicategory inherits the monoidal structure, in the sense specified in Definition 4.9.We thus obtain the following result as the culmination of the abstract development of the paper thus far.This will be the result we use to obtain the monoidal structure on the bicategory of coloured symmetric sequences in Section 10.Corollary 9.9.Let C be a monoidal double category and T be a pseudomonoidal vertical double monad.Under the assumptions of Proposition 9.7, the horizontal bicategory of KlpT q admits a normal oplax monoidal structure.

The arithmetic product of coloured symmetric sequences
In this section, we apply the theory developed in the previous sections to our intended application, namely coloured symmetric sequences.Throughout this section, we fix a cocomplete cartesian closed category V, considered as a symmetric monoidal closed category with respect to its cartesian closed structure.The restriction to a cartesian monoidal structure was already made in [DH14,GL16] and indeed it is essential for some of our results, as we explain further below.
In order to help readers follow our development, let us display the main double categories to be considered in this section in a commutative diagram of inclusions: On the left-hand side of the diagram, Mat V is the double category of matrices of Example 2.4 and Prof V is the double category of profunctors of Example 2.5.On the right-hand side of the diagram, CatSym V is the double category of categorical symmetric sequences which arises from Prof V as a Kleisli double category, and Sym V is its full double subcategory spanned by discrete V-categories-much like the double category Mat V is a full double subcategory of Prof V .We will define the double categories CatSym V and Sym V explicitly in Theorem 10.5, but in order to do so, we must first introduce the relevant double monad for the Kleisli construction.Let X be a small V-category.For n P N, let us define the V-category S n pXq as follows.The objects of S n pXq are n-tuples x " px 1 , . . ., x n q of objects of X.Given two such n-tuples x " px 1 , . . ., x n q and x 1 " px 1 1 , . . ., x 1 n q, the hom-object of maps between them is defined by S n pXqr x, x 1 s :" where S n is the n-th symmetric group, and where Ů and Ű denote coproduct and product respectively.
We then let SX be the following coproduct in Cat V : SX " ğ nPN S n pXq.
The V-category SX admits a symmetric strict monoidal structure in which the tensor product, written as x, y Þ Ñ x b y, is given by concatenation of sequences; the tensor unit is given by the empty sequence, written p q; and the symmetry is given by the evident permutations.The operation mapping X to SX extends to a 2-functor S : Cat V Ñ Cat V , which is part of a 2-monad whose strict algebras are symmetric strict monoidal V-categories.The multiplication of this 2-monad has components m X : SSX Ñ SX, for X P Cat V , defined by taking a list of lists to its flattening: The unit of the 2-monad has components e X : X Ñ SX, for X P Cat V , defined by taking an object x P X to the singleton list pxq P SX.
We now show that, firstly, S can be made into a vertical double monad on Prof V , and secondly, that this vertical double monad can be turned into a horizontal double monad.To say that S can be made into a vertical double monad is equivalently to say that the underlying 2-functor of S extends along the inclusion of bicategories Cat V Ñ Prof V -see Remark 10.2 below-while to say that this vertical monad can be turned into a horizontal one amounts to saying that the whole 2-monad S extends from Cat V to Prof V .This is a known result, and there are two approaches in the literature to proving it.The first uses the theory of pseudo-distributive laws; see, for example [FGHW18].The second, which we follow here, is essentially a categorification of the approach of [Bar70].
Proposition 10.1.The free symmetric strict monoidal category 2-monad S on Cat V extends in an essentially unique way to a vertical double monad on Prof V .The multiplication and unit vertical transformations of this double monad are special.
Here, we say that a vertical double monad T on a double category C extends a 2-monad R on the vertical 2-category VpCq, if R is isomorphic (as a 2-monad) to the 2-monad VpT q on VpCq induced by T .By saying that an extension of R is essentially unique, we mean to assert the contractibility of the category in which objects are vertical double monads T on C endowed with an isomorphism of 2-monads VpT q -R, and morphisms are vertical double monad morphisms compatible with the isomorphisms to R.
