The Gray monoidal product of double categories
Gabriella B\"ohm
TL;DR
This paper establishes a symmetric closed monoidal structure on the category of double categories, defining an internal hom functor that captures double functors, pseudotransformations, and modifications, with practical functors shown to be compatible.
Contribution
It introduces a new symmetric closed monoidal structure on double categories and characterizes the internal hom functor with compatibility results for key functors.
Findings
Defines the symmetric closed monoidal structure for double categories.
Constructs the internal hom functor with detailed cell structures.
Shows compatibility of well-known functors with this monoidal structure.
Abstract
The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category , the corresponding internal hom functor sends a double category to the double category whose 0-cells are the double functors , whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.
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The Gray monoidal product of double categories
Gabriella Böhm
Wigner Research Centre for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary
(Date: Jan 2019)
Abstract.
The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category , the corresponding internal hom functor sends a double category to the double category whose 0-cells are the double functors , whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.
Introduction
The category of 2-categories and 2-functors carries different monoidal structures. The simplest one is given by the Cartesian product. It is symmetric and closed. For any 2-category , the internal hom functor sends a 2-category to the 2-category of 2-functors , 2-natural transformations, and modifications. This is, however, often too restrictive. For example, important examples of 2-categories which are intuitively monoidal, fail to be monoids for that [1, 14, 15]. A well established generalization is the so-called Gray monoidal product [13]. It is also symmetric and closed and for any 2-category the corresponding internal hom functor sends a 2-category to the 2-category of 2-functors , pseudonatural transformations, and modifications. The Cartesian monoidal structure is more restrictive than the Gray one in the sense that the identity functor on is a monoidal functor from the former to the latter one.
The category of double categories and double functors is also symmetric closed monoidal via the Cartesian product . For any double category , the corresponding internal hom functor sends a double category to the double category whose 0-cells are the double functors , whose horizontal and vertical 1-cells are the horizontal and vertical transformations, respectively, and whose 2-cells are the modifications; see [9]. The analogue of the Gray monoidal product on , however, has apparently not yet been discussed in the literature. The current paper addresses this question.
For any double categories and , there is a bigger double category in which the 0-cells are still the double functors . The horizontal and vertical 1-cells are, however, the horizontal and vertical pseudo (or strong) transformations of [9]. The 2-cells are their modifications. In Section 1 we prove that for any double categories and , there is a representing object of the functor . Constructing the associativity and unit constraints, as well as the symmetry, in Section 2 we show that equips with a symmetric monoidal structure. In order to support this choice of monoidal structure on , in Section 3 monoidality of the following functors is checked.
- •
The identity functor .
- •
The functors sending double categories to their horizontal – or vertical – 2-categories (for the Gray monoidal product on ).
- •
The square (or quintet) construction functor due to Ehresmann [5].
- •
The functor , sending a double category to the double category of its monads by Fiore, Gambino and Kock [6].
We also give an explicit description of monoids in which generalize the strict monoidal double categories of [4]; that is, the monoids in .
Acknowledgement
Financial support by the Hungarian National Research, Development and Innovation Office – NKFIH (grant K124138) is gratefully acknowledged.
1. Existence
In this section we construct an adjunction of endofunctors on the category of double categories, for any double category . Our line of reasoning is similar to [3, Proposition 3.10]. The occurring double functor is our candidate Gray monoidal product on . Mac Lane’s coherence conditions are checked in Section 2.
1.1. The category of double categories
We begin with introducing the category of double categories and double functors, and recording some of its basic properties.
Definition 1.1**.**
A double category is an internal category in the category of categories and functors. A double functor in an internal functor in . Double categories are the objects, and double functors are the morphisms of the category .
So a double category consists of 0-cells, also called objects, (interpreted as the objects of the category of objects), vertical 1-cells (which are the morphisms of the category of objects), horizontal 1-cells (the objects of the category of morphisms) and 2-cells (the morphisms of the category of morphisms). They can be composed vertically (in the category of objects and the category of morphisms, respectively) and horizontally (via the composition functor of the double category). As usual in the literature (see e.g. [9]), we denote 2-cells as squares surrounded by the appropriate horizontal and vertical source and target 1-cells. We denote by both horizontal and vertical identity 1-cells; and also identity 2-cells for the horizontal or vertical composition. Usually we neither make notational difference between the compositions of horizontal and vertical 1-cells; both are denoted by a dot (if not a diagram is rather drawn).
By [7, Theorem 4.1] and its proof, is locally finitely presentable — so in particular cocomplete — and complete. Its terminal object is the double category of a single object and only identity higher cells.