Proof.Because any double functor preserves companions (Lemma 3.3), any extension of S to Prof V must satisfy Sp p F q -y SF for a V-functor F .Because p F % q F in HpProf V q, and any double functor preserves adjunctions in the horizontal bicategory, we must also have Sp q F q -} SF .Since by [Str80, §6], every V-profunctor M : X ÝÞÑ Y admits a globular isomorphism to one of the form q G ˝p F for a suitable cospan of V-functors F : X Ñ Z Ð Y : G, the preceding conditions determine the action of S on horizontal 1-cells of Prof V to within isomorphism.Using this idea, one obtains the following explicit definition: given M : X ÝÞÑ Y , SM : SX ÝÞÑ SY is defined as the N-indexed coproduct in pProf V q 1 of S n pM q : S n pXq ÝÞÑ S n pY q, where S n pM qp y, xq :" The action of S on 2-morphisms of Prof V is now forced by Proposition 2.9(iii) and the fact that any double functor preserves globularity; the reader will easily guess an explicit formula, and this guess is the correct one.This completes the extension of S to a double functor on Prof V ; that these data are indeed double functorial is verified, for example, in [Gar06b, Proposition 55], to which we refer for further details.Note also that the manner in which we defined the action of S on horizontal 1-cells and 2-morphisms means that this extension is essentially unique, in the sense set out above.We now extend the unit e and multiplication m of the 2-monad S; the missing data are the 2-morphism components e M and m M (52, 53) associated to a horizontal 1-cell M : X ÝÞÑ Y .Writing M -q G ˝p F as before, we see that e M and m M are determined by e p F , e q G , m p F and m q G : e To the left, Lemma 3.12 implies that e p F is the companion transpose of the identity of vertical 1-cells e Z ˝F " SF ˝eX ; while a suitable dual of Lemma 3.12 implies that e q F is the conjoint transpose of the identity e Z ˝G " SG ˝eY .In a similar way, the 2-morphisms m p F and m q G are forced.An explicit verification that these data satisfy the vertical double monad axioms is, again, given in [Gar06b,Proposition 55].So we have extended S to a vertical double monad on Prof V ; and like before, the manner in which we made this extension forces it to be essentially unique.
It remains only to verify that the unit and multiplication of our extended S are special.Before doing so, we note a fact which will be used repeatedly in what follows.Suppose given profunctors N : X Y and M : SY Z. Then for any z P Z and x " px 1 , . . ., x n q P SX, the value at pz, xq of the composite M ˝SN : SX SY Z, as to the left below, is equally given as to the right: Indeed, we can immediately reduce the left-hand coend to one over y P S n Y ; and for such a y, we have M pz, yq ˆSN p y, xq -Ů σPSn M pz, yq ˆŰ1ďiďn N py σpiq , x i q.On the σ-summand of this coproduct, we define the component of the desired isomorphism (69) to be where pσ ˚ yq i " y σpiq and where σ -1 : y Ñ σ ˚ y is the evident symmetry isomorphism in SY .
We Starting to the left, let z P Z and y " py 1 , . . ., y n q P SY .To within isomorphism, using the formulas for composition and companions for profunctors (8, 11) as well as (67), the profunctor at the bottom of the square sends p y, zq to SM p y, pzqq " On the other hand, the profunctor around the top sends p y, zq to: ż y 1 PY SY r y, py 1 qs ˆM py 1 , zq " # ş y 1 PY Y ry 1 , y 1 s ˆM py 1 , zq if n " 1; In the only non-trivial case where n " 1, the comparison 2-cell p e M from (71) to ( 70) is given by composition: and this is invertible by the Yoneda lemma.We proceed similarly for p m M .Let z " p z 1 , . . ., z n q P SSZ with z i " pz mi-1`1 , . . ., z mi q for some 0 " m 0 ď m 1 ď ¨¨¨ď m n , and let y " py 1 , . . ., y m q P SY .The only non-trivial case is when m n " m so we immediately restrict to that.This time, the profunctor in the bottom row sends p y, zq to SM p y, â On the other hand, the profunctor around the top sends p y, zq to ż wPSSY SY r y, â 1ďiďm w i s ˆSSM p w, zq -ż w 1 ,..., w n PSY SY r y, â 1ďiďm w i s ˆę 1ďjďn SM p w j , z j q ż w1,...,wmPY SY r y, ws ˆę 1ďjďn mi-1`1ďiďmi M pw i , z i q ğ σPSm ż w1,...,wmPY ę 1ďiďm Y ry σpiq , w i s ˆM pw i , z i q, (73) using (69) once at the first step, and n times at the second step.The comparison 2-morphism p m M from (73) to (72) is again given by composition, and this is again invertible by the Yoneda lemma.