Consider the double category which is freely generated by a single 2-cell. In more detail, has four objects, we denote them by , , and . There are identity horizontal and vertical identity 1-cells for each object as well as non-identity horizontal and vertical 1-cells
[TABLE]
There are vertical identity 2-cells at each horizontal 1-cell, horizontal identity 2-cells at each vertical 1-cell, and a single non-identity 2-cell
[TABLE]
The functor sends a double category to the set of double functors , which can be identified with the set of 2-cells in . A double functor is sent to its 2-cell part, which is an isomorphism in if and only if is bijective on the 2-cells. Since this includes bijectivity also on the identity 2-cells of various kinds, it is equivalent to being bijective on all kinds of cells; that is, its being an isomorphism in . By the so obtained conservativity of the functor we conclude that is a strong generator of the finitely complete category with coproducts, see [2, Proposition 4.5.10].
1.2. The double categories of double functors
Using similar constructions to those in [9, Section 7], any double categories and determine a double category as follows.
The 0-cells are the double functors .
The horizontal 1-cells are the horizontal pseudotransformations (called strong horizontal transformations in [9, Section 7.4]). A horizontal pseudotransformation consists of the following data.
- •
For any 0-cell of , a horizontal morphism in on the left;
- •
for any vertical 1-cell in , a 2-cell in in the middle;
- •
for any horizontal 1-cell in , a vertically invertible 2-cell in on the right:
[TABLE]
These ingredients are subject to the following axioms.
- (i)
Vertical functoriality, saying that for the identity vertical 1-cell on any object in , is equal to the vertical identity 2-cell on the left; and for any composable vertical 1-cells and in , the equality on the right holds:
[TABLE] 2. (ii)
Horizontal functoriality, saying that for the identity horizontal 1-cell on any object , is equal to the same vertical identity 2-cell on the left; and for any composable horizontal 1-cells and in , the equality on the right holds:
[TABLE] 3. (iii)
Naturality, saying that for any 2-cell in ,
[TABLE]
The vertical 1-cells are the vertical pseudotransformations (called strong vertical transformations in [9, Section 7.4]). A vertical pseudotransformation consists of the following data.
- •
For any 0-cell of , a vertical 1-cell in on the left;
- •
for any horizontal 1-cell in , a 2-cell in in the middle;
- •
for any vertical 1-cell in , a horizontally invertible 2-cell in on the right:
[TABLE]
These ingredients are subject to the following axioms.
- (i)
Horizontal functoriality, saying that for the identity horizontal 1-cell on any object in , is equal to the horizontal identity 2-cell on the left; and for any composable horizontal 1-cells and in , the equality on the right holds:
[TABLE] 2. (ii)
Vertical functoriality, saying that for the identity vertical 1-cell on any object , is equal to the same horizontal identity 2-cell on the left; and for any composable vertical 1-cells and in , the equality on the right holds:
[TABLE] 3. (iii)
Naturality, saying that for any 2-cell in ,
[TABLE]
The 2-cells are the modifications. A modification on the left is given by a collection of 2-cells in on the right, for all 0-cells of :
[TABLE]
satisfying the following axioms.
- (i)
For any horizontal 1-cell in ,
[TABLE] 2. (ii)
For any vertical 1-cell in ,
[TABLE]
The identity horizontal pseudotransformation has the components
[TABLE]
while the composite of some horizontal pseudotransformations
\textstyle{\mathsf{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathsf{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z}$$\textstyle{\mathsf{H}}
has the components
[TABLE]
for any horizontal 1-cell and vertical 1-cell in . Symmetric formulae apply to the vertical pseudotransformations. The components of the horizontal composite of modifications are the horizontal composites of their components, and the components of the vertical composite of modifications are the vertical composites of their components.
Throughout, we identify any double category with the isomorphic double category .
1.3. The functor
In this section we interpret the map, sending a pair of double categories and to the double category of Section 1.2, as the object map of a functor in the title. So we need to construct its morphism map, sending a pair of double functors and to a double functor .
Its 0-cell part sends a double functor
\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{H}}$$\textstyle{\mathbb{B}}
to the composite
\textstyle{\mathbb{A}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{F}}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{H}}$$\textstyle{\mathbb{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{G}}$$\textstyle{\mathbb{B}^{\prime}.}
The horizontal 1-cell part sends a horizontal pseudotransformation
\textstyle{\mathsf{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathsf{H}^{\prime}}
to the horizontal pseudotransformation with the components
[TABLE]
for any horizontal 1-cell and vertical 1-cell in .
Symmetrically, the vertical 1-cell part sends a vertical pseudotransformation to the vertical pseudotransformation with the components
[TABLE]
for any horizontal 1-cell and vertical 1-cell in .
Finally, the 2-cell part sends a modification in the first diagram to the modification with components in the second diagram:
[TABLE]
1.4. The extranatural transformation
In this section we construct an extranatural transformation
[TABLE]
where in the top row denotes the unique functor to the terminal category, and in the left column the depicted symmetry natural isomorphism — that is, the appropriate flip map — occurs. We denote by lower indices that is ordinary natural in the first two arguments, and an upper index reminds us that it is extranatural in the last two factors. At any object of the form , it is given by the following double functor .
The 0-cell part sends a double functor to the double functor of Section 1.3.
The horizontal 1-cell part sends a horizontal pseudotransformation
\textstyle{\mathsf{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathsf{G^{\prime}}}
to the horizontal pseudotransformation with the following components, for any horizontal pseudotransformation and vertical pseudotransformation between double functors .
- •
The horizontal pseudotransformation with components
[TABLE]
for any horizontal 1-cell and vertical 1-cell in .
- •
The modification with components
[TABLE]
for any 0-cell in .
- •
The vertically invertible modification with components
[TABLE]
for any 0-cell in .
Symmetrically, the vertical 1-cell part sends a vertical pseudotransformation from to to the vertical pseudotransformation from to with the following components, for any horizontal pseudotransformation and vertical pseudotransformation between double functors .
- •
The vertical pseudotransformation with components
[TABLE]
for any horizontal 1-cell and vertical 1-cell in .
- •
The modification with components
[TABLE]
for any 0-cell in .
- •
The horizontally invertible modification with components
[TABLE]
for any 0-cell in .
The 2-cell part sends a modification in the first diagram to the modification with components in the second diagram, for any 0-cell in :
[TABLE]
For any double categories , , and , direct computation verifies the commutativity of
[TABLE]
and the equality of
[TABLE]
to the identity double functor, where is the double functor sending the single object of to the identity double functor .
1.5. The extranatural transformation
In this section we construct another extranatural transformation
[TABLE]
where again, in the top row denotes the unique functor to the terminal category, and in the left column the depicted symmetry natural isomorphism — that is, the appropriate flip map — occurs. As in Section 1.4, we denote by lower indices that is ordinary natural in the first two arguments, and an upper index reminds us that it is extranatural in the last two factors. At any object of the form , it is given by the following double functor .
The 0-cell part sends a double functor to the double functor in Section 1.3.
The horizontal 1-cell part sends a horizontal pseudotransformation \textstyle{\mathsf{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathsf{F^{\prime}}}
to the horizontal pseudotransformation with the following components, for any horizontal pseudotransformation and vertical pseudotransformation between double functors .
- •
The horizontal pseudotransformation with components
[TABLE]
for horizontal 1-cells and vertical 1-cells in .
- •
The modification with the components
[TABLE]
for any 0-cell in .
- •
The vertically invertible modification with the components
[TABLE]
for any 0-cell in .
Symmetrically, the vertical 1-cell part sends a vertical pseudotransformation from to to the vertical pseudotransformation to with the following components, for any horizontal pseudotransformation and vertical pseudotransformation between double functors .
- •
The vertical pseudotransformation with components
[TABLE]
for horizontal 1-cells and vertical 1-cells in .
- •
The modification with the components
[TABLE]
for any 0-cell in .
- •
The horizontally invertible modification with the components
[TABLE]
for any 0-cell in .
The 2-cell part sends a modification in the first diagram to the modification with components in the second diagram, for any 0-cell in :
[TABLE]
Direct computation verifies the commutativity of
[TABLE]
and the equality of
[TABLE]
to the identity double functor, for any double categories , and . It follows from these properties that the double functors
[TABLE]
constitute an idempotent natural transformation
[TABLE]
for any double category . This is natural in its lower indices by the naturality of . It is natural in the upper index as well (here the upper index no longer refers to extranaturality). In order to see that, both naturality and extranaturality of are needed.
The extranatural transformation in this section and in Section 1.4 together render commutative the following diagram, for any double categories , and .
[TABLE]
1.6. Representability of the functor
In this section we investigate the functor in the title, for the double category of Section 1.1.
By the description of in Section 1.1, for any double category the double functors correspond bijectively to the 2-cells of . The 0-cells at the corners of such a 2-cell are double functors denoted as , for all 0-cells of . The top and bottom horizontal 1-cells are horizontal pseudotransformations labelled by the horizontal 1-cells in . We denote their components by
[TABLE]
for any horizontal 1-cell and vertical 1-cell in . They satisfy the naturality condition
[TABLE]
Symmetrically, the left and right vertical 1-cells are vertical pseudotransformations labelled by the vertical 1-cells of ; with components denoted by
[TABLE]
for any horizontal 1-cell and vertical 1-cell in . They satisfy the naturality condition
[TABLE]
Finally, the 2-cell itself is a modification with components denoted by
[TABLE]
for all 0-cells of . They satisfy the horizontal compatibility conditions
[TABLE]
for all horizontal 1-cells in ; and the vertical compatibility conditions
[TABLE]
for all vertical 1-cells in .
From all that we can read off that the functor is represented by the following double category.
- •
The 0-cells are pairs of 0-cells in .
- •
There are two kinds of non-identity horizontal 1-cells and ; both are ordered pairs of a 0-cell , and a horizontal 1-cell in .