Remark 10.2.In fact, the above argument shows that, any 2-monad R on Cat V which extends to Prof V will have an essentially-unique such extension; for indeed, the action on horizontal 1-cells must be given as RpM q " } RG ˝y RF , where M " q G ˝p F , and the components of the extended unit and multiplication at a horizontal 1-cell M must be determined similarly.The only non-trivial point is verifying that composition of horizontal 1-cells is preserved to within globular isomorphism-and this comes to the same thing as asking that the underlying 2-functor of R extends along the homomorphism of bicategories x p-q : Cat V Ñ Prof V .Thus, to give an extension of the 2-monad R to Prof V is equally to give an extension of the underlying 2-functor along x p-q : Cat V Ñ Prof V , as claimed above.
Corollary 10.3.The 2-monad S : Proof.Apply Theorem 7.4 to the vertical double monad of Proposition 10.1 to obtain the horizontal double monad pS, p m, p eq.
We are now ready to recall the definition of categorical and coloured symmetric sequences.
‚ Let X and Y be small V-categories.A categorical symmetric sequence M : X ù Y is a profunctor M : X ÝÞÑ SY , i.e. a V-functor M : SY op ˆX Ñ V.
‚ Let X and Y be sets.A coloured symmetric sequence M : X ù Y is a categorical symmetric sequence from X to Y , considered as discrete V-categories.
Categorical symmetric sequences and coloured symmetric sequences are the horizontal 1-cells of double categories that we denote CatSym V and Sym V , which we may now obtain by forming the horizontal Kleisli double category (Theorem 9.1) of the horizontal double monad S : Prof V Ñ Prof V .
(i) There exists a double category CatSym V having small V-categories as objects, V-functors as vertical 1-cells and categorical symmetric sequences as horizontal 1-cells.(ii) There exists a double subcategory Sym V having sets as objects, functions as vertical 1-cells and coloured symmetric sequences as horizontal 1-cells.
Proof.For Theorem 10.5(i), it suffices that we apply Theorem 9.1 to the horizontal double monad pS, p m, p eq : Prof V Ñ Prof V of Corollary 10.3, where m and e are as in (68).Indeed, a categorical symmetric sequence M : X ù Y is precisely a horizontal Kleisli 1-cell, and so by (59), the composition of categorical symmetric sequences M : X ÝÞÑ SY and N : Y ÝÞÑ SZ is the profunctor N ˝Kl M : X ÝÞÑ SZ found as the composite Using (8), ( 11) and (67), this composite has value at z " pz 1 , . . ., z m q P SZ and x P X given by pN ˝Kl M qp z, xq " ż wPSSZ, yPSY SZr z, â i w i s ˆSN p w, yq ˆM p y, xq which by applying (69) simplifies to the following well-known formula (c.f.[FGHW08, eq. ( 10)]): which is a generalisation of the substitution monoidal structure for symmetric sequences [Kel05].Theorem 10.5(ii) follows immediately and the formula for composition does not actually simplify significantly, since SY and SZ are genuine categories even when Y and Z are sets.
Remark 10.6.Even if the primary focus of our interest is the double category of coloured symmetric sequences Sym V , it is useful to consider the larger double category of categorical symmetric sequences CatSym V .The reason is that the latter arises naturally from the double category of profunctors as a Kleisli double category and enjoys better closure properties than the former, since the free symmetric strict monoidal category on a discrete V-category is not discrete.
We now wish to apply the theory developed in the previous sections in order to obtain the desired oplax monoidal structures on CatSym V and Sym V .First note that, by Example 4.7, the double category Prof V has a monoidal structure induced from that on V. Thus, by Proposition 9.7, it suffices to show that the vertical double monad S : Prof V Ñ Prof V has well-behaved pseudomonoidal structure.
The key fact which allows us to do this is that, as a 2-monad on Cat V , S is pseudomonoidal [HP02,Kel74].Indeed, [HP02, Section 3.3] shows that S can be equipped with the structure of a pseudocommutative 2-monad, while [HP02, Theorem 7] states that every pseudo-commutative 2-monad is pseudomonoidal, cf. also [Koc72, Theorem 2.3].For our purposes, it will be convenient to describe the relevant structure explicitly.First of all, S admits a strength [Koc72] given by: as well as a costrength κ 1 : SX ˆY Ñ SpX ˆY q given dually.Note that, because the formula for κpx, yq repeats the variable x, we can only make the assignment of (74) V-functorial when V is cartesian monoidaland this explains why we made this restriction in the first place.In this situation, the 2-functor S acquires two canonical lax monoidal structures built from the strength, the costrength and the monad multiplication as in [Koc72, eqs.(2.1) and (2.2)].In our case, one of these lax monoidal structures has S 0 : 1 Ñ S1 given by the monad unit, and S 2 X,Y : SX ˆSY Ñ SpX ˆY q defined by lexicographic product: `px 1 , . . ., x m q , py 1 , . . ., y n q ˘Þ Ñ `px 1 , y 1 q, px 1 , y 2 q, . . ., px 1 , y n q, px 2 , y 1 q, . . ., px m , y n q ˘, (75) which we sometimes also denote by x b y as in (5).The other lax monoidal structure has the same S 0 and binary constraints SX ˆSY Ñ SpX ˆY q given by colexicographic product.Evidently, these two lax monoidal structures are isomorphic, and this is the key aspect of S being pseudo-commutative in the sense of [HP02].In this situation, by [HP02, Theorem 7], which is a higher-dimensional adaptation of [Koc72, Theorem 2.3], S becomes a pseudomonoidal 2-monad with respect to the lax monoidal structure S 2 .It is not hard to see that the monad unit e : 1 ñ S is in fact a genuine monoidal transformation; however, the multiplication m : SS ñ S is not monoidal, but only a pseudomonoidal transformation; which is to say that the two sides of the diagram are not equal, but only coherently isomorphic via a 2-cell as displayed.We now describe this 2-cell m 2 X,Y explicitly.To this end, let us take a typical element of SSX ˆSSY , say `p x 1 , . . ., x k q, p y 1 , . . .y ℓ q ˘where x i " px i 1 , . . ., x i mi q for 1 ď i ď k and y j " py j 1 , . . ., y j nj q for 1 ď j ď ℓ.On the one hand, around the lower side of (76), this element is sent first to `px 1 1 , . . ., x k m k q, py 1 1 , . . ., y ℓ n ℓ q ˘and then to `px 1 1 , y 1 1 q, px 1 1 , y 1 2 q, . . ., px 1 1 , y ℓ n ℓ q, px 1 2 , y 1 1 q, . . ., px 1 2 , y ℓ n ℓ q, . . ., This is the lexicographic order on four indices.On the other hand, around the upper side of (76) we obtain first `p x 1 , y 1 q, p x 1 , y 2 q, . . ., p x k , y ℓ q ˘and then, applying (75) to each pair, we get This is a twisted lexicographic order on the indices, with the significance order being 1-3-2-4 rather than 1-2-3-4.Clearly, there is a unique bijection θ which exchanges these orderings, giving the components of the desired natural isomorphism m 2 X,Y filling (76).This establishes the binary pseudomonoidality of m; the corresponding nullary pseudomonoidality constraint m 0 is in fact the identity.That these data satisfy the necessary coherences to form a pseudomonoidal monad is now asserted in [HP02, Theorem 7], but one could also establish this directly, following a reasoning similar to that used in [HP02, Section 3.3] to establish pseudocommutativity.
The next lemma extends the pseudomonoidal structure of the 2-monad S : Cat V Ñ Cat V to a pseudomonoidal structure in the sense of Definition 8.3 on the vertical double monad S : Prof V Ñ Prof V of Proposition 10.1.Before doing this, let us note that under our assumption that V is cartesian monoidal, the induced monoidal structure on Prof V is also cartesian, in the sense that the ordinary monoidal structures on pProf V q 0 and pProf V q 1 that underlie it are both cartesian monoidal; as such, we will continue to write ˆrather than b for this tensor product, in particular, for its action on horizontal 1-cells of Prof V .