- •
Symmetrically, there are two kinds of non-identity vertical 1-cells and ; both are ordered pairs of a 0-cell , and a vertical 1-cell in .
- •
There are non-identity 2-cells of ordered pairs
and , for all 0-cells in
- -
and for horizontal 1-cells and vertical 1-cells in
- -
vertically invertible 2-cells for horizontal 1-cells in
- -
horizontally invertible 2-cells for vertical 1-cells in .
All further cells are generated by their compositions modulo the associativity and unitality conditions, the middle four interchange law, and the identities (1.5), (1.6), (1.7) and (1.8).
1.7. Representability of the functors
In this section we investigate the functors in the title, for any double categories and .
Consider a functor between locally presentable categories. Bourke and Gurski’s [3, Lemma 3.9] says that the functor is representable for all objects — that is, possesses a left adjoint — if and only if is representable for all in a strong generator of .
The category is locally presentable by [7, Theorem 4.1]. The double category in Section 1.6 is a strong generator of ; see Section 1.1. So from the representability result of Section 1.6 we conclude by Bourke and Gurski’s lemma that the functor is representable for any double category .
The 0-cell part of the iso double functor (1.3) yields a bijection for any double categories and ; which is natural in by the extranaturality and naturality of . Therefore also the functor is representable for any double category . Applying again Bourke and Gurski’s lemma, we obtain the representability of for any double categories and .
In other words, the functor possesses a left adjoint for any double category which we denote by . By the functoriality in , it gives raise to a double functor . For any double categories and , an explicit description of can be given analogously to Section 1.6.
2. Coherence
This section is devoted to the proof that the double functor of Section 1.7 renders a symmetric monoidal category (which is then closed with the internal hom functors , for all double categories ).
In what follows, the unit of the adjunction will be denoted by , and the counit will be denoted by , for all double categories and .
2.1. The associativity natural isomorphism
For any double category , consider the natural transformation
[TABLE]
with the components
[TABLE]
at any double categories and . It is natural in and by the naturality of and (the upper index of no longer refers to extranaturality). It is natural in as well which follows by the extranaturality of and together with the naturality of .
The 0-cell parts of the iso double functors in (1.3) yield bijections in the columns of the commutative diagram
[TABLE]
Since all of the occurring maps but the top row are known to be bijections, we conclude that so is the top row. Whence by Yoneda’s lemma is a natural isomorphism. Using the adjunction isomorphisms in the first and last steps, we obtain a natural isomorphism
[TABLE]
By Yoneda’s lemma again, it determines a natural isomorphism which is our candidate associativity natural isomorphism.
2.2. The pentagon condition
By Yoneda’s lemma, Mac Lane’s pentagon condition on the natural isomorphism of Section 2.1 is equivalent to the commutativity of the exterior of the diagram of Figure 1; hence also to the commutativity of the diagram of Figure 2.
The left column in Figure 2 is equal to . The triangles marked by commute by the naturality of and a triangle condition on the adjunction , yielding the commutative diagram
[TABLE]
for any double categories , and . The region marked by commutes by (1.1) and extranaturality of , yielding the commutative diagram
[TABLE]
for any double categories , , and .
2.3. The unitality natural isomorphisms
For any double categories and there are natural isomorphisms
[TABLE]
where denotes the 0-cell part of the iso double functor (1.3). By Yoneda’s lemma, they induce respective natural isomorphisms and with the components
[TABLE]
at any double category (where is the double functor which sends the single object of to the identity double functor ). They are our candidate unitality natural isomorphisms.
2.4. The triangle conditions
By Yoneda’s lemma, Mac Lane’s triangle condition on the natural isomorphisms of Section 2.1 and of Section 2.3 is equivalent to the commutativity of the exterior of Figure 3 for any double categories , and ; hence also to the commutativity of
[TABLE]
whose left-bottom path is (1.2); that is, the canonical (usually omitted) isomorphism.
2.5. The symmetry
The natural isomorphism
[TABLE]
constructed from the 0-cell part of in (1.3) induces a natural isomorphism with the components
[TABLE]
at any double categories and . It is our candidate symmetry.
2.6. The hexagon condition
By Yoneda’s lemma, the hexagon condition on the natural isomorphisms of Section 2.1 and of Section 2.5 is equivalent to the commutativity of the exterior of Figure 4 for any double categories , , and . Hence it follows by the commutativity of the diagram of Figure 5 whose left-bottom path is equal to . In order to see that the left column of Figure 5 is equal to , apply twice (1.4) to obtain the commutative diagram
[TABLE]
for any double categories , and . The region of Figure 5 marked by commutes by the extranaturality of .
3. Examples
Although our notions of (horizontal and vertical) pseudotransformations and of corresponding modification in Section 1.2 may look quite natural, admittedly no higher principle fixes their choice. Therefore there is no a’priori good resulting Gray monoidal product of double categories. In this final section we support our construction by relating it to existing structures. Namely, we verify the monoidality of some well-known functors between our monoidal category and some other monoidal categories which occur in the literature quite frequently.
3.1. Monoidal functors between closed monoidal categories
In any closed monoidal category we may take the mate
[TABLE]
of the associativity isomorphism under the adjunctions for being the objects , and ; and its mate
[TABLE]
Consider now a functor between closed monoidal categories. Some natural transformation (for both monoidal products denoted by ) and a morphism (for both monoidal units denoted by ) render monoidal if and only if the mate
[TABLE]
of under the adjunctions and makes the following diagrams commute.
The left and right unitality conditions translate to the commutativity of the respective diagrams in
[TABLE]
for all objects of the domain category. The associativity condition translates to the commutativity of
[TABLE]
for all objects , and of the domain category.
3.2. The closed monoidal category
For the closed monoidal category of Section 1 and Section 2,
[TABLE]
is equal to ; that is, the double functor sending the single object of to the identity double functor for any double category . The double functor
[TABLE]
is the canonical isomorphism for any double category . The double functor of (3.1) is equal to that in Section 1.4.
3.3. Monoids in
Monoidal 2-categories can be defined at different levels of generality. The most restrictive one in the literature is a monoid in the category of 2-categories and 2-functors with respect to the Cartesian monoidal structure. This is known as a strict monoidal 2-category. The most general one is a single object tricategory [8]; known as a monoidal bicategory. In between them are the so-called Gray monoids; these are again monoids in the category of 2-categories and 2-functors, but in this case with respect to the Gray monoidal structure [13]. Their importance stems from the coherence result of [8], proving that any monoidal bicategory is equivalent to a Gray monoid (as a tricategory).
Analogously, a strict monoidal double category [4] is a monoid in the category of double categories and double functors with respect to the Cartesian monoidal structure. In [16, 12] it was generalized to a pseudomonoid in the 2-category of (pseudo) double categories and pseudo double functors and, say, vertical transformations. However, no double category analogues of Gray monoids and of monoidal bicategories seem to be available in the literature. While the considerations in this paper do not promise any insight how to define most general monoidal (pseudo) double categories, monoids in are natural candidates for the double category analogue of Gray monoid. In this section we give their explicit characterization, similar to the characterization of Gray monoids in [1, Lemma 4].
A monoid in is equivalently a monoidal functor from the terminal double category (with the trivial monoidal structure) to . It can be described in terms of the data in Section 3.1. Namely, a monoid structure on a double category translates to double functors and which render commutative the diagrams of (3.2) and (3.3). Spelling out the details, this amounts to the following data.
- •
A distinguished 0-cell .
- •
For any 0-cells and , a 0-cell .
- •
For any 0-cell and any horizontal 1-cell on the left below, horizontal 1-cells on the right:
[TABLE]
- •
For any 0-cell and any vertical 1-cell on the left below, vertical 1-cells on the right:
[TABLE]
- •
For any 0-cell and any 2-cell on the left below, 2-cells on the right:
[TABLE]
- •
For any horizontal 1-cell and any vertical 1-cell , 2-cells
[TABLE]
- •
For any horizontal 1-cells and , a vertically invertible 2-cell
[TABLE]
- •
For any vertical 1-cells and , a horizontally invertible 2-cell
[TABLE]
One can memorize this as the rule that a pair of an dimensional and an dimensional cell is sent by the operation to an dimensional cell. These data are subject to the following conditions.
- (i)
For any 0-cell , and are double functors . 2. (ii)
. 3. (iii)
For any 0-cells and , the following equalities of double functors hold.
[TABLE] 4. (iv)
For any 0-cell , any horizontal 1-cells and any vertical 1-cells , the following equalities of 2-cells hold.
\begin{array}[]{lll}h\ast(X\ast q)=(h\ast X)\ast q&(X\ast h)\ast q=X\ast(h\ast q)&h\ast(q\ast X)=(h\ast q)\ast X\\ v\ast(X\ast p)=(v\ast X)\ast p&(X\ast v)\ast p=X\ast(v\ast p)&v\ast(p\ast X)=(v\ast p)\ast X\\ h\ast(X\ast p)=(h\ast X)\ast p&(X\ast h)\ast p=X\ast(h\ast p)&h\ast(p\ast X)=(h\ast p)\ast X\\ v\ast(X\ast q)=(v\ast X)\ast q&(X\ast v)\ast q=X\ast(v\ast q)&v\ast(q\ast X)=(v\ast q)\ast X\end{array} 5. (v)
For any 0-cell , we denote by the horizontal identity 1-cell; and by the vertical identity 1-cell on . For any horizontal 1-cell and any vertical 1-cell , the following equalities of 2-cells hold.