Proof.We need to check that S is a lax monoidal double functor as in Definition 5.1 and that m and e are pseudomonoidal vertical transformations as in Definition 5.3.Let us begin by showing that S admits a lax monoidal structure.For X 1 , X 2 P Prof V , the vertical 1-cells S 2 X1,X2 : SX 1 ˆSX 2 Ñ SpX 1 ˆX2 q and S 0 : 1 Ñ S1 are given in (75).For M 1 : X 1 ÝÞÑ Y 1 and M 2 : X 2 ÝÞÑ Y 2 , the squares

SpM1ˆM2q
(77) can be constructed following the same reasoning as in the proof of Proposition 10.1, i.e. reducing to the cases where M " p F , N " p G and M " q F , N " q G, and using that S is lax monoidal on Cat V .We already observed that the unit e is genuinely monoidal at the 2-monad level, and the same is true for e qua vertical transformation.As for the vertical transformation m, the axioms for a pseudomonoidal vertical transformation that concern only the vertical fragment are exactly those expressing that m is a pseudomonoidal natural transformation in Cat V , which we have discussed above.The only axiom not of this form is (40), and this can be verified using the construction of the squares in (77) via a reduction to companions and conjoints and the modification axiom for the 2-cells filling (76).
So S extends to a pseudomonoidal vertical double monad S : Prof V Ñ Prof V ; and the last step required to establish the normal oplax monoidal structure of coloured symmetric sequences is to verify that this vertical double monad satisfies the additional conditions of Proposition 9.7.
Lemma 10.8.The pseudomonoidal vertical double monad S : Prof V Ñ Prof V has the properties that: Proof.Item (i) was already shown as part of Proposition 10.1.Item (ii) is immediate since every vertical 1-cell in Prof V has a companion, see Example 2.8.For Item (iii), the two cases are dual, so we only provide details for one.According to Proposition 9.7, the strength in question is defined from the lax monoidal structure S 2 of S via (65); but because we originally obtained S 2 using the strength κ of (74) and the dual costrength, it follows as in [Koc72, Theorem 2.3] that the strength of (65) is this same κ, and similarly for the costrength.Thus, the condition we must prove e.g. for the costrength is that, for any M : X ÝÞÑ Y and N : W ÝÞÑ Z, the companion transpose 2-morphism is invertible.To this end, let u " `py 1 , z 1 q, . . ., py m , z m q ˘in SpY ˆZq and let p x, wq P SX ˆW where x " px 1 , . . ., x m q.Note we assume that u and x have the same length; that we may do so without loss of generality will be clear from the formulae which follow.Now, the profunctor along the bottom of (78) has value at p u, p x, wqq given by SpM ˆN q ` u, κp x, wq ˘" ğ σPSm ę 1ďiďm pM ˆN q `py σpiq , z σpiq q, px i , wq " ğ σPSm ę 1ďiďm M py σpiq , x i q ˆN pz σpiq , wq.
On the other hand, the profunctor across the top of (78) has value at p u, p x, wqq given by ż vPSY,z 1 PZ SpY ˆZqr u, κp v, z 1 qs ˆpSM ˆN q `p v, z 1 q, p x, wq " where at the first isomorphism we use (69) and at the final one we use the Yoneda lemma.By tracing it through we may see that the isomorphism constructed in this way is exactly p κ M,N , which is thus invertible.As noted above, the specialness of the costrength follows by an identical dual argument.
Theorem 10.9.The double category CatSym V of categorical symmetric sequences admits a normal oplax monoidal structure, given by arithmetic product of categorical symmetric sequences.Moreover, this restricts to a normal oplax monoidal structure on the double category Sym V of coloured symmetric sequences.
Proof.CatSym V is the horizontal Kleisli double category of the horizontal double monad induced by the vertical double monad S : Prof V Ñ Prof V , as seen in the proof of Theorem 10.5.Moreover, by Lemma 10.7, the vertical double monad S is pseudomonoidal, and by Lemma 10.8, it satisfies the further hypotheses of Corollary 9.5 and Proposition 9.7.Applying these results, we see that the monoidal structure of Prof V extends to a normal oplax monoidal structure on CatSym V , which clearly restricts back to the full sub-double-category Sym V .
It remains to show that the tensor product of horizontal 1-cells computes the arithmetic product of categorical symmetric sequences.For categorical symmetric sequences M 1 : X 1 ù Y 1 and M 2 : X 2 ù Y 2 , the tensor product M 1 b M 2 : X 1 ˆX2 ù Y 1 ˆY2 is defined as the profunctor: We now unfold this expression explicitly.First, by the definition of a companion in (11) applied to S 2 Y1,Y2 , the second of these profunctors is given by x S 2 Y1,Y2 p y, p y 1 , y 2 qq " SpY 1 ˆY2 qr y, y 1 b y 2 s, where y 1 b y 2 is given by the lexicographic ordering (75).Thus, using the definition of tensor product of profunctors (26) and of composition of profunctors in (8), we obtain pM 1 b M 2 qp y, px 1 , x 2 qq " ż y1, y2 SpY 1 ˆY2 qr y, y 1 b y 2 s ˆM1 p y 1 , x 1 q ˆM2 p y 2 , x 2 q, (79) which is the formula for the arithmetic product of categorical and coloured symmetric sequences and in particular gives the formula in (4) for coloured symmetric sequences.
The final step is to obtain the desired oplax monoidal structures at the level of bicategories rather than double categories.Indeed, the horizontal bicategories of CatSym V and Sym V are the bicategories of categorical and coloured symmetric sequences introduced in [FGHW18] for V " Set and in [GJ17] for a general V. Thus, we can apply Corollary 9.9 to obtain: Theorem 10.10.The bicategory of categorical symmetric sequences CatSym V admits a normal oplax monoidal structure, given by the arithmetic product of categorical symmetric sequences.Furthermore, this normal oplax monoidal structure restricts to the bicategory of coloured symmetric sequences Sym V .
Apart from its intrinsic interest, and the construction of an example of a sophisticated kind of lowdimensional categorical structure by purely algebraic means, without any appeal to homotopy theory, Theorem 10.9 and Theorem 10.10 will be essential for subsequent work on the Boardman-Vogt tensor product of bimodules between symmetric coloured operads, extending that in [DH14] for bimodules between symmetric operads.Appendix A.
A.1.Coherence axioms for an oplax monoidal double category.In this appendix, we spell out in detail the axioms for an oplax monoidal category as in Definition 4.1.In [HS19], the explicit axioms for a monoidal double category can be found, which are analogous but differ in that we use the opposite orientation for our (non-invertible) structure maps, and also provide additional non-invertible structure cells δ and ι as in (24), which in the pseudo case can be chosen to be identities and so omitted.
The 2-cells τ and η as in (23) satisfy the following axioms, for M i : X i ÝÞÑ Y i , N i : Y i ÝÞÑ Z i and P i : Z i ÝÞÑ U i :

M1bM2
These axioms make b into an oplax double functor.Notice that at the bottom of the left diagram of the first axiom there is a composition associativity constraint implied.The 2-cells δ and ι as in (24) satisfy the following axioms Next, for horizontal 1-cells M i : X i ÝÞÑ Y i and N i : Y i ÝÞÑ Z i , the following axioms hold " idX 1 bpidX 2 bidX 3 q ó1bη idX 1 bidX 2 bX 3 óη id X 1 bpX 2 bX 3 q which make associativity into a vertical transformation of oplax double functors (together with the naturality of components which comes from pD 1 , b 1 , I 1 q being a monoidal category).Moreover, the following axioms hold A.2. Proof of Theorem 9.4.In this section, we illustrate the main ideas in the proof of Theorem 9.4.We begin by stating a couple of technical lemmas which, along with Lemma 3.12, are used repeatedly in the proof.
for any 2-morphisms φ, ψ of the right shape.
Proof.By pasting the 2-morphism F pp 2 ˝1M q on the top of the left diagram, we obtain where the first equality is due to naturality of components of ξ, and the second one (up to pasting with appropriate coherence isomorphisms) follows from the definition of the transpose p ψ and the unitality axiom for F .Lemma A.2. Let β : F G be a horizontal transformation and f : X Ñ X 1 a vertical 1-cell in C. If f has a companion p f , then the globular coherence 2-isomorphism β p f is vertically inverse to the transpose of the 2-morphism component β f , i.e. : Proof.We will show that if we vertically compose the transpose of β f with β p f on both sides, it produces a vertical identity (up to coherence isomorphisms).Indeed, We now provide a sample verification of one of the axioms needed in the proof of Theorem 9.4.We will show that the left unit constraint of the monoidal structure of a Kleisli double category KlpT q for a monoidal horizontal double monad is compatible with horizontal composition, namely the top left axiom of (81).Now using the fact that pT 2 , T 0 q and pm 2 , m 0 q are the structure maps of lax monoidal functors, namely which is equal to (82), since λ is a vertical transformation of double functors (18).