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0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-3.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 5.05914pt\raise-35.43054pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces}}}}\ignorespaces}=h\ast 1_{X} and v\ast 1_{X}=\raisebox{17.0pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 19.49281pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-19.49281pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern 12.5pt\raise-15.0pt\hbox{\hbox{\kern 0.0pt\raise-3.22223pt\hbox{}}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 30.0pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.0pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces}}}}\ignorespaces}=v\ast 1^{X}
1^{X}\ast h=\raisebox{17.0pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 5.05914pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern 12.5pt\raise-15.0pt\hbox{\hbox{\kern 0.0pt\raise-3.22223pt\hbox{}}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-3.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 5.05914pt\raise-35.43054pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces}}}}\ignorespaces}=1_{X}\ast h and 1_{X}\ast v=\raisebox{17.0pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 19.49281pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-19.49281pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern 12.5pt\raise-15.0pt\hbox{\hbox{\kern 0.0pt\raise-3.22223pt\hbox{}}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 30.0pt\raise-15.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.0pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 27.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{}}}}}}}\ignorespaces}}}}\ignorespaces}=1^{X}\ast v 6. (vi)
We denote the vertical composition (of vertical 1-cells and of 2-cells) by an upper dot, and the horizontal composition (of horizontal 1-cells and of 2-cells) by a lower dot. For any composable pairs of horizontal 1-cells and and for any composable pairs of vertical 1-cells and , the following hold.
h\ast(q^{\prime}\raisebox{3.0pt}{\cdot}q)=(h\ast q^{\prime})\raisebox{3.0pt}{\cdot}(h\ast q) and
and (v^{\prime}\raisebox{3.0pt}{\cdot}v)\ast p=(v^{\prime}\ast p)\raisebox{3.0pt}{\cdot}(v\ast p)
and are equal to the respective 2-cells
[TABLE]
v\ast(q^{\prime}\raisebox{3.0pt}{\cdot}q) and (v^{\prime}\raisebox{3.0pt}{\cdot}v)\ast q are equal to the respective 2-cells
[TABLE] 7. (vii)
For every 2-cell , horizontal 1-cell and vertical 1-cell , the following naturality conditions hold.
[TABLE]
[TABLE]
[TABLE]
[TABLE]
In part (vii) we see the naturality conditions on the horizontal and vertical pseudotransformations obtained as the images of 1-cells under the double functor ; and the compatibility conditions on the modifications obtained as the images of 2-cells under . The conditions in (i-v-vi) come from two sources: from the requirements that , and the image of any 0-cell under it, are double functors; and from the functoriality conditions on the horizontal and vertical pseudotransformations obtained as the images of 1-cells under . Condition (i) expresses the unitality, and (iii-iv) express the associativity of the monoid.
There seems to be no evident way to interpret a monoid in like above as a monoidal double category in the sense of [16, Definition 2.9] and [12, Section 3.1]. Using the notation of this section, one can easily introduce multiplication maps on the various cells. They are associative with the unit :
- •
A pair of 0-cells and is sent to .
- •
A pair of horizontal 1-cells and is sent to the horizontal 1-cell on the left; and a pair of vertical 1-cells and is sent to the vertical 1-cell on the right:
[TABLE]
- •
A pair of 2-cells and is sent to
[TABLE]
However, these maps do not constitute a double functor or at least a pseudo double functor in the sense of [10, Section 2.1], [16, Definition 2.7]. Recall that a pseudo double functor in these references is defined to strictly preserve the composition in one direction; and up-to a coherent natural family of invertible 2-cells in the other direction. These maps above, however, do not preserve any of the horizontal and vertical compositions in the strict sense, only up-to coherent natural families
[TABLE]
of vertically, respectively, horizontally invertible 2-cells.
Succinctly, monoids in determine monoids in the Cartesian monoidal category whose objects are double categories and whose morphisms are pseudo-pseudo double functors (rather than strict–pseudo double functors in [10]). By this reason, there seems to be no easy way to regard a monoid in in this section as a suitably degenerate intercategory [11, 12].
3.4. Monoidality of the functor due to Fiore, Gambino and Kock [6]
To any double category , the double category of monads in was associated in [6]. This construction can be seen as the object map of the functor in the title, which sends a morphism; that is, a double functor to the double functor of ‘componentwise’ action.
As the nullary part of the candidate monoidal structure, we take the evident iso double functor
\textstyle{\mathbbm{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{M}\mathsf{nd}(\mathbbm{1})} . For any double categories and , for the double functors encoding the binary part, we propose the following.
A 0-cell in is by definition a monad ((\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 7.22223pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{\scriptstyle{\mathsf{T}}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 18.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 18.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\mathbb{B}}}}}}}}}\ignorespaces}}}}\ignorespaces,\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.61111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-6.61111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{\mathsf{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 8.34723pt\raise 5.15279pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.15279pt\hbox{\scriptstyle{t}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 18.61111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 18.61111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\mathsf{T}}}}}}}}}\ignorespaces}}}}\ignorespaces),\theta,\tau) in the horizontal 2-category of . We have to associate to it a 0-cell in ; that is, a double functor .
Evaluation at any 0-cell of gives a 0-cell in . The image of any 0-cell of under the double functor is a 0-cell in . Between these monads in the horizontal 2-category of , there is a distributive law . It induces a 0-cell in ,
[TABLE]
It will be the image of the 0-cell of under the double functor . On the horizontal 1-cells acts as
[TABLE]
and on the vertical 1-cells it acts as
[TABLE]
Finally, sends a 2-cell in to the 2-cell of .
A horizontal 1-cell in is a monad morphism
[TABLE]
in the horizontal 2-category of (so that in particular is a horizontal pseudotransformation and is a modification). The double functor should send it to a horizontal 1-cell in ; that is, the following horizontal pseudotransformation .
It consists of the horizontal 1-cell in
[TABLE]
for all 0-cells in together with the 2-cell in for all vertical 1-cells in , and the vertically invertible 2-cell in for all horizontal 1-cells in .
A vertical 1-cell
[TABLE]
of is sent by to the vertical pseudotransformation from to which consists of the following data.
For all 0-cells of the vertical 1-cell in
[TABLE]
for all horizontal 1-cells of the 2-cell in ; and for all vertical 1-cells of the horizontally invertible 2-cell in .
The double functor sends a 2-cell in to the modification whose component at every 0-cell of is .
These double functors constitute a natural transformation . It is not hard (although a bit long) to see that the double functors of Section 3.2, the trivial isomorphism and the double functors constructed above, satisfy the conditions of (3.2) and . This proves the monoidality of the functor in the title of the section (which sends then monoids as in Section 3.3 to monoids in the same sense).
3.5. The closed monoidal category
Recall that for any double category , the internal hom functor of the closed monoidal category in the title sends an object; that is, a double category to the following double subcategory of .
- •
The 0-cells are still the double functors .
- •
The horizontal 1-cells are the horizontal transformations of [9]. That is, those horizontal pseudotransformations (see Section 1.2) whose 2-cell parts are vertical identity 2-cells for all horizontal 1-cells in .
- •
Symmetrically, the vertical 1-cells are the vertical transformations of [9]. That is, those vertical pseudotransformations (see Section 1.2) whose 2-cell parts are horizontal identity 2-cells for all vertical 1-cells in .
- •
Finally, the 2-cells are the modifications of [9] (this is the same notion as in Section 1.2).
The functor sends a morphism; that is, a double functor to the restriction of the double functor in Section 1.3.
For any double category , the double functor
[TABLE]
is , sending the single object of to the identity double functor ; and the double functor
[TABLE]
is the canonical isomorphism .
For any double categories , and , the double functor of (3.1) is constituted by the following maps.
- •
It sends a 0-cell; that is, a double functor to the double functor .
- •
It sends a horizontal 1-cell; that is, a horizontal transformation
\textstyle{\mathsf{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathsf{G}}
to the horizontal transformation whose component at any vertical transformation on the left — between double functors — is the modification on the right:
[TABLE]
- •
Symmetrically, it sends a vertical 1-cell; that is, a vertical transformation on the left — between double functors — to the vertical transformation whose component at any horizontal transformation
\textstyle{\mathsf{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathsf{H^{\prime}}} — between double functors — is the modification on the right:
[TABLE]
- •
Finally, it sends a modification on the left to the modification whose component at a double functor is the modification on the right:
[TABLE]
3.6. Monoidality of the identity functor
The evident inclusion double functors , for all double categories and , define a natural transformation . Together with the double functors in Section 3.2 and those in Section 3.5, and the identity double functor
\textstyle{\mathbbm{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbbm{1}}
as the nullary part of the stated monoidal structure, they clearly satisfy the conditions in (3.2) and (3.3). With this we infer the monoidality of the functor in the title of the section.
In particular, a strict monoidal double category [4] – which is a monoid in – gives rise to a monoid in – described in Section 3.3.
3.7. The closed monoidal category
In this section we regard the category of 2-categories and 2-functors as a closed monoidal category via the Gray monoidal product in [13]. Recall that, for any 2-category , the internal hom functor sends an object; that is, a 2-category to the 2-category of 2-functors , pseudonatural transformations, and modifications. It sends a morphism; that is, a 2-functor to the 2-functor given by postcomposition with .
The 2-functor
[TABLE]
is , the 2-functor sending the single object of the terminal 2-category
1
to the identity 2-functor .
The 2-functor
[TABLE]
is the canonical isomorphism.
For any 2-categories , and , the 2-functor in (3.1) has the following maps.
- •
It sends a 0-cell; that is, a 2-functor to the 2-functor .
- •
It sends a 1-cell; that is, a pseudonatural transformation to the pseudonatural transformation whose component at any pseudonatural transformation between 2-functors is
[TABLE]
- •
It sends a 2-cell; that is, a modification to the modification whose component at any 2-functor is .
3.8. Monoidality of the functors sending double categories to their horizontal – or vertical – 2-categories
Consider the functor which sends a double category to its so-called horizontal 2-category. The 0-cells of are the 0-cells of , the 1-cells of are the horizontal 1-cells of and the 2-cells of are those 2-cells of which are surrounded by identity vertical 1-cells (and arbitrary horizontal 1-cells). Compositions in are inherited from . The functor sends a morphism; that is, a double functor to the 2-functor which acts on the various cells as does.
The horizontal 2-category of the terminal double category is the terminal 2-category
1
. So we may choose the nullary part of the candidate monoidal structure on to be the identity 2-functor \scalebox{1.15}{\textit{1}}\to\scalebox{1.15}{\textit{1}}. As the 2-functor for any double categories and , encoding the binary part, we propose the following.
- •
A 0-cell; that is, a double functor is sent to the 2-functor .
- •
A 1-cell; that is, a horizontal pseudotransformation is sent to the pseudonatural transformation whose component at any 1-cell of — that is, horizontal 1-cell of — is the 2-cell of .
- •
A 2-cell; that is, a modification of the form
\textstyle{\mathsf{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{\displaystyle\omega}$$\textstyle{\mathsf{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z}$$\textstyle{\mathsf{G}} , is sent to the modification whose component at any 0-cell of — that is, 0-cell of — is the 2-cell of .
The so defined 2-functors constitute a natural transformation and satisfy the conditions in (3.2) and (3.3). Hence there is a corresponding monoidal structure on . In particular, applying to a monoid in as in Section 3.3, we obtain a monoid in (known as a Gray monoid [8]).
Symmetric considerations verify monoidality of the functor , sending a double category to its vertical 2-category.
3.9. Monoidality of the functor due to Ehresmann
Ehresmann’s square- or quintet construction [5] sends a 2-category to the following double category . The 0-cells of are the 0-cells of . Both the horizontal and the vertical 1-cells of are the 1-cells of . A 2-cell of with boundaries
\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{l}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{D}
is a 2-cell of . For any 2-functor there is a double functor which acts on the various cells as does.
Applying the so defined functor to the terminal 2-category
1
, we obtain the terminal double category . So as the nullary part of the candidate monoidal structure on , we may choose the identity double functor . For any 2-categories and , for the double functor encoding the binary part, the following choices can be made.
- •
A 0-cell; that is, a 2-functor is sent to the double functor .
- •
A horizontal 1-cell; that is, a pseudonatural transformation is sent to the horizontal pseudotransformation whose components at any 1-cell of are the 2-cells in
[TABLE]
- •
A vertical 1-cell; which is again a pseudonatural transformation is sent to the vertical pseudotransformation whose components at any 1-cell of are the 2-cells in
[TABLE]
- •
A 2-cell; that is, a modification of 2-functors on the left, is sent to the modification of double functors whose component at any 0-cell of — that is, any 0-cell of — is the 2-cell of on the right:
[TABLE]
The resulting double functors constitute a natural transformation and satisfy the conditions in (3.2) and (3.3). So they render monoidal the functor in the title of the section. In particular, applying the functor to a Gray monoid — that is, a monoid in — a monoid in is obtained (which may not be a monoidal bicategory in the sense of [16, Definition 2.9]).
Appendix A Diagrams
Large size diagrams, used in the earlier sections, are collected in this appendix, after the bibliography.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] John C. Baez and Martin Neuchl, Higher-dimensional algebra. I. Braided monoidal 2-categories, Advances in Mathematics 121 no. 2 (1996) 196–244.
- 2[2] Francis Borceux, Handbook of Categorical Algebra: Volume 1, Basic category theory , Cambridge University Press, 1994.
- 3[3] John Bourke and Nick Gurski, The Gray Tensor Product Via Factorisation, Applied Categorical Structutes 25 no. 4 (2017) 603–624.
- 4[4] Roberto Bruni, José Meseguer and Ugo Montanari, Symmetric monoidal and cartesian double categories as a semantic framework for tile logic, Mathematical Structures in Computer Science 12 no. 1 (2002) 53–90.
- 5[5] Charles Ehresmann, Catégories structurées III. Quintettes et applications covariantes, in: “Séminaire Ehresmann. Topologie et géométrie différentielle” 5 (1963) 1–21.
- 6[6] Thomas M. Fiore, Nicola Gambino and Joachim Kock, Monads in double categories, Journal of Pure and Applied Algebra 215 no. 6 (2011) 1174–1197.
- 7[7] Thomas M. Fiore, Simona Paoli and Dorette Pronk, Model structures on the category of small double categories, Algebraic & Geometric Topology 8 (2008) 1855–1959.
- 8[8] Robert Gordon, Anthony John Power and Ross Street, Coherence for tricategories, Memoirs of the American Mathematical Society 117 no. 558 (1995).