Definition 4.1 (Oplax monoidal double category, monoidal double category).Let C be a double category.An oplax monoidal structure on C consists of the following data: ‚ an oplax double functor b : C ˆC Ñ C; ‚ an oplax double functor I : 1 Ñ C; ‚ invertible vertical transformations α : b ˝p1 ˆbq ñ b ˝pb ˆ1q, λ : b ˝pI ˆ1q ñ 1 and ρ : b ˝p1 ˆIq ñ 1 satisfying the usual Mac Lane coherence axioms for α, λ and ρ.Said another way, this structure amounts to the following: Definition 5.1 (Lax monoidal double functor).Let C and D be monoidal double categories.A lax monoidal double functor F : C Ñ D is a (pseudo) double functor equipped with:

Definition 6. 1 .
Let C be a monoidal double category.A horizontal monoid in C consists of: ‚ an object A; ‚ horizontal 1-cells m : A b A ÝÞÑ A and e : I ÝÞÑ A; ‚ invertible cells pA b Aq b A A b A A A b pA b Aq A b A A required to satisfy the coherence axioms that: ppA b Aq b Aq b A pA b Aq b A A b A A pA b pA b Aqq b A pA b Aq b A A b A A A b ppA b Aq b Aq A b pA b Aq A b Aq b Aq b A pA b Aq b A A b A A pA b Aq b pA b Aq A b pA b Aq A b A A pA b Aq b pA b Aq pA b Aq b A A b Remark 6.2.As discussed in Section 4, under fairly mild conditions the horizontal bicategory HpCq of a monoidal double category C will have the structure of a monoidal double category, whose monoidal associativity and unit constraint 1-cells are the companions of the corresponding constraints for C. In this situation, horizontal monoids in C correspond to pseudomonoids in HpCq by taking the companion transposes of the coherence data (43).Definition 6.3 (Vertical monoid).Let C be a monoidal double category.A vertical monoid in C is a monoid in the monoidal category C 0 .Explicitly, it is an object A endowed with vertical 1-cells m : A b A Ñ A and e : I Ñ A satisfying the usual associativity and unit laws.
Proposition 8.1 (Composition monoidal structure on MonDblCatrC, Cs).Let C be a monoidal double category.The composition monoidal structure of the double category DblCatrC, Cs lifts to a monoidal structure on the double category MonDblCatrC, Cs of monoidal endofunctors, pseudomonoidal vertical transformations, monoidal horizontal transformations and monoidal modifications.
(i) The multiplication m : SS ñ S and unit e : 1 ñ S are special vertical transformations.(ii) The vertical 1-cells S 2 X,Y : SX ˆSY Ñ SpX ˆY q and S 0 : 1 Ñ S1 have companions.(iii) The strength and costrength of S are special vertical transformations.
I : 1 Ñ C into an oplax double functor.
left unit constraint λ : b ˝pI ˆ1q ñ 1 and the right unit constraint ρ : b ˝p1 ˆIq ñ 1 into vertical transformations of oplax double functors (together with the naturality of components coming from pD 1 , b 1 , I 1 q being a monoidal category).

Lemma A. 1 .
Let F : C Ñ D be a double functor.For any horizontal 1-cell M : X ÝÞÑ Y and vertical 1-cell f : Y Ñ Y 1 in C, a 2-morphism in D of the form on the left corresponds, under transpose operations, to a 2-morphism of the form on the right

FirstT
of all, the left unit constraint components J b M ñ M as in (61) bijectively correspond to cells Lemma A.1, since τ is the composition comparison structure map for the double functor b of the monoidal double category C. Using appropriate transpose operations like (12), we can therefore transform the axiom at question to one that does not involve companions of 1-cells and 2-morphisms as follows: the right-hand side of (81) becomes the left-hand side, using naturality of τ , becomes pI b Y q T pI b T Zq T pI b T Zq T T I b T T Z T I b T Z T pT I b T Zq T pT I b T Zq T T pI b Zq T T pI b Zq T pI b Zq now show that the extended vertical transformations e and m are special, i.e. that the following companion transpose 2-morphisms are invertible: