Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 17, 2005, 61 – 148

Per K. Jakobsen, Valentin V. Lychagin
QUANTIZATIONS IN A CATEGORY OF RELATIONS

Abstract. In this paper we develops a categorical theory of relations and use this formulation to deﬁne the notion of quantization for relations. Categories of relations are deﬁned in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of $A - A$ bicomodules. Properties of relations are deﬁned in terms of the symmetric monoidal structure. Equivalence relations are shown to be commutative monoids in the category of relations. Quantization in our view is a property of functors between monoidal categories. This notion of quantization induce a deformation of all algebraic structures in the category, in particular the ones deﬁning properties of relations like transitivity and symmetry.

Contents
1.  Introduction
2.  Categorical framework
   2.1.  Symmetric monoidal categories
   2.2.  Symmetries and group action
   2.3.

σ

-commutative comonoids in symmetric monoidal categories
   2.4.  C-categories and M-categories
3.  Categorical theory of relations
   3.1.  Relations
   3.2.  Categories of relations
   3.3.  Relations in terms of

A - A

bicomodules.
3.4. The

⊠^{A}

product of relations
   3.5.  Semimonoidal structures on the category of relations
   3.6.  The tensor product of relations
   3.7.  Monoidal structures on the category of relations
   3.8.  Symmetries for the category of relations
   3.9.  Commutative monoids in the category of relation
4.  Quantization of relations
   4.1.  Quantized functors
   4.2.  Quantization of algebraic structures
References

1. Introduction

The concept of quantization is somewhat mysterious and rather ill deﬁned. It ﬁrst appeared in a rudimentary form in the work of Max Planck [12] . Its role there was as a purely technical device to solve a problem central to the physics of radiation at the time, the so called ultraviolet catastrophe for the blackbody radiation spectrum. Planck’s original idea was shortly thereafter used by Einstein to explain the photoelectric effect [5] and was further developed by N. Bohr into what we today call the Old Quantum Theory. This theory explained with greater precision than ever before the position of the spectral lines for the hydrogen atom. The theory was however rather ad hoc and it was difficult to generalize the theory to more complicated atomic systems. The next step forward was introduced by Louise De Broglie [2], [3],[4]. He generalized the already well known wave-particle duality for light to matter and postulated that electrons conﬁned to an atom would display wavelike properties. The idea of wave-particle duality inspired E. Schrødinger in 1926 to write down a wave equation for matter waves. A different view on the notion of quantization was introduced by Heisenberg [6][14] in 1925 through his matrix mechanics. These two approaches was soon shown to be equivalent. From a modern point of view the difference in the two approaches lies in Schrødingers use of the Hamiltonian formulation of classical mechanics and of Heisenbergs use of a formulation of classical mechanics in terms of Poisson brackets. Schrødinger’s approach gave rise to the canonical quantization procedure. This procedure has been applied successfully to many systems but contain ambiguities, like variable ordering, and has invariance problems. The method of Geometric Quantization [7] was introduced in order to resolve these problems. Heisenbergs approach to quantization although equivalent to Schrødingers approach at an elementary level, has a distinctly more algebraic ﬂavor than the wave mechanics of Schrødinger. Here the structure of a physical system is represented in terms of an algebra of observables. Representations of this algebra of observables are possible models of the system in question. Whereas algebras derived from a classical description of the system are commutative, the algebras representing quantized systems are in general noncommutative although still associative. Deformation quantization [1],[13] is a collection of tools and methods that have been developed in order to ﬁnd quantized version of classical systems by deforming the algebraic description of the system within some class of algebras. What is clear from the existence of all these different approaches is that the notion of quantization is not well deﬁned. The various approaches agree for simple systems, but they have different domains of applicability and even for a single approach several possible quantizations are possible for a given system. What are the properties, or constraints, a system need in order for the notion of quantization to be applicable? Is quantization one thing or several different things? What is the relation between constraints and quantizations? These are just some of the questions that comes to mind. This paper will not give a deﬁnite answer to any of these questions but will introduce a mathematical framework that emphasize the idea that quantization is something that depends on constraints and that these constraints may not belong to the domain of mechanics or not even to physics. In fact we believe that quantization has its natural description in terms of a theory of representation for constraints. We also believe that at the present time the only mathematical framework with the right kind of generality for the formulation of a representation theory of constraints is Category Theory [8]. Constraints will in this framework take the form of relations between natural transformations and a representation of the constraints will be a category that supports all given functors and natural transformation with the assumed relations. Quantizations will be related to morphisms in the category of possible representations of a given set of constraints. What we describe here is of course a lot of bones with very little ﬂesh. The goal of this paper is to put a little more ﬂesh on the bones. This we will do by developing a theory for the quantization of relations along the lines described above. This theory illustrate our view of quantization, but is also of independent interest since it gives a framework for the quantization of logic and machines as described in the classical theory of computing. In these days when the whole domain of classical computing is in the process of being quantized a wider point of view on the process of quantization is certainly needed. The categorical approach to quantization has been introduced by one of the authors in [9],[10],[11].

2. Categorical framework

In this ﬁrst chapter we formulate the basic categorical machinery that we need in order to categorize the notion of relation. In the ﬁrst subsection we introduce the notion of a semimonoidal and a monoidal category. In line with our general ideas of constraints and representations both notions are deﬁned entirely in terms of functors and natural transformations. This leads to a slightly more general notion of monoidal category than the usual one although we does not pursue this here. Symmetries for monoidal categories is introduced as a further set of constraints on monoidal categories. A certain derived relation for the natural transformations deﬁning a symmetric monoidal category is described and shown to be equivalent to the usual Yang-Baxter equation. This new formulation of the Yang-Baxter equation is essential when we later in this paper introduce a generalization of the usual notion of symmetry that we need in order to formulate commutativity in the context of relations. We lay the groundwork for this generalization by showing how the Yang-Baxter equation is intimately connected to an action by a certain $S_{2}$ -graded group. In the last subsection in this part of the paper we introduce the notion of M-categories and C-categories. These categories have exactly the constraints needed in order to formulate and develop a theory of relations.

2.1. Symmetric monoidal categories. A semimonoidal category is a category that has a product that is associative up to a natural isomorphism. A semimonoidal category is a monoidal category if there is an object that is a unit for the product up to a natural isomorphism. Properties of categories are most clearly expressed in terms of functors and natural transformations. We now review this formulation. On any category we have deﬁned the identity functor $1_{C}$ . Let us assume that there also is a bifunctor $\otimes :$ $C \times C \to C$ deﬁned on $C$ .

Deﬁnition 1. A semimonoidal category is a triple $〈 C, \otimes, α 〉$ where $C$ is a category, $\otimes :$ $C \times$ $C \to$ $C$ is a bifunctors,

α : \otimes \circ (1_{C} \times \otimes) \to \otimes \circ (\otimes \times 1_{C})

is a natural isomorphism and where the following relation holds

\begin{array}{l} (α \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes}) & = (1_{\otimes} \circ (α \times 1_{1_{C}})) \\ \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α)) . \end{array}

A semimonoidal category is strict if $\otimes \circ (1_{C} \times \otimes) = \otimes \circ (\otimes \times 1_{C})$ and $α = 1_{\otimes \circ (1_{C} \times \otimes)}$ . The relation on $α$ given in the previous deﬁnition is the object-free formulation of the usual MacLane coherence condition for the associativity constraint $α$ .

For any category $C$ we have deﬁned two bifunctors $P : C$ $\times C$ $\to C$ and $Q : C$ $\times C$ $\to C$ . These are the projection on the ﬁrst and second factor, $P (X, Y) = X$ and $Q (X, Y) = Y$ with obvious extension to arrows. Let $e$ be a ﬁxed object in the category $C$ and deﬁne a constant functor $K_{e} : C$ $\to C$ by $K_{e} (X) = e$ and $K_{e} (f) = 1_{e}$ . Using these functors we can give a deﬁnition of a monoidal category entirely in terms of functors and natural transformations.

Deﬁnition 2. A monoidal category is a 6-tuple $〈 C, \otimes, K_{e}, α, β, γ 〉$ such that $〈 C, \otimes, α 〉$ is a semimonoidal category and where

\begin{array}{l} β & : \otimes \circ (K_{e} \times 1_{C}) \to Q, \\ γ & : \otimes \circ (1_{C} \times K_{e}) \to P, \end{array}

are natural isomorphisms such that the following relations holds

\begin{array}{l} (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) & = (1_{\otimes} \circ (1_{1_{C}} \times β)), \\ β \circ 1_{1_{C} \times K_{e}} & = γ \circ 1_{K_{e} \times 1_{C}} . \end{array}

A monoidal category is strict if $〈 C, \otimes, α 〉$ is a strict semimonoidal category and if $\otimes \circ (K_{e} \times 1_{C}) = Q$ , $\otimes \circ (1_{C} \times K_{e}) = P$ and $β = 1_{Q}$ , $γ = 1_{P}$ .

Note that $〈 C, P, 1_{P \circ (1_{C,} \times P)} 〉$ and $〈 C, Q, 1_{Q \circ (1_{C,} \times Q)} 〉$ both are strict semimonoidal categories. None of them can be made into a monoidal category by selecting a unit $e$ . However if $\otimes$ is part of a monoidal structure on $C$ then we can reduce the product to projections by ﬁxing the ﬁrst and second argument to be the unit object.

Our deﬁnition in fact deviate somewhat from the standard formulation in terms of objects. Recall that a monoidal category in the usual sense is a 6-tuple $〈 C, \otimes, e, α^{'}, β^{'}, γ^{'} 〉$ where $α_{X, Y, Z}^{'} : X \otimes (Y \otimes Z) \to (X \otimes Y) \otimes Z$ , $β_{X}^{'} : e \otimes X \to X$ and $γ_{X}^{'} : X \otimes e \to X$ are isomorphisms in $C$ that are natural in $X, Y$ , and $Z$ and where the following MacLane Coherence [8] conditions are satisﬁed

$X⊗(e⊗Y )-α′X,e,Y- (X⊗ e)⊗Y \ / 1X ⊗β′Y\\ / /γ′X ⊗ 1Y A ⊗ B$

It is easy to see that if we deﬁne

\begin{array}{l} β_{X, Y} & = β_{Y}^{'}, \\ γ_{X, Y} & = γ_{X}^{'}, \\ α_{X, Y, Z} & = α_{X, Y, Z}^{'} . \end{array}

for all objects $X$ and $Y$ in $C$ , then $〈 \otimes, K_{e}, α, β, γ 〉$ is a monoidal category as deﬁned in 2. If we assume that $C$ is a category such that for all pairs of objects there exists at least one arrow $f : X \to X^{'}$ . Then $K_{e} (f) = 1_{e}$ and naturality of $β$ implies the commutativity of the following diagram

βX,Y
e⊗|Y -------Y|
| |
1e⊗1Y | |1Y
| |
e⊗ Y -------Y
βX′,Y

We thus get $β_{X, Y} = β_{X^{'}, Y}$ . In a similar way we ﬁnd $γ_{X, Y} = γ_{X, Y^{'}}$ . This gives us a monoidal category in the usual sense if we deﬁne $β_{X}^{'} = β_{Y, X}$ and $γ_{X}^{'} = γ_{X, Y}$ . Our aim in this paper is not to investigate generalizations of the notion of a monoidal category and we will therefore assume that solutions to the relations in 2 satisfy $β_{X, Y} = β_{X^{'}, Y}$ and $γ_{X, Y} = γ_{X, Y^{'}}$ .

We will need to express categorically the process of changing order in a product with several factors. For any category $C$ we have the transposition functor $τ : C \times C \to C \times C$ deﬁned by $τ (X, Y) = (Y, X)$ and $τ (f, g) = (g, f)$ . A symmetry for a monoidal category is expressed using the functor $τ$ .

Deﬁnition 3. A symmetric monoidal category is a 7-tuple $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ such that $〈 C, \otimes, K_{e}, α, β, γ 〉$ is a monoidal category and where

σ : \otimes \to \otimes \circ τ

is a natural isomorphism such that the following relations holds

\begin{array}{l} σ \circ 1_{\otimes \times 1_{C}} & = (α^{- 1} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}})) \\ \cdot (α \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \cdot α^{- 1}, \\ σ \circ 1_{1_{C} \times \otimes} & = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{τ \times 1_{C}}) \\ \cdot (α^{- 1} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}})) \cdot α, \\ β & = (γ \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times 1_{C}}), \\ γ & = (β \circ 1_{τ}) \cdot (σ \circ 1_{1_{C} \times K_{e}}), \\ σ \circ 1_{τ} & = σ^{- 1} . \end{array}

A symmetric monoidal category is strict if the underlying monoidal category $〈 C, \otimes, K_{e}, α, β, γ 〉$ is strict.

The conditions in the deﬁnition are not independent.

Proposition 4. Let $〈 C, \otimes, K_{e}, α, β, γ 〉$ be a monoidal category and let $σ : \otimes \to \otimes \circ τ$ be a natural isomorphism such that $σ \circ 1_{τ} = σ^{- 1}$ . Then the following two conditions are equivalent

\begin{array}{l} σ \circ 1_{\otimes \times 1_{C}} & = (α^{- 1} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}})) \\ \cdot (α \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \cdot α^{- 1}, \\ σ \circ 1_{1_{C} \times \otimes} & = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{τ \times 1_{C}}) \\ \cdot (α^{- 1} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}})) \cdot α . \end{array}

Proof. We have the following relations $τ \circ τ = 1_{C \times C}$ and $τ \circ (1_{C} \times \otimes) = (\otimes \times 1_{C}) \circ (1_{C} \times τ) \circ (τ \times 1_{C})$ . Using these functorial relations we have

\begin{array}{l} σ \circ 1_{1_{C} \times \otimes} \\ = σ \circ 1_{τ} \circ 1_{τ} \circ 1_{1_{C} \times \otimes} \\ = σ \circ 1_{τ} \circ 1_{\otimes \times 1_{C}} \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \\ = {(σ \circ 1_{\otimes \times 1_{C}})}^{- 1} \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C} .} \end{array}

We thus have a relations between $σ \circ 1_{1_{C} \times \otimes}$ and $σ \circ 1_{\otimes \times 1_{C}}$ . The equivalence of the two conditions stated in the proposition follows directly from this relation. □

The third and fourth relations are also equivalent

Proposition 5. Let $〈 C, \otimes, K_{e}, α, β, γ 〉$ be a monoidal category and let $σ : \otimes \to \otimes \circ τ$ be a natural isomorphism such that $σ \circ 1_{τ} = σ^{- 1}$ . Then the following two conditions are equivalent

\begin{array}{l} β & = (γ \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times 1_{C}}), \\ γ & = (β \circ 1_{τ}) \cdot (σ \circ 1_{1_{C} \times K_{e}}) . \end{array}

Proof. Let the ﬁrst condition be given. Then we have

\begin{array}{l} β \circ 1_{τ} \\ = ((γ \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times 1_{C}})) \circ 1_{τ} \\ = (γ \circ 1_{τ} \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times 1_{C}} \circ 1_{τ}) \\ = γ \cdot (σ \circ 1_{τ} \circ 1_{1_{C} \times K_{e}}) \\ = γ \cdot (σ^{- 1} \circ 1_{1_{C} \times K_{e}}) . \end{array}

and this is equivalent to the last condition. □

The symmetry conditions have a consequence that will play an important role.

Proposition 6. Let $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ be a symmetric monoidal category. Then the following equation holds

(α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (σ \circ (σ \times 1_{1_{C}})) \cdot α = σ \circ (1_{1_{C}} \times σ) .

Proof. We have

\begin{array}{l} σ \circ (1_{1_{C}} \times σ) \\ = (σ \circ (1_{1_{C}} \times 1_{\otimes \circ τ})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \\ = (σ \circ 1_{1_{C} \times \otimes} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \\ = ((α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times τ}) \cdot (α^{- 1} \circ 1_{τ \times 1_{C}}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}})) \cdot α) \circ 1_{1_{C} \times τ} \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \\ = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (α^{- 1} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (σ \times 1_{1_{C}}) \circ 1_{1_{C} \times τ}) \cdot (α \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ)) \\ = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (σ \circ 1_{\otimes \times 1_{C}}) \cdot α \\ = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (1_{\otimes \circ τ} \circ (σ \times 1_{1_{C}})) \cdot (σ \circ 1_{\otimes \times 1_{C}}) \cdot α \\ = (α \circ 1_{1_{C} \times τ} \circ 1_{τ \times 1_{C}} \circ 1_{1_{C} \times τ}) \cdot (σ \circ (σ \times 1_{1_{C}})) \cdot α . \end{array}

□

If we introduce the expressions for $σ \circ 1_{\otimes \times 1_{C}}$ and $σ \circ 1_{1_{C} \times \otimes}$ into the equation from the previous proposition we get an equation that is cubic in $σ$ . This equation is the well known Yang-Baxter equation. In terms of object it takes in the strict case the following form

\begin{array}{l} (1_{Z} \otimes σ_{X, Y}) \circ (σ_{X, Z} \otimes 1_{Y}) \circ (1_{X} \otimes σ_{Y, Z}) \\ = (σ_{Y, Z} \otimes 1_{X}) \circ (1_{Y} \otimes σ_{X, Z}) \circ (σ_{X, Y} \otimes 1_{Z}) . \end{array}

The equation from the previous proposition is clearly equivalent to the Yang-Baxter equation in a symmetric monoidal category. We will call this equation also for the Yang-Baxter equation. A certain generalization of this equation will play a fundamental role in our theory of relations. This generalization is based on characterization of symmetries in terms of a group action.

2.2. Symmetries and group action. Let $S_{2}$ be the group of permutation of two elements with the single generator given by $t$ . Let $τ : C \times C \to C \times C$ be the transposition bifunctor. The functors $T_{1} = 1_{C}$ , $T_{2} = τ$ and $T_{3} = (1_{C} \times τ) \circ (τ \times 1_{C}) \circ (1_{C} \times τ)$ deﬁnes action of the group $S_{2}$ on the categories $C, C^{2} = C \times C$ and $C^{3} = C \times C \times C$ . Let $[C^{2}, C]$ and $[C^{3}, C]$ be the category of bifunctors and trifunctors on $C$ with natural transformations as arrows. We can induce an action of $S_{2}$ on the functor categories $[C^{2}, C]$ and $[C^{3}, C]$ in the usual way by deﬁning for objects $F$ and arrows $α$ in $[C^{i}, C], i = 2, 3$

\begin{array}{l} t F & = F \circ T_{i}, \\ t a & = α \circ 1_{T_{i}} . \end{array}

It is easy to see that this really deﬁnes an action of $S_{2}$ . Let us ﬁrst consider the case when $C$ is a semimonoidal category with product $\otimes$ and associativity constraint $α$ . Note that

\begin{array}{l} t (\otimes \circ (1_{C} \times \otimes)) \\ = \otimes \circ (1_{C} \times \otimes) \circ (τ \times 1_{C}) \circ (1_{C} \times τ) \circ (τ \times 1_{C}) \\ = t \otimes \circ (\otimes \times 1_{C}) \circ (τ \times 1_{C}) \\ = t \otimes \circ (t \otimes \times 1_{C}) . \end{array}

In a similar way we ﬁnd that $t (\otimes \circ (\otimes \times 1_{C})) = t \otimes \circ (1_{C} \times t \otimes)$ . We have here used the fact that $(1_{C} \times τ) \circ (τ \times 1_{C}) \circ (1_{C} \times τ) = (τ \times 1_{C}) \circ (1_{C} \times τ) \circ (τ \times 1_{C})$ . We therefore have a natural isomorphism

t α^{- 1} : t \otimes \circ (1_{C} \times t \otimes) \to t \otimes \circ (t \otimes \times 1_{C}) .

This is in fact an associativity constraint as the next proposition show

Proposition 7. $〈 C, t \otimes, t α^{- 1} 〉$ is a semimonoidal category

Proof. Let $g = (1_{C} \times τ \times 1_{C}) \circ (τ \times τ) \circ (1_{C} \times τ \times 1_{C}) \circ (τ \times τ)$ . Then we have

\begin{array}{l} (t α^{- 1} \circ 1_{t \otimes \times 1_{C} \times 1_{C}}) \cdot (t α^{- 1} \circ 1_{1_{C} \times 1_{C} \times t \otimes}) \\ = (α^{- 1} \circ 1_{T_{3}} \circ 1_{t \otimes \times 1_{C} \times 1_{C}}) \cdot (α^{- 1} \circ 1_{T_{3}} \circ 1_{1_{C} \times 1_{C} \times t \otimes}) \\ = (α^{- 1} \circ 1_{t \otimes \times 1_{C} \times 1_{C}} \circ 1_{g}) \cdot (α^{- 1} \circ 1_{1_{C} \times 1_{C} \times t \otimes} \circ 1_{g}) \\ = {[(α \circ 1_{1_{C} \times 1_{C} \times t \otimes} \circ 1_{g}) \cdot (α \circ 1_{t \otimes \times 1_{C} \times 1_{C}} \circ 1_{g})]}^{- 1} \\ = {[((α \circ 1_{1_{C} \times 1_{C} \times t \otimes}) \cdot (α \circ 1_{t \otimes \times 1_{C} \times 1_{C}})) \circ 1_{g}]}^{- 1} \\ = {[((1_{\otimes} \circ (α \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α))) \circ 1_{g}]}^{- 1} \\ = ((1_{\otimes} \circ (1_{1_{C}} \times α^{- 1})) \cdot (α^{- 1} \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (α^{- 1} \times 1_{1_{C}}))) \circ 1_{g} \\ = (1_{\otimes} \circ (1_{1_{C}} \times α^{- 1}) \circ 1_{g}) \cdot (α^{- 1} \circ 1_{1_{C} \times \otimes \times 1_{C}} \circ 1_{g}) \cdot (1_{\otimes} \circ (α^{- 1} \times 1_{1_{C}}) \circ 1_{g}) \\ = (1_{t \otimes} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (t α^{- 1} \circ 1_{1_{C} \times t \otimes \times 1_{C}}) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times t α^{- 1})) . \end{array}

□

Let us assume that there exists a natural isomorphism $σ : \otimes \to t \otimes$ and let $α$ be an associativity constraint for a semimonoidal category $〈 C, \otimes, α 〉$ . Then $t α^{- 1} : t \otimes \circ (1_{C} \times t \otimes) \to t \otimes \circ (t \otimes \times 1_{C})$ is an associativity constraint for a semimonoidal category $〈 C, \otimes, t α^{- 1} 〉$ . On the other hand we have natural isomorphisms

\begin{array}{l} σ \circ (1_{1_{C}} \times σ) & : t \otimes \circ (1_{C} \times t \otimes) \to \otimes \circ (1_{C} \times \otimes), \\ σ \circ (σ \times 1_{1_{C}}) & : t \otimes \circ (t \otimes \times 1_{C}) \to \otimes \circ (\otimes \times 1_{C}) . \end{array}

We therefore have a natural isomorphism $\hat{α} : \otimes \circ (1_{C} \times \otimes) \to \otimes \circ (\otimes \times 1_{C})$ where we have deﬁned

\hat{α} = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \cdot t α^{- 1} \cdot (σ \circ (1_{1_{C}} \times σ)) .

This new isomorphism also an associativity constraint.

Proposition 8. $〈 C, \otimes, \hat{α} 〉$ is a semimonoidal category.

Proof. We only need to show that the MacLane coherence condition hold for $\hat{α}$ . Let us ﬁrst observe that

\begin{array}{l} (σ \circ (1_{1_{C}} \times σ) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (σ \circ (1_{1_{C}} \times σ) \circ (1_{\otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \\ \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{\otimes})) \\ = (σ \circ ((1_{1_{C}} \circ 1_{\otimes}) \times (σ \circ (1_{1_{C}} \times 1_{1_{C}})))) \cdot (σ^{- 1} \circ ((σ^{- 1} \circ (1_{1_{C}} \times 1_{1_{C}})) \times (1_{1_{C}} \circ 1_{\otimes}))) \\ = (σ \circ (1_{\otimes} \times σ)) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{\otimes})) \\ = (1_{t \otimes} \circ (σ^{- 1} \times σ)) \\ = (1_{t \otimes} \circ (σ^{- 1} \times 1_{t \otimes})) \cdot (1_{t \otimes} \circ (1_{t \otimes} \times σ)) \\ = (1_{t \otimes} \circ ((1_{1_{C}} \circ σ^{- 1}) \times (1_{t \otimes} \circ (1_{1_{C}} \times 1_{1_{C}})))) \cdot (1_{t \otimes} \circ ((1_{t \otimes} \circ (1_{1_{C}} \times 1_{1_{C}})) \times (1_{1_{C}} \circ σ))) \\ = (1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) . \end{array}

Let $g = (1_{C} \times τ \times 1_{C}) \circ (τ \times τ) \circ (1_{C} \times τ \times 1_{C}) \circ (τ \times τ)$ . Using the previous identity we have for the left hand side of the coherence condition

\begin{array}{l} (\hat{α} \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \circ (\hat{α} \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (t α^{- 1} \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times 1_{C} \times \otimes}) (t α^{- 1} \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (t α^{- 1} \circ (1_{\otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{\otimes})) \cdot (σ \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (1_{\otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{\otimes})) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (α^{- 1} \circ 1_{T_{3}} \circ (1_{t \otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (α^{- 1} \circ 1_{T_{3}} \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{t \otimes})) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (α^{- 1} \circ 1_{1_{C} \times 1_{C} \times \otimes} \circ 1_{g}) \cdot (α^{- 1} \circ 1_{\otimes \times 1_{C} \times 1_{C}} \circ 1_{g}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot ({[(α \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes})]}^{- 1} \circ 1_{g}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot ({[(1 t \circ (α \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α))]}^{- 1} \circ 1_{g}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α^{- 1}) \circ 1_{g}) \cdot (α^{- 1} \circ 1_{1_{C} \times \otimes \times 1_{C}} \circ 1_{g}) \cdot (1_{\otimes} \circ (α^{- 1} \times 1_{1_{C}}) \circ 1_{g}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) . \end{array}

For evaluating the right-hand side of the MacLane condition we need the two identities

\begin{array}{l} (1_{\otimes} \circ ((σ \circ (1_{1_{C}} \times σ)) \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times \otimes \times 1_{C}}) \\ = (1_{\otimes} \circ (σ \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \\ = (σ^{- 1} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times σ \times 1_{1_{C}})) \\ = (σ^{- 1} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) . \\ \cdot (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \end{array}

and

\begin{array}{l} (σ \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})))) \\ = (σ \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \\ \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ^{- 1}) \circ (1_{1_{C}} \times σ^{- 1} \times 1_{1_{C}})) \\ = (σ \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times σ^{- 1} \times 1_{1_{C}})) \\ = (1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times σ^{- 1} \times 1_{1_{C}})) \cdot (σ \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) . \end{array}

Using these identities we have for the right-hand side of the MacLane condition

\begin{array}{l} (1_{\otimes} \circ (\hat{α} \times 1_{1_{C}})) \cdot (\hat{α} \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \times (1_{1_{C}} \times \hat{α})) \\ = (1_{\otimes} \circ ([(σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \cdot t α^{- 1} \cdot (σ \circ (1_{1_{C}} \times σ))] \times 1_{1_{C}})) \\ \cdot ([(σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \cdot t α^{- 1} \cdot (σ \circ (1_{1_{C}} \times σ))] \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times [(σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \cdot t α^{- 1} \cdot (σ \circ (1_{1_{C}} \times σ))])) \\ = (1_{\otimes} \circ (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}}) \cdot (1_{\otimes} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((σ \circ (1_{1_{C}} \times σ)) \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (t α^{- 1} \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (σ \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})))) \cdot (1_{\otimes} \circ (1_{1_{C}} \times t α^{- 1})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times (σ \circ (1_{1_{C}} \times σ)))) \\ = (1_{\otimes} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times σ \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times σ^{- 1} \times 1_{1_{C}})) \cdot (σ \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times t α^{- 1})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (1_{\otimes} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (σ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (σ^{- 1} \circ ((1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes})) \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times σ \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times 1_{t \otimes}) \circ (1_{1_{C}} \times σ^{- 1} \times 1_{1_{C}})) \cdot (σ \circ (1_{1_{C}} \times (1_{t \otimes} \circ (1_{t \otimes} \times 1_{1_{C}})))) \cdot (1_{\otimes} \circ (1_{1_{C}} \times t α^{- 1})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ (1_{1_{C}} \times 1_{1_{C}} \times σ)) \\ = (1_{\otimes} \circ ((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) \cdot (σ \circ (1_{1_{C}} \times t α^{- 1})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times (σ \circ (1_{1_{C}} \times σ)))) \\ = (σ^{- 1} \circ ((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (t α^{- 1} \times 1_{1_{C}})) \cdot (t α^{- 1} \circ (1_{1_{C}} \times 1_{t \otimes} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times t α^{- 1})) \cdot (σ \circ (1_{1_{C}} \times (σ \circ (1_{1_{C}} \times σ)))) \\ = (σ^{- 1} \circ ((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α^{- 1}) \circ 1_{g}) \cdot (α^{- 1} \circ 1_{1_{C} \times \otimes \times 1_{C}} \circ 1_{g}) \cdot (1_{\otimes} \circ (α^{- 1} \times 1_{1_{C}}) \circ 1_{g}) \cdot (σ \circ (1_{1_{C}} \times (σ \circ (1_{1_{C}} \times σ)))) . \end{array}

The left-hand side and the right-hand side are thus equal and this proves the proposition. □

Let us deﬁne $S_{C, \otimes} = {α ∣ 〈 C, \otimes, α 〉 is a semimonoidal category}$ . Then the previous proposition show that for each natural isomorphism $σ : \otimes \to t \otimes$ we have a mapping of $S_{C, \otimes}$ to itself.

Let us next consider the case of a monoidal category $〈 C, \otimes, K_{e}, α, β, γ 〉$ . Using the natural isomorphism $σ$ we can deﬁne new natural isomorphisms

\hat{β} = (t γ) \cdot (σ \circ 1_{K_{e} \times 1_{C}}) : \otimes \circ (K_{e} \times 1_{C}) \to Q,

\hat{γ} = (t β) \cdot (σ \circ 1_{1_{C} \times K_{e}}) : \otimes \circ (1_{C} \times K_{e}) \to P .

For $\hat{α}$ and the two natural isomorphisms $\hat{β}$ and $\hat{γ}$ we have

Proposition 9. $〈 C, \otimes, K_{e}, \hat{α}, \hat{β}, \hat{γ} 〉$ is a monoidal category

Proof. The First MacLane coherence condition has already been veriﬁed. For the second MacLane condition we need the identities

\begin{array}{l} (1_{\otimes} \circ ((σ \circ 1_{1_{C} \times K_{e}}) \times 1_{1_{C}})) \cdot ((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (σ \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}})) \\ = (σ^{- 1} \circ (1_{t \otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}})) \\ = (σ^{- 1} \circ (1_{t \otimes} \circ (1_{1_{C}} \times 1_{K_{e}}) \times 1_{1_{C}})) \end{array}

and

\begin{array}{l} (1_{t \otimes} \circ (t β \times 1_{1_{C}})) \cdot (t α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (1_{1_{C}} \times β) \circ 1_{T_{3}}) \cdot (α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}} \circ 1_{T_{3}}) \\ = (((1_{\otimes} \circ (1_{1_{C}} \times β)) \cdot (α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}})) \circ 1_{T_{3}}) \\ = (1_{\otimes} \circ (γ \times 1_{1_{C}}) \circ 1_{T_{3}}) . \end{array}

Using these two identities we have

\begin{array}{l} (1_{\otimes} \circ (\hat{γ} \times 1_{1_{C}})) \cdot (\hat{α} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (1_{\otimes} \circ ((t β \cdot (σ \circ 1_{1_{C} \times K_{e}})) \times 1_{1_{C}})) \cdot (((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \cdot t α^{- 1} \cdot (σ \circ (1_{1_{C}} \times σ))) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (1_{\otimes} \circ (t β \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((σ \circ 1_{1_{C} \times K_{e}}) \times 1_{1_{C}})) \cdot ((σ^{- 1} \circ (σ^{- 1} \times 1_{1_{C}})) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (t α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot ((σ \circ (1_{1_{C}} \times σ)) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (1_{\otimes} \circ (t β \times 1_{1_{C}})) \cdot (σ^{- 1} \circ (1_{t \otimes} \circ (1_{1_{C}} \times 1_{K_{e}}) \times 1_{1_{C}})) \cdot (t α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot ((σ \circ (1_{1_{C}} \times σ)) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (σ^{- 1} \circ (1_{P} \times 1_{1_{C}})) \cdot (1_{t \otimes} \circ (t β \times 1_{1_{C}})) \cdot (t α^{- 1} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot ((σ \circ (1_{1_{C}} \times σ)) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (σ^{- 1} \circ (1_{P} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}}) \circ 1_{T_{3}}) \cdot ((σ \circ (1_{1_{C}} \times σ)) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (σ^{- 1} \circ 1_{P \times 1_{C}}) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times t γ)) \cdot (σ \circ (1_{1_{C}} \times σ) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \end{array}

\begin{array}{l} = (σ^{- 1} \circ 1_{1_{C} \times Q}) \cdot (1_{t \otimes} \circ (1_{1_{C}} \times t γ)) \cdot (σ \circ (1_{1_{C}} \times (σ \circ (1_{K_{e}} \times 1_{1_{C}})))) \\ = (1_{\otimes} \circ (1_{1_{C}} \times [t γ \cdot (σ \circ 1_{K_{e} \times 1_{C}})])) \\ = (1_{\otimes} \circ (1_{1_{C}} \times \hat{β})) . \end{array}

For the last MacLane condition we have

\begin{array}{l} \hat{β} \circ 1_{1_{C} \times K_{e}} \\ = (t γ \circ 1_{1_{C} \times K_{e}}) \cdot (σ \circ 1_{K_{e} \times 1_{C}} \circ 1_{1_{C} \times K_{e}}) \\ = (γ \circ 1_{τ} \circ 1_{1_{C} \times K_{e}}) \cdot (σ \circ 1_{K_{e} \times K_{e}}) \\ = (γ \circ 1_{K_{e} \times 1_{C}} \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times K_{e}}) \\ = (β \circ 1_{1_{C} \times K_{e}} \circ 1_{τ}) \cdot (σ \circ 1_{K_{e} \times K_{e}}) \\ = (t β \circ 1_{K_{e} \times 1_{C}}) \cdot (σ \circ 1_{1_{C} \times K_{e}} \circ 1_{K_{e} \times 1_{C}}) \\ = (t β \cdot (σ \circ 1_{1_{C} \times K_{e}})) \circ 1_{K_{e} \times 1_{C}} \\ = \hat{γ} \circ 1_{K_{e} \times 1_{C}} . \end{array}

□

Let $M_{C, \otimes, e} = {(α, β, γ) ∣ 〈 C, \otimes, K_{e}, α, β, γ 〉 is a monoidal category}$ . Then the previous proposition show that for each natural isomorphism $σ : \otimes \to t \otimes$ we have a map

T_{t} (σ) : M_{C, \otimes, e} \to M_{C, \otimes, e}

deﬁned by $T_{t} (σ) (α, β, γ) = (\hat{α}, \hat{β}, \hat{γ})$ . Let us next for each $ρ : \otimes \to \otimes$ deﬁne a map on elements in $M_{C, \otimes, e}$

T_{1} (ρ) (α, β, γ) = (\tilde{α}, \tilde{β}, \tilde{γ}),

where we have

\begin{array}{l} \tilde{α} & = (ρ^{- 1} \circ (ρ^{- 1} \times 1_{1_{C}})) \cdot α \cdot (ρ \circ (1_{1_{C}} \times ρ)), \\ \tilde{β} & = β \cdot (ρ \circ 1_{K_{e} \times 1_{C}}), \\ \tilde{γ} & = γ \cdot (ρ \circ 1_{1_{C} \times K_{e}}) . \end{array}

For this map we have

Proposition 10. $T_{1} (ρ) : M_{C, \otimes, e} \to M_{C, \otimes, e} .$

The proof of this proposition is similar to the one for the map $T_{1} (σ)$ and is not reproduced here.

Let

G_{C, \otimes, e} = {T_{t} (σ), T_{1} (ρ) ∣ σ : \otimes \to t \otimes, ρ : \otimes \to \otimes, σ, ρ natural isomorphisms} .

From the construction it is evident that all maps in $G_{C, \otimes, e}$ are bijections. The next proposition show that $G_{C, \otimes, e}$ is closed under composition of maps.

Proposition 11. Let $σ_{1}, σ_{2} : \otimes \to t \otimes$ and $ρ_{1}, ρ_{2} : \otimes \to \otimes$ be natural isomorphisms. Then we have

\begin{array}{l} T_{t} (σ_{2}) \circ T_{t} (σ_{1}) & = T_{1} (t σ_{1} \cdot σ_{2}), \\ T_{t} (σ_{1}) \circ T_{1} (ρ_{1}) & = T_{t} (ρ_{1} \cdot t σ_{1}), \\ T_{1} (ρ_{1}) \circ T_{t} (σ_{1}) & = T_{t} (σ_{1} \cdot ρ_{1}), \\ T_{1} (ρ_{2}) \circ T_{1} (ρ_{1}) & = T_{1} (ρ_{1} \cdot ρ_{2}) . \end{array}

The proof of this proposition is routine and is left out. The set $G_{C, \otimes, e}$ is thus closed under composition and contains the identity map $T_{1} (1_{\otimes}) = 1_{M_{C, \otimes, e}}$ .All maps in the set $G_{C, \otimes, e}$ are invertible by construction and $G_{C, \otimes, e}$ is closed under the operation of taking the inverse of a map. We have

\begin{array}{l} T_{1} (ρ) \circ T_{1} (ρ^{- 1}) & = 1_{M_{C, \otimes, e}}, \\ T_{t} (σ) \circ T_{t} ({(t σ)}^{- 1}) & = 1_{M_{C, \otimes, e}} . \end{array}

The previous propositions can now be restated in the following way.

Corollary 12. The set $M_{C, \otimes, e}$ of monoidal structures on $C$ corresponding to a ﬁxed product $\otimes$ and unit $e$ is invariant under the action of the $S_{2}$ -graded group $G_{C, \otimes, e}$ .

We can use the $S_{2}$ -graded group $G_{C, \otimes, e}$ to give an interpretation of the notion of a symmetric monoidal category.

Proposition 13. Let $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ be a symmetric monoidal category. Then $H = {T_{t} (σ), 1_{\otimes}}$ is a $S_{2}$ graded subgroup of $G_{C, \otimes, e}$ and $(α, β, γ) \in M_{C, \otimes, e}$ is a ﬁxed-point for the action of $H$ .

This gives an interpretation of the Yang-Baxter equation and the two unit conditions in terms of invariance with respect to the action by the group $H$ . No such interpretation appears to be possible for the ﬁrst two conditions from the deﬁnition 3, of a symmetry. These two conditions appear to be of a technical nature.

2.3. $σ$ -commutative comonoids in symmetric monoidal categories. Recall that a comonoid in a monoidal category is a triple $〈 A, δ_{A}, ε_{A} 〉$ where $A$ is an object in the category and $δ_{A} : A \to A \otimes A$ and $ε_{A} : A \to e$ are morphisms in the category such that the following diagrams commute

1 ⊗ δ δ
A⊗(A⊗ A) -A--A- A⊗ A A----A
|
αA,A,A| //
| /
| / / δA ⊗ 1A
|/
(A⊗ A)⊗ A

$e⊗A εA-⊗1A-A ⊗A 1A-⊗εA-A ⊗e | \\ |δA / / βA \ | / γA A$

The simpler structure $〈 A, δ_{A} 〉$ is called a cosemigroup. The morphism $ε_{A}$ is the counit for the comonoid and $δ_{A}$ is called the coproduct.

Before we proceed with formal developments we will ﬁrst consider some examples of these constructions. Let us ﬁrst consider the case of sets. The category $S e t s$ is a monoidal category with Cartesian product, $\times$ as bifunctor. The neutral object is the one point set $e = {*}$ . The associativity constraints $α_{A, B, C} : A \times (B \times C) \to (A \times B) \times C$ and unit constraints $β_{A} : e \otimes A \to A$ and $γ_{A} : A \otimes e \to A$ given by

\begin{array}{l} α_{A, B, C} (x, (y, z)) & = ((x, y), z), \\ β_{A} (*, x) & = x, \\ γ_{A} (x, *) & = x . \end{array}

Finite sets offer many examples of cosemigroups. Let $A = {a, b, c}$ and deﬁne a map $δ_{A} : A \to A \times A$ by

\begin{array}{l} δ_{A} (a) & = (a, a), \\ δ_{A} (b) & = (b, a), \\ δ_{A} (c) & = (a, c) . \end{array}

A direct calculation show that $〈 A, δ_{A} 〉$ is a cosemigroup. There is only one possible map $ε_{A} : A \to e$ since $e = {*}$ is terminal is $S e t s$ and this is the map $ε_{A} (x) = *$ for all $x \in A$ . But for this map we ﬁnd

[β_{A} \circ (ε_{A} \otimes 1_{A}) \circ δ_{A}] (b) = [β_{A} \circ (ε_{A} \otimes 1_{A})] (b, a) = β_{A} (*, a) = a,

so $〈 A, δ_{A}, ε_{A} 〉$ is not a comonoid.

Let $A$ be any set. Deﬁne the map $δ_{A} : A \to A \times A$ by

δ_{A} (x) = (x, x) .

This is the diagonal map in $S e t s$ . We then have

\begin{array}{l} [α_{A, A, A} \circ (1_{A} \times δ_{A}) \circ δ_{A}] (x) & = [α_{A, A, A} \circ (1_{A} \times δ_{A})] (x, x) = ((x, x), x), \\ [(δ_{A} \times 1_{A}) \circ δ_{A}] (x) & = (δ_{A} \times 1_{A}) (x, x) = ((x, x), x), \end{array}

so $〈 A, δ_{A} 〉$ is a cosemigroup. The only possible counit satisfy

\begin{array}{l} [β_{A} \circ (ε_{A} \otimes 1_{A}) \circ δ_{A}] (x) & = [β_{A} \circ (ε_{A} \otimes 1_{A})] (x, x) = β_{A} (*, x) = x, \\ [γ_{A} \circ (1_{A} \otimes ε_{A}) \circ δ_{A}] (x) & = [γ_{A} \circ (1_{A} \otimes ε_{A})] (x, x) = γ_{A} (x, *) = x, \end{array}

so $〈 A, δ_{A}, ε_{A} 〉$ is a comonoid. Let $δ_{A} : A \to A \times A$ , $ε_{A} : A \to {*}$ be any comonoid structure on $A$ . We have $δ_{A} (a) = (f (a), g (a))$ and $ε_{A} (a) = *$ . The ﬁrst counit condition $β_{A} \circ (ε_{A} \times 1_{A}) \circ δ_{A} = 1_{A}$ gives $g (a) = a$ for all $a$ . Similarly the second counit condition gives $f (a) = a$ for all $a$ . So the previous example in fact gives the only possible comonoid structure in this category. We will always assume that the objects in $S e t s$ are comonoid with this structure.

As our next example let us consider a pointed set. This is a set $A$ with a chosen point $x_{0} \in A$ . Deﬁne a map $δ_{A} : A \to A \times A$ by $δ_{A} (x) = (x_{0}, x)$ . Then we have

\begin{array}{l} [(1_{A} \times δ_{A}) \circ δ_{A}] (x) & = (1_{A} \times δ_{A}) (x_{0}, x) = (x_{0}, x_{0}, x), \\ [(δ_{A} \times 1_{A}) \circ δ_{A}] (x) & = (δ_{A} \times 1_{A}) (x_{0}, x) = (x_{0}, x_{0}, x), \end{array}

so $〈 A, δ_{A} 〉$ is a cosemigroup. It is not a comonoid because the only possible map $ε_{A} : A \to e$ gives

[γ_{A} \circ (1_{A} \otimes ε_{A}) \circ δ_{A}] (x) = [γ_{A} \circ (1_{A} \otimes ε_{A})] (x_{0}, x) = γ_{A} (x_{0}, *) = x_{0},

so if there are any elements in $A$ different from $x_{0}$ then $A$ is not a comonoid. This construction only gives a comonoid when $A = e$ . This fact is true for any monoidal category.

Let us next consider the category $V e c t_{k}$ . This is the category of vector spaces over a ﬁeld $k$ with morphisms given by linear maps. This category is monoidal with product bifunctor given by the tensor product of vector spaces $\otimes = \otimes_{k}$ . The neutral object is $k$ . The associativity constraint $α$ and unit constraints $β$ and $γ$ for this case are the linear maps $α_{A, B, C} : A \otimes (B \otimes C) \to (A \otimes B) \otimes C$ , $β_{A} : k \otimes A \to A$ and $γ_{A} : A \otimes k \to A$ given on generators by

\begin{array}{l} α_{A, B, C} (x \otimes (y \otimes z)) & = (x \otimes y) \otimes z, \\ β_{A} (r \otimes x) & = r x, \\ γ_{A} (x \otimes r) & = r x . \end{array}

Let $A$ be any ﬁnite dimensional vector space in $V e c t_{k}$ . Let $Ω$ be a ﬁnite index set and let ${a_{i}}_{i \in Ω}$ be a basis for $A$ indexed by $Ω$ . Then ${a_{i} \otimes a_{i^{'}}}_{i, i^{'} \in Ω}$ is a basis for $A \otimes A$ . Deﬁne a linear map $δ_{A} : A \to A \otimes A$ by

δ_{A} (a_{i}) = a_{i} \otimes a_{i} .

Then evidently $〈 A, δ_{A} 〉$ is a cosemigroup. Deﬁne a linear map $ε_{A} : A \to k$ on generators by $ε_{A} (a_{i}) = 1 \in k$ . Then we have

\begin{array}{l} [β_{A} \circ (ε_{A} \otimes 1_{A}) \circ δ_{A}] (a_{i}) & = [β_{A} \circ (ε_{A} \otimes 1_{A})] (a_{i}, a_{i}) = β_{A} (1 \otimes a_{i}) = a_{i}, \\ [γ_{A} \circ (1_{A} \otimes ε_{A}) \circ δ_{A}] (a_{i}) & = [γ_{A} \circ (1_{A} \otimes ε_{A})] (a_{i}, a_{i}) = γ_{A} (a_{i} \otimes 1) = a_{i}, \end{array}

so $〈 A, δ_{A}, ε_{A} 〉$ is a comonoid. In contrast to the case of $S e t s$ we can have many nonisomorphic comonoid structures on a given object in $V e c t_{k}$ . Let $δ_{A} : A \to A \otimes A$ and $ε_{A} : A \to k$ be linear maps. We have thus

\begin{array}{l} δ_{A} (a_{i}) & = \sum_{j, k} r_{j, k}^{i} a_{j} \otimes a_{k}, \\ ε_{A} (a_{i}) & = q_{i}, \end{array}

where all indices run from $1$ to $m$ , the dimension of $A$ .

Then $〈 A, δ_{A}, ε_{A} 〉$ is a comonoid if ${r_{j, k}^{i}}$ and ${q_{i}}$ are solutions of the following system of quadratic equations.

\begin{array}{l} \sum_{j} (r_{j, k}^{i} r_{l, n}^{j} - r_{l, j}^{i} r_{n, k}^{j}) & = 0 for all i, k, l, n, \\ \sum_{j} r_{j, k}^{i} q_{j} & = δ_{i, k} for all i, k, \\ \sum_{j} r_{k, j}^{i} q_{j} & = δ_{i, k} for all i, k . \end{array}

For $m = 2$ this system have four different families of solutions. One of these families is the following

\begin{array}{l} δ_{A} (a_{1}) & = a_{1} \otimes a_{1}, \\ δ_{A} (a_{2}) & = - x a_{1} \otimes a_{1} + a_{1} \otimes a_{2} + a_{2} \otimes a_{1}, \\ q_{A} (a_{1}) & = 1, \\ q_{A} (a_{2}) & = x, \end{array}

where $x$ is an arbitrary element of $k$ .

Let now $G$ be a ﬁnite group and let $A = ℱ (G)$ be the vector space of $k$ valued functions on $G$ .

Note that since $G$ is ﬁnite we have $ℱ (G \times G) \approx ℱ (G) \otimes_{k} ℱ (G)$ . Deﬁne a linear map $δ_{ℱ (G)} : ℱ (G) \to ℱ (G) \otimes_{k} ℱ (G)$ by

δ_{ℱ (G)} (f) (x, y) = f (x y) .

This clearly makes $ℱ (G)$ into a cosemigroup. The linear map $ε_{ℱ (G)} : ℱ (G) \to k$

ε_{ℱ (G)} (f) = f (e),

where $e \in G$ is the unit of the group $G$ , makes $〈 ℱ (G), δ_{ℱ (G)}, ε_{ℱ (G)} 〉$ into a comonoid. Note that this conclusion depends strongly on the identiﬁcation $ℱ (G \times G) \approx ℱ (G) \otimes_{k} ℱ (G)$ . For inﬁnite groups this relation does not hold in general but for some inﬁnite groups it does. For these cases we also get comonoids.

The tensor product is not the only monoidal structure on $V e c t_{k}$ . Let $\oplus$ be the direct sum of vector spaces. This is a monoidal structure with the neutral object given by the zero dimensional vector space $e = {0}$ . The maps $α, β$ and $γ$ are the standard identiﬁcations used for the direct sum. The symmetry is the linear map $σ (u, v) = (v, u)$ . These structures deﬁnes the structure of a symmetric monoidal category on $V e c t_{k}$ . A cosemigroup is a pair $〈 A, δ_{A} 〉$ with $δ_{A} : A \to A \oplus A$ a coassociative linear map. Any such map is determined by a pair of linear maps $f, g : A \to A$ through $δ_{A} (a) = (f (a), g (a))$ . The coassociativity gives the following conditions on the maps $f$ and $g$ .

\begin{array}{l} f \circ f & = f, \\ g \circ g & = g, \\ f \circ g & = g \circ f . \end{array}

So any pair of commuting projectors on $A$ deﬁne the structure of a cosemigroup on $A$ . There are thus in general many nontrivial cosemigroup structures on a linear space. The comonoid structure is however much more restrictive. This is because the neutral object for $\oplus$ is also the terminal object for the category. This means that there is only one possible counit for any comonoid. It is straight forward to see that the counit property for the only possible counit gives $f = g = 1_{A}$ . So there is only one comonoid structure on $A$ and this is the diagonal map

δ_{A} (a) = (a, a) .

In all the examples we have seen that coproduct for the comonoids have been monomorphisms. This is true in general

Proposition 14. Let $〈 B, δ_{B}, ε_{B} 〉$ be a comonoid. Then the coproduct is a monomorphism.

Proof. Let $D$ be any object in $C$ and let $ϕ, ψ : D \to B$ be two morphisms in $C$ such that $δ_{B} \circ ϕ = δ_{B} \circ ψ$ . Then we have

\begin{array}{l} ψ & = 1_{B} \circ ψ \\ = β_{B} \circ (ε_{B} \otimes 1_{B}) \circ δ_{B} \circ ψ \\ = β_{B} \circ (ε_{B} \otimes 1_{B}) \circ δ_{B} \circ ϕ \\ = 1_{B} \circ ϕ \\ = ϕ, \end{array}

so $δ_{B}$ is by deﬁnition mono. □

We will in general only be interested in comonoids where the coproduct has the additional property of being commutative. Only such comonoids carry enough structure to support a full theory of relations. We express this property by using the symmetry $σ$ .

Deﬁnition 15. A comonoid $〈 A, δ_{A}, ε_{A} 〉$ in a symmetric monoidal category is $σ$ -commutative if $σ_{A, A} \circ δ_{A} = δ_{A}$ .

2.4. C-categories and M-categories. In $S e t s$ each object is a $σ$ -commutative comonoid in one and only one way. For the case of a general symmetric monoidal category we have seen that objects may have several $σ$ -commutative comonoid structures deﬁned on them. We need to preserve the unique relation between objects and structures when we generalize from $S e t s$ . This relation is expressed in terms of functors and natural transformations. To any category $C$ we have associated a set of functors. These are the projection functors $P :$ $C \times C \to C$ and $Q : C \times C \to C$ ,the diagonal functor $Δ :$ $C \to C \times C$ deﬁned by $Δ (X) = (X, X)$ and the transposition functor $τ : C \times C \to C \times C$ . Let $e$ be a ﬁxed object in the category $C$ . To this object we associate the constant functor $K_{e} : C \to C$ . Finally let us assume that $\otimes :$ $C \times C \to C$ is a bifunctor and let $H = (1_{C} \times τ \times 1_{C}) \circ (Δ \times Δ)$ . We are now ready to deﬁne the notion of a C-category.

Deﬁnition 16. A C-category is a collection $〈 C, \otimes, K_{e}, α, β, γ, σ, δ, ε 〉$ where $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ is a symmetric monoidal category and where $δ, ε$ are natural transformations

\begin{array}{l} δ & : 1_{C} \to \otimes \circ Δ, \\ ε & : 1_{C} \to K_{e}, \end{array}

such that the following relations holds

\begin{array}{l} (1_{\otimes} \circ (δ \times 1_{1_{C}}) \circ 1_{Δ}) \cdot δ & = (α \circ 1_{(1_{C} \times Δ) \circ Δ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times σ) \circ 1_{Δ}) \cdot δ, \\ 1_{1_{C}} & = (β \circ 1_{Δ}) \cdot (1_{\otimes} \circ (ε \times 1_{1_{C}}) \circ 1_{Δ}) \cdot δ, \\ 1_{1_{C}} & = (γ \circ 1_{Δ}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times ε) \circ 1_{Δ}) \cdot δ, \\ δ & = (σ^{- 1} \circ 1_{Δ}) \cdot δ, \\ δ \circ 1_{\otimes} & = (α^{- 1} \circ 1_{\otimes \times 1_{C} \times 1_{C}} \circ 1_{H}) \cdot (1_{\otimes} \circ (α \times 1_{1_{C}}) \circ 1_{H}) \\ \cdot (1_{\otimes \circ (\otimes \times 1_{C})} \circ (1_{1_{C}} \times σ \times 1_{1_{C}}) \circ 1_{Δ \times Δ}) \\ \cdot (1_{\otimes} \circ (α^{- 1} \times 1_{1_{C}}) \circ 1_{Δ \times Δ}) \cdot (α \circ 1_{\otimes \times 1_{C} \times 1_{C}} \circ 1_{Δ \times Δ}) \\ \cdot (1_{\otimes} \circ (δ \times δ)), \\ ε \circ 1_{\otimes} & = (β \circ 1_{1_{C} \times K_{e}}) \cdot (1_{\otimes} \circ (ε \times ε)) . \end{array}

The four ﬁrst relations ensure that for each object in $C$ there is ﬁxed a unique commutative comonoid structure. The last two relations say that if an object $X$ can be decomposed as $X = A \otimes B$ , then we can express the unique comonoid structure on $X$ in terms of the comonoid structures on $A$ and $B$ . For the strict case they take the following form in terms of objects

\begin{array}{l} δ_{A \otimes B} & = (1_{A} \otimes σ_{A, B} \otimes 1_{B}) \circ (δ_{A} \otimes δ_{B}), \\ ε_{A \otimes B} & = ε_{A} \otimes ε_{B} . \end{array}

A M-category is the dual of a C-category. We get its deﬁning equations by reversing all arrows. It is a category where for each object there is ﬁxed a unique monoid structure and where the monoid structure on an object of the form $X = A \otimes B$ can be expressed in terms of the structures on $A$ and $B$ .

3. Categorical theory of relations

In this part of the paper we use the categorical framework described in the previous section to deﬁne a category of relations and develop its properties. We ﬁrst deﬁne the notion of a relation and a corelation in a C-category. In a similar way relations and corelations can be developed in a M-category. The notions of C-categories and M-categories are dual concepts so that any deﬁnitions made or propositions proved in one of them hold in a dualized version in the other. Since the notion of relation and corelation also are dual of each other it is clear that it is enough to develop the theory of relations in C-categories. The other cases follow by duality. We start this section by deﬁning relations on an object $A$ in a C-category $C$ in terms of arrows and collect such arrows into a category of relations $ℛ^{A} (C)$ . This category of relations is then shown to be isomorphic to the category $S^{A} (C)$ of $A - A$ bicomodules in $C$ . A semimonoidal structure $⊠^{A}$ is introduced in this category and by isomorphism into the category of relations. This semimonoidal structure is then used to introduce a bifunctor $\otimes^{A}$ on $S^{A} (C)$ and by isomorphism on $ℛ^{A} (C)$ . This bifunctor is used to introduce a monoidal structure on the category of relations. Certain properties of relations like transitivity and reﬂexivity are formulated in algebraic terms using the monoidal structure. In the ﬁnal sections a generalized notion of symmetry is introduced, this notion of symmetry use in an essential way the formulation of the Yang-Baxter equation in terms of action of a $S_{2}$ graded group. The new notion of symmetry is then used to further categorize properties of relations. Equivalence relations appears as commutative and associative algebras with units.

3.1. Relations. Let $〈 C, \otimes, k_{e}, α, β, γ, σ, δ, ε 〉$ be a C-category and let $A$ be an object in $C$ .

Deﬁnition 17. A relation on $A$ is an arrow in $C$ with codomain $A \otimes A$ .

Note that we will use the same symbol for an arrow in $C$ and the corresponding morphism of relations. Also note that a given arrow $f : B \to B^{'}$ in $C$ can give rise to more than one morphism of relations. This can happen because we might have $r_{1} = r_{1}^{'} \circ f$ and $r_{2} = r_{2}^{'} \circ f$ where $r_{1}, r_{2} : B \to A \otimes A$ and $r_{1}^{'}, r_{2}^{'} : B^{'} \to A \otimes A$ are two pairs of relations on $A$ . In this sense we can write $1_{r} = 1_{B}$ where $B$ is the domain of the arrow $r$ . Let us now consider a few examples of this construction.

Let us ﬁrst consider the case of $S e t s$ . This is a C-category with $δ_{X} (x) = (x, x)$ and $ε_{X} (x) = *$ for all objects $X$ $\in C$ . Let $A$ and $B$ be sets and let $r : B \to A \times A$ be a map of sets. We can write $r (b) = (f (b), g (b))$ . We have

\begin{array}{l} [(r \times r) \circ δ_{B}] (b) \\ = (f (b), f (b), g (b), g (b)) \\ = (δ_{A} \times δ_{A}) (f (b), g (b)) \\ = [δ_{A \times A} \circ r] (b), \end{array}

so $r$ is an arrow in the C-category $S e t s$ and is therefore a relation in $S e t s$ in our sense. A relation on $A$ in the usual sense is a subset of $A \times A$ . This is equivalent to assuming that the map $r$ is a monomorphism. In general the map $r$ assign to each element in $B$ a source and a target. Several elements in $B$ can be assigned the same source and target. In fact we observe that in $S e t s$ a relation in our sense is the same as a directed labelled graph.

Let us next consider the C-category $V e c t_{k}$ with direct sum as monoidal structure and $δ$ and $ε$ deﬁned as for $S e t s$ . A relation on a linear space $A$ is any linear map $r : B \to A \oplus A$ . Let $L : A \to A$ be an endomorphism on $A$ . Let $B = A$ and deﬁne $r : B \to A \oplus A$ by

r (a) = (a, L (a)) .

Then $r$ is a linear map and therefore deﬁnes a relation on $A$ in our sense. Note that the image of $A$ under $r$ is by deﬁnition the graph of the linear map $L$ . More generally, let $L$ be a linear subspace of $A \oplus A$ . Let $B = L$ and $r : B \to A \oplus A$ the inclusion map. Then $r$ is evidently a relation on $A$ . In general a relation on $A$ is like a graph, where the set of vertices and the set of labels have a vector space structure and the source and target maps respect these structures.

As with any categorical concept the notion of a relation has a dual.

Deﬁnition 18. A corelation on a $A$ is an arrow in $C$ with domain $A \otimes A$ .

Let $r : S \to Ω \times Ω$ be a relation on $Ω$ in $S e t s$ . We assume now that the sets $S$ and $Ω$ are ﬁnite. The algebraic description of the sets $S$ and $Ω$ are given by the space of $k$ valued functions $B = ℱ (S)$ on $S$ and $A = ℱ (Ω)$ on $Ω$ . Let $c$ deﬁned by $c (f \otimes g) (x) = (f \otimes g) (r (x))$ . Then $c : A \otimes A \to B$ is a linear map and by duality a morphism of the induced algebra structures on $B$ and $A \otimes A$ .

Therefore the algebraic image of the relation $r$ in $S e t s$ is a corelation $c$ in $V e c t_{k}$ . This example show that corelations arise naturally by algebraization of relations in $S e t s$ . Note that in general a corelation $c : A \otimes A \to B$ in $V e c t_{k}$ with the tensor product as monoidal structure is in algebra usually called a $A \otimes A$ algebra.

3.2. Categories of relations. Let $r : B \to A \otimes A$ and $r^{'} : B^{'} \to A \otimes A$ be two relations on $A$ . A morphism $f : r \to r^{'}$ is an arrow $f : B \to B^{'}$ in $C$ such that the following diagram commute

$f B--------------- B′ \ / \ / ′ r \ / r A⊗ A$

Let $ℛ^{A} (C)$ be the category of relations on $A$ . This is a category whose objects are relations and morphisms are morphisms of relations as just deﬁned. It is evident that to each diagram in $ℛ^{A} (C)$ there is a corresponding diagram of arrows in $C$ and commutativity of diagrams in $ℛ^{A} (C)$ follows from commutativity of the corresponding diagrams in $C$ . For now there is no restriction on the object $A$ or the arrows that are relations on $A$ . We will introduce further restrictions as we develop the properties of the category of relations.

Morphisms of corelations are deﬁned by dualizing the corresponding diagrams for morphisms of relations. Corelations and morphisms of corelations form the category of corelations on $A,$ $ℛ_{A} (C)$ .

We will now proceed to develop some formal properties of the category $ℛ^{A} (C)$ . The corresponding dualized properties holds for the category $ℛ_{A} (C)$ .

Let $r$ be an object in $ℛ^{A} (C)$ with domain $B$ . Deﬁne two arrows $δ^{l} : B \to A \otimes B$ and $δ^{r} : B \to B \otimes A$ in $C$ by

\begin{array}{l} δ^{l} & = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B}, \\ δ^{r} & = (1_{B} \otimes β_{A}) \circ (1_{B} \otimes (ε_{A} \otimes 1_{A})) \circ (1_{B} \otimes r) \circ δ_{B} . \end{array}

Deﬁne $θ^{l} : B \to (A \otimes A) \otimes B$ and $θ^{r} : B \to B \otimes (A \otimes A)$ by

\begin{array}{l} θ^{l} & = (r \otimes 1_{B}) \circ δ_{B}, \\ θ^{r} & = (1_{B} \otimes r) \circ δ_{B} . \end{array}

We ﬁrst prove the identities

Lemma 19.

\begin{array}{l} (δ_{A \otimes A} \otimes 1_{B}) \circ θ^{l} & = α_{A \otimes A, A \otimes A, B} \circ (1_{A \otimes A} \otimes θ^{l}) \circ θ^{l}, \\ (1_{B} \otimes δ_{A \otimes A}) \circ θ^{r} & = α_{B, A \otimes A, A \otimes A} \circ (θ^{r} \otimes 1_{A \otimes A}) \circ θ^{r}, \\ (θ^{l} \otimes 1_{A \otimes A}) \circ θ^{r} & = α_{A \otimes A, B, A \otimes A} \circ (1_{A \otimes A} \otimes θ^{r}) \circ θ^{l} . \end{array}

Proof. Since $r$ is a morphism in $C$ we have the diagram

r⊗ r
B×B ---- A⊗ A ⊗A ⊗A
| |
δB |δA⊗A
| |
B ---------A ⊗A
r

But then we have

\begin{array}{l} α_{A \otimes A, A \otimes A, B} \circ (1_{A \otimes A} \otimes θ^{l}) \circ θ^{l} \\ = α_{A \otimes A, A \otimes A, B} \circ (1_{A \otimes A} \otimes (r \otimes 1_{B})) \circ (1_{A \otimes A} \otimes δ_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = α_{A \otimes A, A \otimes A, B} \circ (r \otimes (r \otimes 1_{B})) \circ (1_{B} \otimes δ_{B}) \circ δ_{B} \\ = α_{A \otimes A, A \otimes A, B} \circ (r \otimes (r \otimes 1_{B})) \circ α_{B, B, B}^{- 1} \circ (δ_{B} \otimes 1_{B}) \circ δ_{B} \\ = ((r \otimes r) \circ δ_{B} \otimes 1_{B}) \circ δ_{B} \\ = (δ_{A \otimes A} \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} . \end{array}

The proof of the second relation proceeds in a similar way. For the third we have

\begin{array}{l} (θ^{l} \otimes 1_{A \otimes A}) \circ θ^{r} \\ = ((r \otimes 1_{B}) \otimes 1_{A \otimes A}) \circ (δ_{B} \otimes 1_{A \otimes A}) \circ (1_{B} \otimes r) \circ δ_{B} \\ = ((r \otimes 1_{B}) \otimes r) \circ (δ_{B} \otimes 1_{B}) \circ δ_{B} \\ = ((r \otimes 1_{B}) \otimes r) \circ α_{B, B, B} \circ (1_{B} \otimes δ_{B}) \circ δ_{B} \\ = α_{A \otimes A, B, A \otimes A} \circ (r \otimes (1_{B} \otimes r)) \circ (1_{B} \otimes δ_{B}) \circ δ_{B} \\ = α_{A \otimes A, B, A \otimes A} \circ (r \otimes θ^{r}) \circ δ_{B}, \end{array}

□

we can now prove the following

Proposition 20. Let $r$ be a relation. Then the following diagrams commute

$A ⊗B δA-⊗1B (A⊗ A)⊗ B | δl / | / / | | B αA,A,B \ l | \ δ | \ | A ⊗B -----l A ⊗(A ⊗B) 1A⊗ δ$ $e ⊗B εA-⊗1B-A ⊗B | \ | l β\B |δ \ | B$

$1 ⊗δ B ⊗A -B---A B ⊗(A|⊗A) | δ/r / | / | B αB,A,A | \ δr | \ | \ B ⊗A -δr-⊗1A (B⊗ A)⊗ A$ B ⊗ A 1B ⊗-εA B⊗ e
|
δr | /
| / /γB
B|

$r A ⊗B 1A-⊗δ- A⊗ (B⊗ A) | δl / | // | | B |αA,B,A \ r | \ δ | \ | B ⊗A -l---- (A⊗ B)⊗ A δ⊗ 1A$

Proof. Using the lemma and the naturality of $α$ and $δ$ we have

\begin{array}{l} α_{A, A, B} \circ (1_{A} \otimes δ^{l}) \circ δ^{l} \\ = α_{A, A, B} \circ (1_{A} \otimes (γ_{A} \otimes 1_{B})) \circ (1_{A} \otimes ((1_{A} \otimes ε_{A}) \otimes 1_{B})) \\ \circ (1_{A} \otimes θ^{l}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ θ^{l} \\ = (((γ_{A} \circ (1_{A} \otimes ε_{A})) \otimes (γ_{A} \circ (1_{A} \otimes ε_{A}))) \otimes 1_{B}) \\ \circ α_{A \otimes A, A \otimes A, B} \circ (1_{A \otimes A} \otimes θ^{l}) \circ θ^{l} \\ = (((γ_{A} \circ (1_{A} \otimes ε_{A})) \otimes (γ_{A} \circ (1_{A} \otimes ε_{A}))) \otimes 1_{B}) \\ \circ (δ_{A \otimes A} \otimes 1_{B}) \circ θ^{l} \\ = (δ_{A} \otimes 1_{B}) \circ ((γ_{A} \circ (1_{A} \otimes ε_{A})) \otimes 1_{B}) \circ θ^{l} \\ = (δ_{A} \otimes 1_{B}) \circ δ^{l} \\ = (δ_{A} \otimes 1_{B}) \circ (1_{A} \otimes ε_{A} \otimes 1_{B}) \circ θ^{l} \\ = (δ_{A} \otimes 1_{B}) \circ δ^{l} . \end{array}

Since $ε$ is natural and $r$ a morphism in $C$ we have the identities

\begin{array}{l} ε_{A \otimes e} & = ε_{A} \circ γ_{A}, \\ ε_{A \otimes A} & = ε_{A \otimes e} \circ (1_{A} \otimes ε_{A}), \\ ε_{B} & = ε_{A \otimes A} \circ r . \end{array}

But then we have

\begin{array}{l} β_{B} \circ (ε_{A} \otimes 1_{B}) \circ δ^{l} \\ = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = β_{B} \circ (ε_{A \otimes e} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = β_{B} \circ (ε_{A \otimes A} \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = β_{B} \circ (ε_{B} \otimes 1_{B}) \circ δ_{B} \\ = 1_{B}, \end{array}

so the ﬁrst pair of diagrams are commutative. The proof of the commutativity of the second pair of diagrams is similar. For the last diagram we have

\begin{array}{l} α_{A, B, A} \circ (1_{A} \otimes δ^{r}) \circ δ^{l} \\ α_{A, B, A} \circ (1_{A} \otimes (1_{B} \otimes β_{A})) \circ (1_{A} \otimes (1_{B} \otimes (ε_{A} \otimes 1_{A}))) \\ \circ (1_{A} \otimes θ^{r}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ θ^{l} \end{array}

\begin{array}{l} = ((1_{A} \otimes 1_{B}) \otimes β_{A}) \circ ((1_{A} \otimes 1_{B}) \otimes (ε_{A} \otimes 1_{A})) \\ \circ ((γ_{A} \otimes 1_{B}) \otimes 1_{A \otimes A}) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes 1_{A \otimes A}) \\ \circ α_{A \otimes A, B, A \otimes A} \circ (1_{A \otimes A} \otimes θ^{r}) \circ θ^{l} \end{array}

\begin{array}{l} = (((γ_{A} \circ (1_{A} \otimes ε_{A})) \otimes 1_{B}) \otimes (β_{A} \circ (ε_{A} \otimes 1_{A}))) \\ \circ (θ^{l} \otimes 1_{A \otimes A}) \circ θ^{r} \\ = (δ^{l} \otimes 1_{A}) \circ (1_{B} \otimes (β_{A} \circ (ε_{A} \otimes 1_{A}))) \circ θ^{r} \\ = (δ^{l} \otimes 1_{A}) \circ δ^{r}, \end{array}

so this diagram is also commutative. □

The previous proposition show that the pair ${δ^{l}, δ^{r}}$ deﬁne the structure of a $A - A$ bicomodule on $B$ .

Deﬁnition 21. Let $δ^{l} : B \to A \otimes B, δ^{r} : B \to B \otimes A$ be arrows in $C$ . Then ${δ^{l}, δ^{r}}$ deﬁne the structure of a $A - A$ bicomodule on $B$ if the diagrams in proposition 20 commute

We call the object $B$ in the previous deﬁnition the underlying object for the $A - A$ bicomodule $δ = {δ^{l}, δ^{r}}$ .

Let now $δ$ and $γ$ be $A - A$ bicomodules with underlying objects $B$ and $E$ . A morphism $f : δ \to γ$ of $A - A$ bicomodules is an arrow $f : B \to E$ in $C$ such that the following diagrams commute

1 ⊗ f
A⊗B -A--- A⊗|E
| |
δl |γl
| |
B-------- E
f

f ⊗ 1
B ⊗|A ----A E⊗|A
| |
δr | |γr
| |
B -------- E
f

We now form a new category where objects are $A - A$ bicomodules and where morphisms are morphisms of $A - A$ bicomodules. Let this category be named $S^{A} (C)$ .

3.3. Relations in terms of $A - A$ bicomodules.. To each object $r$ in $ℛ^{A} (C)$ there corresponds an object $δ$ in $S^{A} (C)$ . For morphisms of relations we have the following.

Proposition 22. Let $r$ and $s$ be two relations with domains $B$ and $E$ and let $f : r \to s$ be a morphism of relations. Let $δ$ and $γ$ be the objects in $S^{A} (C)$ corresponding to $r$ and $s$ . Then the corresponding arrow $f : B \to E$ in $C$ deﬁnes a morphism $f : δ \to γ$ in $S^{A} (C)$ .

Proof. Since $f$ is a morphism in $C$ and also a morphism from $r$ to $s$ we have the following commutative diagrams

$f B--------------- E \ / \r /s \ / A⊗ A$ f ⊗f
B⊗|B ---- E ⊗E
| |
δB | |δE
| |
B ------- E
f

Using these identities we then have

\begin{array}{l} (1_{A} \otimes f) \circ δ^{l} \\ = (1_{A} \otimes f) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{E}) \circ ((1_{A} \otimes 1_{e}) \otimes f) \circ (s \otimes 1_{B}) \circ (f \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{E}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{E}) \circ (s \otimes 1_{B}) \circ (f \otimes f) \circ δ_{B} \\ = (γ_{A} \otimes 1_{E}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{E}) \circ (s \otimes 1_{B}) \circ δ_{E} \circ f \\ = γ^{l} \circ f . \end{array}

In a similar way we prove the identity $(f \otimes 1_{A}) \circ δ^{r} = γ^{r} \circ f$ . □

The previous deﬁnition show that we have a well deﬁned functor $Φ : ℛ^{A} (C) \to S^{A} (C)$ , where $Φ (r)$ is the $A - A$ bicomodule corresponding to $r$ and where $Φ (f) = f$ . We will next construct a functor from $S^{A} (C)$ to $ℛ^{A} (C)$ .

Let $δ$ be an object in $S^{A} (C)$ . deﬁne a morphism $r : B \to A \otimes A$ by

r = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{B} \otimes 1_{A})) \circ (1_{A} \otimes δ^{r}) \circ δ^{l} .

We have proved that $β_{A}$ and $ε_{B}$ are arrows in $C$ and have therefore the following result.

Proposition 23. $r$ is an object in $ℛ^{A} (C)$ .

Using this result we can deﬁne a map of objects $Ψ : S^{A} (C) \to ℛ^{A} (C)$ by $Ψ (δ) = r$ . For morphisms in $S^{A} (C)$ we have

Proposition 24. Let $δ$ , and $γ$ be two objects in $S^{A} (C)$ and let $f : δ \to γ$ be a morphism. Then the corresponding arrow in $C$ deﬁnes a morphism of the objects $r = Ψ (δ)$ and $s = Ψ (γ)$ in $ℛ^{A} (C)$ .

Proof. Let the domains of $r$ and $s$ be $B$ and $E$ . We have the following commutative diagrams

1A⊗-f
A⊗B A⊗|E
| |
δl |γl
| |
B----f--- E

f ⊗-1A
B ⊗|A E⊗|A
| |
δr | |γr
| |
B ----f--- E

But then we have

\begin{array}{l} s \circ f \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{E} \otimes 1_{A})) \circ (1_{A} \otimes γ^{r}) \circ γ^{l} \circ f \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{E} \otimes 1_{A})) \circ (1_{A} \otimes γ^{r}) \circ (1_{A} \otimes f) \circ δ^{l} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{E} \otimes 1_{A})) \circ (1_{A} \otimes (f \otimes 1_{A})) \circ (1_{A} \otimes δ^{r}) \circ δ^{l} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{E} \circ f \otimes 1_{A})) \circ (1_{A} \otimes δ^{r}) \circ δ^{l} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{B} \otimes 1_{A})) \circ (1_{A} \otimes δ^{r}) \circ δ^{l} \\ = r . \end{array}

so $f$ is a morphism of relations. □

We use this result to extend $Ψ$ to a functor from $S^{A} (C)$ to $ℛ^{A} (C)$ by deﬁning $Ψ (f) = f$ . We will now show that $ℛ^{A} (C)$ and $S^{A} (C)$ are isomorphic categories. We need the following lemma

Lemma 25.

\begin{array}{l} (γ_{A} \otimes β_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ δ_{A \otimes A} & = 1_{A \otimes A}, \\ (δ^{l} \otimes 1_{B}) \circ δ_{B} & = α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l}, \\ (1_{B} \otimes δ^{r}) \circ δ_{B} & = α_{B, B, A}^{- 1} \circ (δ_{B} \otimes 1_{A}) \circ δ^{r} . \end{array}

Proof. For the ﬁrst part of the lemma we have

\begin{array}{l} (γ_{A} \otimes β_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ δ_{A \otimes A} \end{array}

\begin{array}{l} = (γ_{A} \otimes β_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ α_{A \otimes A, A, A}^{- 1} \\ \circ (α_{A, A, A} \otimes 1_{A}) \circ ((1_{A} \otimes σ_{A, A}) \otimes 1_{A}) \circ (α_{A, A, A}^{- 1} \otimes 1_{A}) \\ \circ α_{A \otimes A, A, A} \circ (δ_{A} \otimes δ_{A}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (γ_{A} \otimes (1_{e} \otimes 1_{A})) \circ α_{A \otimes e, e, A}^{- 1} \circ (α_{A, e, e} \otimes 1_{A}) \circ ((1_{A} \otimes σ_{e, e}) \otimes 1_{A}) \circ (α_{A, e, e}^{- 1} \otimes 1_{A}) \circ α_{A \otimes e, A, A} \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ (δ_{A} \otimes δ_{A}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (γ_{A} \otimes (1_{e} \otimes 1_{A})) \circ α_{A \otimes e, e, A}^{- 1} \circ (α_{A, e, e} \otimes 1_{A}) \circ ((1_{A} \otimes σ_{e, e}) \otimes 1_{A}) \circ (α_{A, e, e}^{- 1} \otimes 1_{A}) \circ α_{A \otimes e, A, A} \circ (γ_{A}^{- 1} \otimes β_{A}^{- 1}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ α_{A, e, A}^{- 1} \circ ((1_{A} \otimes β_{e}) \otimes 1_{A}) \circ ((1_{A} \otimes σ_{e, e}) \otimes 1_{A}) \circ (α_{A, e, e}^{- 1} \otimes 1_{A}) \circ α_{A \otimes e, e, A} \circ (γ_{A}^{- 1} \otimes β_{A}^{- 1}) \\ = (1_{A} \otimes β_{A}) \circ α_{A, e, A}^{- 1} \circ ((1_{A} \otimes γ_{e}) \otimes 1_{A}) \circ \circ (α_{A, e, e}^{- 1} \otimes 1_{A}) \circ α_{A \otimes e, e, A} \circ (γ_{A}^{- 1} \otimes β_{A}^{- 1}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (γ_{e} \otimes 1_{A})) \circ \circ α_{A, e \otimes e, A}^{- 1} \circ (α_{A, e, e}^{- 1} \otimes 1_{A}) \circ α_{A \otimes e, e, A} \circ (γ_{A}^{- 1} \otimes β_{A}^{- 1}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (γ_{e} \otimes 1_{A})) \circ (1_{A} \otimes α_{e, e, A}) \circ α_{A, e, e \otimes A}^{- 1} \circ ((1_{A} \otimes 1_{e}) \otimes β_{A}^{- 1}) \circ (γ_{A}^{- 1} \otimes 1_{A}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (1_{e} \otimes β_{A})) \circ (1_{A} \otimes (1_{e} \otimes β_{A}^{- 1})) \circ α_{A, e, A}^{- 1} \circ (γ_{A}^{- 1} \otimes 1_{A}) \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ α_{A, e, A}^{- 1} \circ (γ_{A}^{- 1} \otimes 1_{A}) \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes β_{A}^{- 1}) \\ = 1_{A \otimes A} . \end{array}

For the second part of the lemma we have

\begin{array}{l} (δ^{l} \otimes 1_{B}) \circ δ_{B} \\ = ((γ_{A} \otimes 1_{B}) \otimes 1_{B}) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes 1_{B}) \circ ((r \otimes 1_{B}) \otimes 1_{B}) \circ (δ_{B} \otimes 1_{B}) \circ δ_{B} \end{array}

\begin{array}{l} = ((γ_{A} \otimes 1_{B}) \otimes 1_{B}) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes 1_{B}) \circ ((r \otimes 1_{B}) \otimes 1_{B}) \circ α_{B, B, B} \circ (1_{B} \otimes δ_{B}) \circ δ_{B} \end{array}

\begin{array}{l} = α_{A, B, B} \circ (γ_{A} \otimes 1_{B \otimes B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B \otimes B}) \circ (r \otimes 1_{B \otimes B}) \circ (1_{B} \otimes δ_{B}) \circ δ_{B} \end{array}

\begin{array}{l} = (1_{B} \otimes δ_{B}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (1_{B} \otimes δ_{B}) \circ δ^{l} . \end{array}

The proof of the third part of the lemma is similar to the second part. □

We now use the lemma to prove the following theorem

Theorem 26. The functor $Φ : ℛ^{A} (C) \to S^{A} (C)$ is invertible with inverse $Ψ : S^{A} (C) \to ℛ^{A} (C)$ .

Proof. We only need to prove that $Φ$ is bijective with inverse $Ψ$ on objects since $Ψ$ is obviously the inverse of $Φ$ on morphisms.

Let $r$ be an object in $ℛ^{A} (C)$ with domain $B$ . Using lemma 25 we have

\begin{array}{l} (Ψ \circ Φ) (r) \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{B} \otimes 1_{A})) \circ (1_{A} \otimes {(Φ (r))}^{r}) \circ {(Φ (r))}^{l} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (ε_{B} \otimes 1_{A})) \circ (1_{A} \otimes (1_{B} \otimes β_{A})) \circ (1_{A} \otimes (1_{B} \otimes (ε_{A} \otimes 1_{A}))) \circ (1_{A} \otimes (1_{B} \otimes r)) \circ (1_{A} \otimes δ_{B}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (1_{e} \otimes β_{A})) \circ (1_{A} \otimes (1_{e} \otimes (ε_{A} \otimes 1_{A}))) \circ (1_{A} \otimes (1_{e} \otimes r)) \circ (1_{A} \otimes (ε_{B} \otimes 1_{B}) \circ δ_{B}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \end{array}

\begin{array}{l} = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (1_{e} \otimes β_{A})) \circ (1_{A} \otimes (1_{e} \otimes (ε_{A} \otimes 1_{A}))) \circ (1_{A} \otimes (1_{e} \otimes r)) \circ (1_{A} \otimes β_{B}^{- 1}) \circ (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (1_{A} \otimes β_{A}) \circ (1_{A} \otimes (1_{e} \otimes β_{A})) \circ (1_{A} \otimes (1_{e} \otimes (ε_{A} \otimes 1_{A}))) \circ (1_{A} \otimes β_{A \otimes A}^{- 1}) \circ (γ_{A} \otimes 1_{A \otimes A}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{A \otimes A}) \circ (r \otimes r) \circ δ_{B} \\ = (1_{A} \otimes β_{A} \circ (1_{e} \otimes β_{A}) \circ β_{e \otimes A}^{- 1}) \circ (1_{A} \otimes β_{e \otimes A}^{- 1}) \circ (γ_{A} \otimes (1_{e} \otimes 1_{A})) ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ δ_{A \otimes A} \circ r \\ = (γ_{A} \otimes β_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ δ_{A \otimes A} \circ r \\ = r, \end{array}

Ψ \circ Φ = 1_{ℛ^{A} (C)}

. Next let

δ

be any object in

S^{A} (C)

with underlying object

B

. We have

\begin{array}{l} {(Φ (Ψ (δ)))}^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (Ψ (δ) \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ ((1_{A} \otimes β_{A}) \otimes 1_{B}) \circ ((1_{A} \otimes (ε_{B} \otimes 1_{A})) \otimes 1_{B}) \circ ((1_{A} \otimes δ^{r}) \otimes 1_{B}) \circ (δ^{l} \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ ((1_{A} \otimes β_{A}) \otimes 1_{B}) \circ ((1_{A} \otimes (ε_{B} \otimes 1_{A})) \otimes 1_{B}) \circ ((1_{A} \otimes δ^{r}) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{e \otimes A}) \otimes 1_{B}) \circ ((1_{A} \otimes (ε_{B} \otimes 1_{A})) \otimes 1_{B}) \circ ((1_{A} \otimes δ 〉) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \end{array}

\begin{array}{l} = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{B \otimes A}) \otimes 1_{B}) \circ ((1_{A} \otimes δ^{r}) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{B \otimes e}) \otimes 1_{B}) \circ ((1_{A} \otimes (1_{B} \otimes ε_{A})) \otimes 1_{B}) \circ ((1_{A} \otimes δ^{r}) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{B \otimes e}) \otimes 1_{B}) \circ ((1_{A} \otimes γ_{B}^{- 1}) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{B}) \otimes 1_{B}) \circ α_{A, B, B} \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (γ_{A} \otimes 1_{B}) \circ α_{A, e, B} \circ (1_{A} \otimes (ε_{B} \otimes 1_{B})) \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = (1_{A} \otimes β_{B}) \circ (1_{A} \otimes (ε_{B} \otimes 1_{B})) \circ (1_{A} \otimes δ_{B}) \circ δ^{l} \\ = δ^{l} \end{array}

and similarly we ﬁnd that

{(Φ (Ψ (δ)))}^{r} = δ^{r}

. This proves that

Φ \circ Ψ = 1_{S^{A} (C)}

. □

By deﬁnition $δ_{A} : A \to A \otimes A$ is an object in $ℛ^{A} (C)$ . Its image by $Φ$ is therefore an object in $S^{A} (C)$ .

Proposition 27. $Φ (δ_{A}) = {δ_{A}, δ_{A}} .$

Proof. We have

\begin{array}{l} Φ {(δ_{A})}^{l} \\ = (γ_{A} \otimes 1_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{A}) \circ (δ_{A} \otimes 1_{A}) \circ δ_{A} \\ = (γ_{A} \circ (1_{A} \otimes ε_{A}) \circ δ_{A} \otimes 1_{A}) \circ δ_{A} \\ = (1_{A} \otimes 1_{A}) \circ δ_{A} \\ = δ_{A} . \end{array}

In a similar way we prove that $Φ {(δ_{A})}^{r} = δ_{A}$ . □

The object ${δ_{A}, δ_{A}}$ in $S^{A} (C)$ will play an important role for a product we will deﬁne later and is given a special name.

a = {δ_{A}, δ_{A}} .

In the following we will mostly work in the category $S^{A} (C)$ and use the isomorphism to induce the corresponding structures on the category of relations $ℛ^{A} (C)$ . Note that the category of corelations on $A$ , $ℛ_{A} (C)$ , is by duality isomorphic to the category of $A - A$ bimodules. Denote this category by $S_{A} (C)$ .

3.4. The $⊠^{A}$ product of relations. Let $δ$ and $γ$ be two objects in $S^{A} (C)$ with underlying objects $B$ and $E$ . Deﬁne two arrows in $C$ ${(δ ⊠^{A} γ)}^{l} : B \otimes E \to A \otimes (B \otimes E)$ and ${(δ ⊠^{A} γ)}^{r} : B \otimes E \to (B \otimes E) \otimes A$ by

\begin{array}{l} {(δ ⊠^{A} γ)}^{l} & = α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}), \\ {(δ ⊠^{A} γ)}^{r} & = α_{B, E, A} \circ (1_{B} \otimes γ^{r}) . \end{array}

Then we have

Proposition 28. $δ ⊠^{A} γ = {{(δ ⊠^{A} γ)}^{l}, {(δ ⊠^{A} γ)}^{r}}$ is an object in $S^{A} (C)$ .

Proof. Using the naturality and the MacLane coherence condition for $α$ we have

\begin{array}{l} α_{A, A, B \otimes E} \circ (1_{A} \otimes {(δ ⊠^{A} γ)}^{l}) \circ {(δ ⊠^{A} γ)}^{l} \\ = α_{A, A, B \otimes E} \circ (1_{A} \otimes α_{A, B, E}^{- 1}) \circ (1_{A} (δ^{l} \otimes 1_{E})) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \\ = α_{A, A, B \otimes E} \circ (1_{A} \otimes α_{A, B, E}^{- 1}) \circ α_{A, A \otimes B, E}^{- 1} \circ ((1_{A} \otimes δ^{l}) \circ δ^{l} \otimes 1_{E}) \\ = α_{A, A, B \otimes E} \circ (1_{A} \otimes α_{A, B, E}^{- 1}) \circ α_{A, A \otimes B, E}^{- 1} \circ (α_{A, A, B}^{- 1} \otimes 1_{E}) \circ ((δ_{A} \otimes 1_{B}) \otimes 1_{E}) \circ (δ^{l} \otimes 1_{E}) \\ = α_{A \otimes A, B, E}^{- 1} \circ ((δ_{A} \otimes 1_{B}) \otimes 1_{E}) \circ (δ^{l} \otimes 1_{E}) \\ = (δ_{A} \otimes 1_{B \otimes E}) \circ (α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E})) \\ = (δ_{A} \otimes 1_{B \otimes E}) \circ {(δ ⊠^{A} γ)}^{l} \end{array}

and

\begin{array}{l} β_{B \otimes E} \circ (ε_{A} \otimes 1_{B \otimes E}) \circ {(δ ⊠^{A} γ)}^{l} \\ = β_{B \otimes E} \circ (ε_{A} \otimes (1_{B} \otimes 1_{E})) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \\ = β_{B \otimes E} \circ α_{e, B, E}^{- 1} \circ ((ε_{A} \otimes 1_{B}) \circ δ^{l} \otimes 1_{E}) \\ = β_{B \otimes E} \circ α_{e, B, E}^{- 1} \circ (β_{B}^{- 1} \otimes 1_{E}) \\ = β_{B \otimes E} \circ β_{B \otimes E}^{- 1} \\ = 1_{B \otimes E}, \end{array}

so $B \otimes E$ is a left $A$ comodule. In a similar way we show that $B \otimes E$ is a right $A$ comodule. For the compatibility between the two structures we have

\begin{array}{l} α_{A, B \otimes E, A} \circ (1_{A} \otimes {(δ ⊠^{A} γ)}^{r}) \circ {(δ ⊠^{A} γ)}^{l} \\ = α_{A, B \otimes E, A} \circ (1_{A} \otimes α_{B, E, A}) \circ (1_{A} \otimes (1_{B} \otimes γ^{r})) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \\ = α_{A, B \otimes E, A} \circ (1_{A} \otimes α_{B, E, A}) \circ α_{A, B, E \otimes A}^{- 1} \circ (δ^{l} \otimes (1_{E} \otimes 1_{A})) \circ (1_{B} \otimes γ^{r}) \\ = (α_{A, B, E}^{- 1} \otimes 1_{A}) \circ α_{A \otimes B, E, A} \circ (δ^{l} \otimes (1_{E} \otimes 1_{A})) \circ (1_{B} \otimes γ^{r}) \\ = (α_{A, B, E}^{- 1} \otimes 1_{A}) \circ ((δ^{l} \otimes 1_{E}) \otimes 1_{A}) \circ α_{B, E, A} \circ (1_{B} \otimes γ^{r}) \\ = ({(δ ⊠^{A} γ)}^{l} \otimes 1_{A}) \circ {(δ ⊠^{A} γ)}^{r} . \end{array}

□

Using the previous proposition we can deﬁne an object map $⊠^{A} : S^{A} (C) \times S^{A} (C) \to S^{A} (C)$ by

⊠^{A} (δ, γ) = δ ⊠^{A} γ .

Let $δ, γ, ρ$ and $θ$ be objects in $S^{A} (C)$ with underlying objects $B, E, D$ and $T$ in $C$ and let $f : δ \to ρ$ and $g : γ \to θ$ be two morphisms in $S^{A} (C)$ . Let $f : B \to E$ and $g : D \to T$ be the corresponding arrows in $C$ and let $f \otimes g : B \otimes E \to D \otimes T$ be their product in $C$ . Deﬁne $f ⊠^{A} g = f \otimes g$ .

Proposition 29. $f ⊠^{A} g :^{A} δ ⊠^{A} γ \to ρ ⊠^{A} θ$ is a morphism in $S^{A} (C) .$

Proof. We have previously proved that $f \otimes g$ is an arrow in $C$ . It is also a morphism in $S^{A} (C)$

\begin{array}{l} {(ρ ⊠^{A} θ)}^{l} \circ (f ⊠^{A} g) \\ = α_{A, D, T}^{- 1} \circ (ρ^{l} \otimes 1_{T}) \circ (f \otimes g) \\ = α_{B, D, T}^{- 1} \circ (ρ^{l} \circ f \otimes g) \end{array}

\begin{array}{l} = α_{B, D, T}^{- 1} \circ ((1_{A} \otimes f) \otimes g) \circ (δ^{l} \otimes 1_{E}) \\ = (1_{A} \otimes (f \otimes g)) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \\ = (1_{A} \otimes (f ⊠^{A} g)) \circ {(δ ⊠^{A} γ)}^{l} . \end{array}

The identity ${(ρ ⊠^{A} θ)}^{r} \circ (f \otimes g) = ((f ⊠^{A} g) \otimes 1_{A}) \circ {(δ ⊠^{A} γ)}^{r}$ is proved in a similar way. □

Using this proposition we can extend the object map $⊠^{A}$ to a bifunctor by deﬁning

⊠^{A} (f, g) = f ⊠^{A} g .

In terms of this bifunctor we have the following immediate consequence of lemma 25

Corollary 30. Let $δ$ be an object in $S^{A} (C)$ with underlying object $B$ . Then $δ_{B} : δ \to δ ⊠^{A} δ$ is a morphism in $S^{A} (C)$ .

In general there exists no neutral object for $⊠^{A}$ . This is clearly seen in the case of $S e t s$ . Let $B$ be a set and let $f : B \to A$ be a injective map of sets. Deﬁne a $A - A$ bicomodule structure on $B$ by $δ^{l} (x) = (f (x), x)$ and $δ^{r} (x) = (x, f (x))$ . Assume that $ω$ is a neutral object for $⊠^{A}$ and let the underlying object for $ω$ be $S$ . Then there must exist a isomorphism $h : δ ⊠^{A} ω \to δ$ and therefore bijective map $h : B \times S \to B$ that is a morphism of $A - A$ bicomodules. But this implies that $f (h (x, s)) = f (x)$ for all $x$ and $s$ . But since $f$ is injective we must have $h (x, s) = x$ and this is not possible if there is more than one element in $S$ . A neutral element $ω$ for $⊠^{A}$ therefore must have $e = {*}$ as underlying object. Any $A - A$ bicomodule structure $ω$ on $e$ must be of the form $ω^{l} (*) = (x_{0}, *)$ and $ω^{r} (*) = (*, y_{0})$ for some elements $x_{0}, y_{0} \in A$ . Let $B$ be a set with more than one point and let $f : B \to A$ be a map of sets that is not constant. Deﬁne a $A - A$ bicomodule structure on $B$ by $δ^{l} (x) = (f (x), x)$ and $δ^{r} (x) = (x, f (x))$ . If $ω$ is a neutral object for $⊠^{A}$ there must exist an isomorphism $h : e \times B \to B$ that is a morphism of $A - A$ bicomodules. But this implies that for all $b \in B$ we have $f (h (*, b)) = x_{0}$ and this implies that $f$ is constant since $h$ is bijective. This is a contradiction and this proves that $⊠^{A}$ does not have a neutral object in the case of $S e t s$ .

3.5. Semimonoidal structures on the category of relations. Recall that $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ is a semimonoidal category if $M^{A} : ⊠^{A} \circ (1 \times ⊠^{A}) \to ⊠^{A} \circ (⊠^{A} \times 1)$ is a natural isomorphism such that the ﬁrst of the MacLane Coherence conditions is satisﬁed. We will in general assume that the category $S^{A} (C)$ is a semimonoidal category with respect to some choice of natural isomorphism $M^{A}$ .

At least one semimonoidal structure for $⊠^{A}$ will always exists.

Proposition 31. Let $δ$ and $γ$ be objects in $S^{A} (C)$ with underlying objects $B$ and $E$ in $C$ . Deﬁne

M_{δ, γ, ρ}^{A} = α_{B, E, D},

where $α$ is the associativity constraint for the category $C$ . Then $〈 S^{A} (C), ⊠^{A}, M^{A}, S^{A} 〉$ is a symmetric semimonoidal category.

Proof. We have proved previously that $α_{B, E, D}$ is a $C$ arrow in $C$ . Next we need to show that the associativity constraint for $\otimes$ on $C$ is in fact a morphism in $S^{A} (C)$ . If we use the naturality and the MacLane coherence condition for $α$ we have

\begin{array}{l} ((δ ⊠^{A} γ) ⊠^{A} ρ) \circ α_{B, E, D} \\ = α_{A, B \otimes E, D}^{- 1} \circ (α_{A, B, E}^{- 1} \otimes 1_{D}) \circ ((δ^{l} \otimes 1_{B}) \otimes 1_{D}) \circ α_{B, E, D} \\ = α_{A, B \otimes E, D}^{- 1} \circ (α_{A, B, E}^{- 1} \otimes 1_{D}) \circ α_{A \otimes B, E, D} \circ (δ^{l} \otimes (1_{E} \otimes 1_{D})) \\ = (1_{A} \otimes α_{B, E, D}) \circ α_{A, B, E \otimes D}^{- 1} \circ (δ_{B} \otimes 1_{E \otimes D}) \\ = (1_{A} \otimes α_{B, E, D}) \circ (δ ⊠^{A} (γ ⊠^{A} ρ)) . \end{array}

$M^{A}$ is clearly an isomorphism and is a natural transformation if the following identity holds $((f ⊠^{A} g) ⊠^{A} h) \circ M_{δ, γ, ρ}^{A} = M_{δ^{'}, γ^{'}, ρ^{'}}^{A} \circ (f ⊠^{A} (g ⊠^{A} h))$ for all morphisms $f : δ \to δ^{'}, g : γ \to γ^{'}$ and $h : ρ \to ρ^{'}$ in $S^{A} (C)$ . But the corresponding identity in $C$ is $((f \otimes g) \otimes h) \circ α_{B, E, D} = α_{B^{'}, E^{'}, D^{'}} \circ (f \otimes (g \otimes h))$ and this identity holds because $α$ is a natural transformation. □

The previous proposition leads us to the following deﬁnition

Deﬁnition 32. The semimonoidal $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ category is external if for all objects $δ, γ$ and $ρ$ we have

M_{δ, γ, ρ}^{A} = α_{B, E, D},

where $α$ is the associativity constraint for the product $\otimes$ on the category $C$ and where $B$ , $E$ and $D$ are the underlying objects for $δ, γ$ and $ρ$ .

Since $S^{A} (C)$ is isomorphic to the category of relations, a semimonoidal structure on $S^{A} (C)$ will induce one on the category of relations. Let the product in $ℛ^{A} (C)$ corresponding to $⊠^{A}$ be $⊡^{A} : ℛ^{A} (C) \times ℛ^{A} (C) \to ℛ^{A} (C)$ . We thus have

⊡^{A} = Ψ \circ ⊠^{A} \circ (Φ \times Φ) .

We have the following explicit expression for the product

Proposition 33. Let $r$ and $s$ be two objects in $ℛ^{A} (C)$ . Then we have

r ⊡^{A} s = (γ_{A} \otimes β_{A}) \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \circ (r \otimes s) .

Proof. Since $ϕ$ and $Ψ$ are isomorphisms with $Φ = Ψ^{- 1}$ we only need to verify that

Φ (r ⊡^{A} s) = Φ (r) ⊠^{A} Φ (s),

for all objects $r$ and $s$ in $ℛ^{A} (C)$ . Note that the naturality of $ε$ gives the following relations

\begin{array}{l} ε_{A} \circ β_{A} \circ (ε_{A} \otimes 1_{A}) \\ = ε_{e \otimes A} \circ (ε_{A} \otimes 1_{A}) \\ = ε_{A \otimes A} \\ = ε_{A \otimes e} \circ (1_{A} \otimes ε_{A}) \\ = ε_{A} \circ γ_{A} \circ (ε_{A} \otimes 1_{A}) . \end{array}

We then have

\begin{array}{l} {(Φ (r ⊡^{A} s))}^{l} \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B \otimes E}) \circ ((t ⊡^{A} s) \otimes 1_{B \otimes E}) \circ δ_{B \otimes E} \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B \otimes E}) \circ ((γ_{A} \otimes β_{A}) \otimes 1_{B \otimes E}) \\ \circ (((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \otimes 1_{B \otimes E}) \circ ((r \otimes s) \otimes 1_{B \otimes E}) \circ δ_{B \otimes E} \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B \otimes E}) \circ ((γ_{A} \otimes β_{A}) \otimes 1_{B \otimes E}) \\ \circ (((1_{A} \otimes ε_{A}) \otimes (ε_{A} \otimes 1_{A})) \otimes 1_{B \otimes E}) \circ ((r \otimes s) \otimes 1_{B \otimes E}) \\ \circ α_{B \otimes E, B, E}^{- 1} \circ (α_{B, E, B} \otimes 1_{E}) \circ ((1_{B} \otimes σ_{B, E}) \otimes 1_{E}) \\ \circ (α_{B, B, E}^{- 1} \otimes 1_{E}) \circ α_{B \otimes B, E, E} \circ (δ_{B} \otimes δ_{E}) \\ = α_{A, B, E}^{- 1} \circ ((γ_{A} \otimes 1_{B}) \otimes 1_{E}) \circ (α_{A, e, B} \otimes 1_{E}) \circ ((1_{A} \otimes σ_{B, e}) \otimes 1_{E}) \\ \circ ((1_{A} \otimes (1_{B} \otimes ε_{A})) \otimes 1_{E}) \circ (α_{A, B, A}^{- 1} \otimes 1_{E}) \circ α_{A \otimes B, A, E} \\ \circ ((γ_{A} \otimes 1_{B}) \otimes (β_{A} \otimes 1_{E})) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes ((ε_{A} \otimes 1_{A}) \otimes 1_{E})) \\ \circ ((r \otimes 1_{B}) \otimes (s \otimes 1_{E})) \circ (δ_{B} \otimes δ_{E}) \end{array}

\begin{array}{l} = α_{A, B, E}^{- 1} \circ ((1_{A} \otimes γ_{B}) \otimes 1_{E}) \circ (α_{A, B, e}^{- 1} \otimes 1_{E}) \circ α_{A \otimes B, e, E} \circ (1_{A \otimes B} \otimes (ε_{A} \otimes 1_{E})) \\ \circ ((γ_{A} \otimes 1_{B}) \otimes (β_{A} \otimes 1_{E})) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes ((ε_{A} \otimes 1_{A}) \otimes 1_{E})) \\ \circ ((r \otimes 1_{B}) \otimes (s \otimes 1_{E})) \circ (δ_{B} \otimes δ_{E}) \\ = (1_{A} \otimes (γ_{B} \otimes 1_{E})) \circ α_{A, B \otimes e, E}^{- 1} \circ (α_{A, B, e}^{- 1} \otimes 1_{E}) \circ α_{A \otimes B, e, E} \circ (1_{A \otimes B} \otimes (ε_{A} \otimes 1_{E})) \\ \circ ((γ_{A} \otimes 1_{B}) \otimes (β_{A} \otimes 1_{E})) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes ((ε_{A} \otimes 1_{A}) \otimes 1_{E})) \\ \circ ((r \otimes 1_{B}) \otimes (s \otimes 1_{E})) \circ (δ_{B} \otimes δ_{E}) \\ = (1_{A} \otimes (γ_{B} \otimes 1_{E})) \circ (1_{A} \otimes α_{B, e, E}) \circ α_{A, B, e \otimes E}^{- 1} \circ (1_{A \otimes B} \otimes (ε_{A} \otimes 1_{E})) \\ \circ ((γ_{A} \otimes 1_{B}) \otimes (β_{A} \otimes 1_{E})) \circ (((1_{A} \otimes ε_{A}) \otimes 1_{B}) \otimes ((ε_{A} \otimes 1_{A}) \otimes 1_{E})) \\ \circ ((r \otimes 1_{B}) \otimes (s \otimes 1_{E})) \circ (δ_{B} \otimes δ_{E}) \\ = α_{A, B, E}^{- 1} \circ (Φ {(r)}^{l} \otimes β_{E} \circ (ε_{A} \circ β_{A} \circ (ε_{A} \otimes 1_{A}) \circ s) \otimes 1_{E} \circ δ_{E}) \\ = α_{A, B, E}^{- 1} \circ (Φ {(r)}^{l} \otimes β_{E} \circ (ε_{A} \circ γ_{A} \circ (1_{A} \otimes ε_{A}) \circ s) \otimes 1_{E}) \circ δ_{E} \\ = α_{A, B, E}^{- 1} \circ (Φ {(r)}^{l} \otimes β_{E} \circ (ε_{A} \otimes 1_{E}) \circ Φ {(s)}^{l}) \\ = α_{A, B, E}^{- 1} \circ (Φ {(r)}^{l} \otimes 1_{E}) \\ = Φ {(r)}^{l} ⊠^{A} Φ {(s)}^{l} . \end{array}

The proof of the identity ${(Φ (r ⊡^{A} s))}^{r} = {(Φ (r) ⊠^{A} Φ (s))}^{r}$ is similar. □

By duality the category of $C$ -corelations $ℛ_{A} (C)$ is isomorphic to the category of $A - A$ bimodules $S_{A} (C)$ .

3.6. The tensor product of relations. For categories of bimodules over rings we have a standard construction of a tensor product. This construction is categorical in nature and can in a natural way be generalized to the category of $A - A$ bimodules $S_{A} (C)$ . By dualizing this construction we arrive at our deﬁnition of a tensor product of $A - A$ bicomodules. The isomorphism $Ψ : S^{A} (C) \to ℛ^{A} (C)$ is used to deﬁne the tensor product of relations.

The following lemma is fundamental for the construction of the tensor product.

Lemma 34. Let $δ$ be an object in $S^{A} (C)$ . Then $δ^{l} : δ \to a ⊠^{A} δ$ and $δ^{r} : δ \to δ ⊠^{A} a$ are morphisms in $S^{A} (C)$ and if $f : δ \to γ$ is a morphism in $S^{A} (C)$ the following diagrams in $S^{A} (C)$ are commutative.

1a⊠A-f A
a⊠δ a⊠| γ
| |
δl |γl
| |
δ----f---- γ

A f ⊠A-1a A
δ ⊠|a γ⊠| a
| |
δr| |γr
| |
δ -----f---- γ

Proof. There are four diagrams that need to be commutative for the ﬁrst part of the lemma to be true. It is seen by inspection that this set of diagrams is included in the set of diagrams deﬁning $δ$ to be a $A - A$ bicomodule. The second part of the lemma is clearly true since the diagrams in $C$ corresponding to the given diagrams are the conditions for $f$ to be a morphism of the $A - A$ bicomodules $δ$ and $γ$ . □

Let now $δ$ and $γ$ be any pair of objects in $S^{A} (C)$ . From the previous lemma we can conclude that $P_{δ, γ}^{A}$ given by

$δ⊠A(a⊠A γ)--MAδ,a,γ- (δ ⊠Aa)⊠ γ \ / 1δ⊠A γ\l / δr ⊠A1γ \ / δ⊠γ$

is a diagram in $S^{A} (C)$ . The limit of this diagram, when it exists, is determined by an object in $S^{A} (C)$ denoted by $δ \otimes^{A} γ$ and a morphism $π_{δ, γ}^{A} : δ \otimes^{A} γ \to δ ⊠^{A} γ$ .

Deﬁnition 35. Let $δ$ and $γ$ be two objects in $S^{A} (C)$ . The tensor product of $δ$ and $γ$ is given by

\otimes^{A} (δ, γ) = δ \otimes^{A} γ .

The following property of $π_{δ, γ}^{A}$ is important.

Proposition 36. $π_{δ, γ}^{A}$ is a monomorphism.

Proof. Let $ρ$ be an object in $S^{A} (C)$ and let $f, g : ρ \to δ \otimes^{A} γ$ be a pair of morphisms such that $π_{δ, γ}^{A} \circ f = π_{δ, γ}^{A} \circ g$ . Deﬁne $h = π_{δ, γ}^{A} \circ g$ . Then $〈 ρ, h 〉$ is a cone on $P_{δ, γ}^{A}$ and therefore the equation

π_{δ, γ}^{A} \circ f = h

has a unique solution. But both $f$ and $g$ are solutions and therefore by uniqueness we can conclude that $f = g$ and this proves that $π_{δ, γ}^{A}$ is a monomorphism. □

We now want to extend the tensor product to morphisms. Let $δ$ , $γ$ , $θ$ and $ρ$ be objects in $S^{A} (C)$ and let $f : δ \to θ$ and $g : γ \to ρ$ be morphisms. Then we have

Lemma 37. $(f ⊠^{A} g) \circ π_{δ, γ}^{A}$ is a cone on the diagram $P_{θ, ρ}^{A}$ .

Proof. We have

\begin{array}{l} M_{θ, a, ρ}^{A} \circ (1_{θ} ⊠^{A} ρ^{l}) \circ (f ⊠^{A} g) \circ π_{δ, γ}^{A} \\ = M_{θ, a, ρ}^{A} \circ (f ⊠^{A} (ρ^{l} \circ g)) \circ π_{δ, γ}^{A} \\ = M_{θ, a, ρ}^{A} \circ (f ⊠^{A} (1_{a} ⊠^{A} g) \circ γ^{l}) \circ π_{δ, γ}^{A} \\ = M_{θ, a, ρ}^{A} \circ (f ⊠^{A} (1_{a} ⊠^{A} g)) \circ (1_{δ} ⊠^{A} γ^{l}) \circ π_{δ, γ}^{A} \end{array}

\begin{array}{l} = ((f ⊠^{A} 1_{a}) ⊠^{A} g) \circ M_{δ, a, γ}^{A} \circ (1_{δ} ⊠^{A} γ^{l}) \circ π_{δ, γ}^{A} \\ = ((f ⊠^{A} 1_{a}) ⊠^{A} g) \circ (δ^{r} ⊠^{A} 1_{γ}) \circ π_{δ, γ}^{A} \\ = ((f ⊠^{A} 1_{a}) \circ δ^{r} ⊠^{A} g) \circ π_{δ, γ}^{A} \\ = ((θ^{r} \circ f) ⊠^{A} g) \circ π_{δ, γ}^{A} \\ = (θ^{r} ⊠^{A} 1_{ρ}) \circ (f ⊠^{A} g) \circ π_{δ, γ}^{A} . \end{array}

□

Let $f \otimes^{A} g : δ \otimes^{A} γ \to θ \otimes^{A} ρ$ be the unique morphism that exists by the universality of the cone $θ \otimes^{A} ρ$ . For this morphism we have the commutative diagram

δ⊠A γ f-⊠A-g θ⊠A ρ
| |
πA| πA
δ,γ| |θ,ρ
A ----- A
δ⊗ γf ⊗A g θ⊗ ρ

In general $δ \otimes^{A} γ$ will not exist for all pairs of objects in $S^{A} (C)$ . In order for it to exists and have reasonable properties we need to restrict the notion relation as we have deﬁned it. Our ﬁrst restriction is to assume that $\otimes^{A}$ is deﬁned for all pairs of objects in $S^{A} (C)$ . Our second restriction involves the arrow $π_{δ, γ}^{A}$ . Let $δ, γ$ and $ρ$ be relations. We require that the morphism $π_{δ, γ}^{A} ⊠^{A} 1_{ρ}$ is mono. We have proved that $π_{δ, γ}^{A}$ is always mono, but requiring that $π_{δ, γ}^{A} ⊠^{A} 1_{ρ}$ is mono is a nontrivial restriction in general. It can be thought of a some kind of ”ﬂatness” condition on $A$ .

Given the above restrictions we can deﬁne a map $\otimes^{A} : S^{A} (C) \times S^{A} (C) \to S^{A} (C)$ by

\begin{array}{l} \otimes^{A} (δ, γ) & = δ \otimes^{A} γ, \\ \otimes^{A} (f, g) & = f \otimes^{A} g . \end{array}

Proposition 38. The map $\otimes^{A}$ is a bifunctor

Proof. Let $δ, δ^{'}, δ^{''}, γ, γ^{'}$ and $γ^{''}$ be four objects in $S^{A} (C)$ and let $f : δ \to δ^{'}, f^{'} : δ^{'} \to δ^{''}, g : γ \to γ^{'}$ and $g^{'} : γ^{'} \to γ^{''}$ be morphisms. By universality we know that the equation

π_{δ^{''}, γ^{''}} \circ ϕ = ((f^{'} \circ f) ⊠^{A} (g^{'} \circ g)) \circ π_{δ, γ}^{A}

has a unique solution. One solution is by deﬁnition $(f^{'} \circ f) \otimes^{A} (g^{'} \circ g)$ . But we also have

\begin{array}{l} π_{δ^{''}, γ^{''}}^{A} \circ ((f^{'} \otimes^{A} g^{'}) \circ (f \otimes^{A} g)) \\ = (f^{'} ⊠^{A} g^{'}) \circ π_{δ^{'}, γ^{'}}^{A} \circ (f \otimes^{A} g) \\ = (f^{'} ⊠^{A} g^{'}) \circ (f ⊠^{A} g) \circ π_{δ, γ}^{A} \\ = ((f^{'} \circ f) ⊠^{A} (g^{'} \circ g)) \circ π_{δ, γ}^{A} . \end{array}

By uniqueness we must have

(f^{'} \circ f) \otimes^{A} (g^{'} \circ g) = (f^{'} \otimes^{A} g^{'}) \circ (f \otimes^{A} g) .

Also by universality the following equation has a unique solution.

π_{δ, γ}^{A} \circ ϕ = 1_{δ ⊠^{A} γ} \circ π_{δ, γ}^{A} .

One solution is clearly $1_{δ \otimes^{A} γ}$ . But we have

\begin{array}{l} π_{δ, γ}^{A} \circ (1_{δ} \otimes^{A} 1_{γ}) \\ = (1_{δ} ⊠^{A} 1_{γ}) \circ π_{δ, γ}^{A} \\ = 1_{δ ⊠^{A} γ} \circ π_{δ, γ}^{A}, \end{array}

so by uniqueness we have $1_{δ \otimes^{A} γ} = 1_{δ} \otimes^{A} 1_{γ}$ . □

We will call $δ \otimes^{A} γ$ for the tensor product of the $A - A$ bicomodules $δ$ and $γ$ .

We deﬁned the map $\otimes^{A}$ using universal cones in the category $S^{A} (C)$ but we will now prove that it can be constructed from universal cones in the category $C$ .

Let $δ$ and $γ$ be two objects in $S^{A} (C)$ with underlying objects $B$ and $E$ in $C$ . Let the diagram $P_{B, E}^{A}$ in $C$ be given by

$B⊗(A ⊗ E)--αB,A,E-- (B⊗ A)⊗ E \ / 1B ⊗ γ\l\ / /δr⊗ 1E B⊗ E$

Assume that there exists a universal cone $〈 X, h 〉$ on the diagram $P_{B, E}^{A}$ in $C$ . Deﬁne

\begin{array}{l} Θ^{l} & = (γ_{A} \otimes 1_{X}) \circ ((1_{A} \otimes ε_{B \otimes E}) \otimes 1_{X}) \circ (α_{A, B, E}^{- 1} \otimes 1_{X}) \\ \circ ((δ^{l} \otimes 1_{E}) \otimes 1_{X}) \circ (h \otimes 1_{X}) \circ δ_{X}, \\ Θ^{r} & = (1_{X} \otimes β_{A}) \circ (1_{X} \otimes (ε_{B \otimes E} \otimes 1_{A})) \circ (1_{X} \otimes α_{B, E, A}) \\ \circ (1_{X} \otimes (1_{B} \otimes γ^{r})) \circ (1_{X} \otimes h) \circ δ_{X} . \end{array}

Proposition 39. $Θ = {Θ^{l}, Θ^{r}}$ is a $A - A$ bicomodule with underlying object $X$ .

Proof. Let us deﬁne morphisms $L, M : X \to A$ by

\begin{array}{l} L & = γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \circ h, \\ M & = β_{A} \circ (ε_{B \otimes E} \otimes 1_{A}) \circ α_{B, E, A} \circ (1_{B} \otimes γ^{r}) \circ h . \end{array}

Then $Θ^{l} = (L \otimes 1_{X}) \circ δ_{X}$ and $Θ^{r} = (1_{X} \otimes M) \circ δ_{X}$ and we have for the left structure

\begin{array}{l} α_{A, A, X} \circ (1_{A} \otimes Θ^{l}) \circ Θ^{l} \\ = α_{A, A, X} \circ (1_{A} \otimes (L \otimes 1_{X}) \circ δ_{X}) \circ (L \otimes 1_{X}) \circ δ_{X} \\ = α_{A, A, X} \circ (1_{A} \otimes (L \otimes 1_{X})) \circ (1_{A} \otimes δ_{X}) \circ (L \otimes 1_{X}) \circ δ_{X} \\ = α_{A, A, X} \circ (1_{A} \otimes (L \otimes 1_{X})) \circ (L \otimes (1_{X} \otimes 1_{X})) \circ (1_{X} \otimes δ_{X}) \circ δ_{X} \\ = α_{A, A, X} \circ (1_{A} \otimes (L \otimes 1_{X})) \circ (L \otimes (1_{X} \otimes 1_{X})) \circ α_{X; X; X}^{- 1} \circ (δ_{X} \otimes 1_{X}) \circ δ_{X} \\ = ((1_{A} \otimes L) \otimes 1_{X}) \circ ((L \otimes 1_{X}) \otimes 1_{X}) \circ (δ_{X} \otimes 1_{X}) \circ δ_{X} \\ = ((L \otimes L) \circ δ_{X} \otimes 1_{X}) \circ δ_{X} \\ = (δ_{A} \circ L \otimes 1_{X}) \circ δ_{X} \\ = (δ_{A} \otimes 1_{X}) \circ Θ^{l} . \end{array}

The proof for the right structure is similar. For the compatibility of the left and right structure we have

\begin{array}{l} (Θ^{l} \otimes 1_{A}) \circ Θ^{r} \\ = ((L \otimes 1_{X}) \circ δ_{X} \otimes 1_{A}) \circ (1_{X} \otimes M) \circ δ_{X} \\ = ((L \otimes 1_{X}) \otimes 1_{A}) \circ (δ_{X} \otimes 1_{A}) \circ (1_{X} \otimes M) \circ δ_{X} \\ = ((L \otimes 1_{X}) \otimes M) \circ (δ_{X} \otimes 1_{X}) \circ δ_{X} \\ = ((L \otimes 1_{X}) \otimes M) \circ α_{X; X; X}^{- 1} \circ (1_{X} \otimes δ_{X}) \circ δ_{X} \\ = α_{A, X, A}^{- 1} \circ (L \otimes (1_{X} \otimes M) \circ δ_{X}) \circ δ_{X} \\ = α_{A, X, A}^{- 1} \circ (1_{A} \otimes Θ^{r}) \circ Θ^{l} . \end{array}

□

We will next show that $h$ is a morphism in $S^{A} (C)$ . For this we need the following lemma

Lemma 40.

(γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes (β_{B} \circ (ε_{A} \otimes 1_{B}) \otimes 1_{E}) \circ α_{A, B, E}) \circ δ_{A \otimes (B \otimes E)} = 1_{A \otimes (B \otimes E)} .

Proof. Let $T$ be deﬁned as

\begin{array}{l} T & = ((1_{A} \otimes σ_{e, e}) \otimes 1_{B \otimes E}) \circ (α_{A, e, e}^{- 1} \otimes 1_{B \otimes E}) \circ α_{A \otimes e, e, B \otimes E} \\ \circ ((1_{A} \otimes ε_{A}) \otimes (ε_{B \otimes E} \otimes 1_{B \otimes E})) \circ (δ_{A} \otimes δ_{B \otimes E}) . \end{array}

Then we have

\begin{array}{l} (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes (β_{B} \circ (ε_{A} \otimes 1_{B}) \otimes 1_{E}) \circ α_{A, B, E}) \\ = (γ_{A} \otimes (β_{B} \otimes 1_{E})) \circ ((1_{A} \otimes ε_{B \otimes E}) \otimes ((ε_{A} \otimes 1_{B}) \otimes 1_{E})) \\ \circ (1_{A \otimes (B \otimes E)} \otimes α_{A, B, E}) \circ δ_{A \otimes (B \otimes E)} \end{array}

\begin{array}{l} = (γ_{A} \otimes (β_{B} \otimes 1_{E})) \circ (1_{A \otimes e} \otimes α_{e, B, E}) \circ ((1_{A} \otimes ε_{B \otimes E}) \otimes (ε_{A} \otimes 1_{B \otimes E})) \\ \circ α_{A \otimes (B \otimes E), A, B \otimes E}^{- 1} \circ (α_{A, B \otimes E, A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes σ_{A, B \otimes E}) \otimes 1_{B \otimes E}) \\ \circ (α_{A, A, B \otimes E}^{- 1} \otimes 1_{B \otimes E}) α_{A \otimes A, B \otimes E, B \otimes E} \circ (δ_{A} \otimes δ_{B \otimes E}) \\ = (γ_{A} \otimes (β_{B} \otimes 1_{E})) \circ (1_{A \otimes e} \otimes α_{e, B, E}) \circ α_{A \otimes e, e, B \otimes E} \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \\ = (γ_{A} \otimes (β_{B} \otimes 1_{E})) \circ α_{A \otimes e, η p B, E}^{- 1} \circ (α_{A \otimes e, e, B} \otimes 1_{E}) \circ α_{(A \otimes e) \otimes e, B, E} \\ \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \\ = α_{A, B, E}^{- 1} \circ ((γ_{A} \otimes 1_{B}) \otimes 1_{E}) \circ ((1_{A \otimes e} \otimes β_{B}) \otimes 1_{E}) \circ (α_{A \otimes e, e, B} \otimes 1_{E}) \\ \circ α_{(A \otimes e) \otimes e, B, E} \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \\ = α_{A, B, E}^{- 1} \circ ((γ_{A} \otimes 1_{B}) \otimes 1_{E}) \circ ((γ_{A \otimes e} \otimes 1_{B}) \otimes 1_{E}) \circ α_{(A \otimes e) \otimes e, B, E} \\ \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \end{array}

\begin{array}{l} = (γ_{A} \otimes 1_{B \otimes E}) \circ (γ_{A \otimes e} \otimes 1_{B \otimes E}) \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((γ_{A} \otimes 1_{e}) \otimes 1_{B \otimes E}) \circ (α_{A, e, e} \otimes 1_{B \otimes E}) \circ T \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes β_{e}) \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes σ_{e, e}) \otimes 1_{B \otimes E}) \\ \circ (α_{A, e, e}^{- 1} \otimes 1_{B \otimes E}) \circ α_{A \otimes e, e, B \otimes E} \circ ((1_{A} \otimes ε_{A}) \circ δ_{A} \otimes (ε_{B \otimes E} \otimes 1_{B \otimes E}) \circ δ_{B \otimes E}) \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((1_{A} \otimes γ_{e}) \otimes 1_{B \otimes E}) \circ (α_{A, e, e}^{- 1} \otimes 1_{B \otimes E}) \circ α_{A \otimes e, e, B \otimes E} \\ \circ (γ_{A}^{- 1} \otimes β_{B \otimes E}^{- 1}) \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ ((γ_{A} \otimes 1_{e}) \otimes 1_{B \otimes E}) \circ α_{A \otimes e, e, B \otimes E} \circ (γ_{A}^{- 1} \otimes β_{B \otimes E}^{- 1}) \\ = (γ_{A} \otimes 1_{B \otimes E}) \circ α_{A, e, B \otimes E} \circ (γ_{A} \otimes (1_{e} \otimes 1_{B \otimes E})) \circ (γ_{A}^{- 1} \otimes β_{B \otimes E}^{- 1}) \\ = (γ_{A} \otimes β_{B \otimes E}) \circ (γ_{A}^{- 1} \otimes β_{B \otimes E}^{- 1}) \\ = 1_{A \otimes (B \otimes E)} . \end{array}

□

We can now prove that $h$ is a morphism in $S^{A} (C)$ .

Proposition 41. $h$ deﬁnes a monomorphism in $S^{A} (C)$ with domain $Θ$ and codomain $δ ⊠^{A} γ$

Proof. The fact that $h$ is a monomorphism in $C$ follows from the universality as it did for $π_{δ, γ}^{A}$ in proposition 36.

For the left structure we have

\begin{array}{l} (1_{A} \otimes h) \circ Θ^{l} \\ (1_{A} \otimes h) \circ (L \otimes 1_{X}) \circ δ_{X} \\ = (L \otimes 1_{B \otimes E}) \circ (1_{X} \otimes h) \circ δ_{X} \\ = (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes 1_{B \otimes E}) \circ (α_{A, B, E}^{- 1} \otimes 1_{B \otimes E}) \circ ((δ^{l} \otimes 1_{E}) \otimes 1_{B \otimes E}) \\ \circ (h \otimes h) \circ δ_{X} \end{array}

\begin{array}{l} = (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes 1_{B \otimes E}) \circ (α_{A, B, E}^{- 1} \otimes 1_{B \otimes E}) \circ ((δ^{l} \otimes 1_{E}) \otimes 1_{B \otimes E}) \\ \circ δ_{B \otimes E} \circ h \\ = (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes 1_{B \otimes E}) \circ (α_{A, B, E}^{- 1} \otimes 1_{B \otimes E}) \\ \circ ((1_{A \otimes B} \otimes 1_{E}) \otimes (β_{B} \otimes 1_{E}) \circ ((ε_{A} \otimes 1_{B}) \otimes 1_{E})) \circ ((δ^{l} \otimes 1_{E}) \otimes (δ^{l} \otimes 1_{E})) \\ \circ δ_{B \otimes E} \circ h \\ = (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes (β_{B} \circ (ε_{A} \otimes 1_{B}) \otimes 1_{E}) \circ α_{A, B, E}) \circ (α_{A, B, E}^{- 1} \otimes α_{A, B, E}^{- 1}) \\ \circ δ_{(A \otimes B) \otimes E} \circ (δ^{l} \otimes 1_{E}) \circ h \\ = (γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \otimes (β_{B} \circ (ε_{A} \otimes 1_{B}) \otimes 1_{E}) \circ α_{A, B, E}) \circ δ_{A \otimes (B \otimes E)} \\ \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \circ h \\ = α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \circ h \\ = {(δ ⊠^{A} γ)}^{l} \circ h . \end{array}

In a similar way we show the identity $(h \otimes 1_{A}) \circ Θ^{r} = {(δ ⊠^{A} γ)}^{r} \circ h$ . □

Proposition 42. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, α 〉$ be external. Then $〈 Θ, h 〉$ is a universal cone on $P_{δ, γ}^{A}$ .

Proof. It is evident that $〈 Θ, h 〉$ is a cone on the diagram $P_{δ, γ}^{A}$ . Let $〈 θ, u 〉$ be any cone on $P_{δ, γ}^{A}$ . Let $ϕ, ψ : θ \to Θ$ be two morphisms in $S^{A} (C)$ such that

\begin{array}{l} h \circ ϕ & = θ, \\ h \circ ψ & = θ . \end{array}

Then $h \circ ϕ = h \circ ψ$ and since $h$ is mono we have $ϕ = ψ$ . Therefore the equation $h \circ ϕ = θ$ has at most one solution.

The fact that $〈 θ, u 〉$ is a cone gives us the relation

M_{δ, a, γ}^{A} \circ (1_{δ} ⊠^{A} γ^{l}) \circ u = (δ^{l} ⊠^{A} 1_{γ}) \circ u .

If the underlying objects for $δ, γ$ and $θ$ are $B, E$ and $D$ , then $u : D \to B \otimes E$ and the previous identity corresponds to the following

identity in $C$

(1_{B} \otimes γ^{l}) \circ u = (δ^{l} \otimes 1_{E}) \circ u

and therefore $〈 D, u 〉$ is a cone on the diagram $P_{B, E}^{A}$ in $C$ . By universality there exists a unique morphism $ϕ : D \to X$ in $C$ such that

h \circ ϕ = u .

The fact that $ϕ$ is a morphism in $C$ and $u$ and $δ_{D}$ are morphisms in $S^{A} (C)$ gives us the following four commutative diagrams

u
D-------- B ⊗|E
| |
θl δl⊗ 1E
| |
A⊗D ----- A⊗ B ⊗E
1A⊗ u

δ
D|-----D--- D ⊗|D
| |
θl| δl⊗ 1D
| |
A ⊗ D ------ A⊗ D ⊗D
1A⊗ δD

D ---φ--- X
| |
| |
δD δX
| ---- |
D⊗D φ⊗ φ X ⊗ X

$D -------u-------B ⊗ E \ / εD\ /εB⊗E \ / e$

If we deﬁne $L$ as in proposition 39 we have the following identities

\begin{array}{l} L \circ ϕ \\ = γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \circ h \circ ϕ \\ = γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \circ u \\ = γ_{A} \circ (1_{A} \otimes ε_{B \otimes E}) \circ (1_{A} \otimes u) \circ θ^{l} \\ = γ_{A} \circ (1_{A} \otimes ε_{D}) \circ θ^{l} . \end{array}

But then we have

\begin{array}{l} Θ^{l} \circ ϕ \\ = (L \otimes 1_{X}) \circ δ_{X} \circ ϕ \\ = (L \otimes 1_{X}) \circ (ϕ \otimes ϕ) \circ δ_{D} \\ = (L \circ ϕ \otimes ϕ) \circ δ_{D} \\ = (1_{A} \otimes ϕ) \circ (γ_{A} \otimes 1_{D}) \circ ((1_{A} \otimes ε_{D}) \otimes 1_{D}) \circ (θ^{l} \otimes 1_{D}) \circ δ_{D} \\ = (1_{A} \otimes ϕ) \circ (γ_{A} \otimes 1_{D}) \circ α_{A, e, D} \circ (1_{A} \otimes (ε_{D} \otimes 1_{D})) \circ (1_{A} \otimes δ_{D}) \circ θ^{l} \\ = (1_{A} \otimes ϕ) \circ (1_{A} \otimes β_{D} \circ (ε_{D} \otimes 1_{D}) \circ δ_{D}) \circ θ^{l} \\ = (1_{A} \otimes ϕ) \circ θ^{l} . \end{array}

In a similar way we prove the identity $Θ^{r} \circ ϕ = (ϕ \otimes 1_{A}) \circ θ^{r}$ . This proves that $ϕ$ is a morphism in $S^{A} (C)$ and therefore that the equation

h \circ ϕ = u

has a unique solution in $S^{A} (C)$ . □

This proposition show that $δ \otimes^{A} γ \approx Θ$ since universal cones are determined up to isomorphism. This is the way the tensor product is usually computed.

We now have two bifunctors $⊠^{A}$ and $\otimes^{A}$ deﬁned on $S^{A} (C)$ . These two structures are related at the functorial level as the next proposition show.

Proposition 43. $π_{δ, γ}^{A}$ are the components of a natural monomorphism

π^{A} : \otimes^{A} \to ⊠^{A} .

Proof. The proposition follows directly from the commutative diagram 3.6. □

3.7. Monoidal structures on the category of relations. We have seen that $⊠^{A}$ deﬁnes a semimonoidal structure on $S^{A} (C)$ with associativity constraint $M^{A}$ . We will in the following only consider the case when the product $\otimes^{A}$ deﬁnes a monoidal structure on $S^{A} (C)$ with neutral object $a$ . This is a further restriction on the category $S^{A} (C)$ and thus on the category of relations. Recall that the pair $\otimes^{A}, a$ deﬁnes a monoidal structure on $S^{A} (C)$ if for all objects $δ, γ$ and $ρ$ there exists isomorphisms

\begin{array}{l} m_{δ, γ, ρ}^{A} & : δ \otimes^{A} (γ \otimes^{A} ρ) \to (δ \otimes^{A} γ) \otimes^{A} ρ, \\ l_{δ}^{A} & : a \otimes^{A} δ \to δ, \\ r_{δ}^{A} & : δ \otimes^{A} a \to δ, \end{array}

that are natural in $δ, γ$ and $ρ$ and such that the MacLane coherence conditions are satisﬁed. The coherence conditions are a set of equations for the morphisms $m^{A}, l^{A}$ and $r^{A}$ and these equations may have no solutions, a unique solution or many solutions depending on the category $C$ and the coalgebra $A$ .

Deﬁnition 44. A monoidal structure $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ on the category $S^{A} (C)$ is induced if for all objects $δ, γ$ and $ρ$ in $S^{A} (C)$ the following diagrams commute

$a⊗A δ------lAδ-------- δ \ / πA\ /δl a,δ \ / a⊠A δ$ $δ ⊗Aa ------rAδ-------- δ \ / πA\ /δr δ,a \ / δ⊠Aa$

A A MAδ,γ,ρ A
δ⊠ (γ⊠ ρ) (δ⊠ γ)⊠ρ
| |
1δ⊠A πAγ,ρ| πAδ,γ ⊗ 1ρ

δ⊠A (γ⊗A ρ) (δ⊗A γ)⊠A ρ
| |
πA A | πA A
δ,γ⊗ ρ| |δ⊗ γ,ρ
δ⊗A (γ⊗A ρ) ----- (δ⊗A γ)⊗A ρ
mAδ,γ,ρ

We will in the following derive a necessary and sufficient condition for induced constraints to exist in the external case. Let us assume that the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ is external. Let $P_{a, δ}^{A}$ be the diagram

$A A MAa,a,δ A a⊠ (a⊠ δ)-------- (δ⊠ a)⊠ γ \ / A \l / A 1a⊠ δ \ / δA ⊠ 1δ a⊠δ$

Then $〈 δ, δ^{l} 〉$ is clearly a cone on this diagram since this is equivalent to the condition that $δ^{l}$ is a left comodule structure on the underlying object of $δ$ . But we have also a stronger condition.

Proposition 45. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ be external. Then $〈 δ, δ^{l} 〉$ is a universal cone on $P_{a, δ}^{A}$ .

Proof. In order to prove that $〈 δ, δ^{l} 〉$ is a universal cone we must show that the equation

δ^{l} \circ ϕ = f,

has a unique solution $ϕ : γ \to δ$ for any $f : γ \to a ⊠^{A} δ$ such that

(δ_{A} \otimes 1_{δ}) \circ f = M_{a, a, δ}^{A} \circ (1_{a} \otimes δ^{l}) \circ f .

Since $β_{B} \circ (ε_{A} \otimes 1_{B}) \circ δ^{l} = 1_{B}$ the equation can have only one solution and this solution must be

ϕ = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ f .

The universality is proved if we can show that this is in fact a solution and also a morphism in $S^{A} (C) .$

\begin{array}{l} δ^{l} \circ ϕ \\ = δ^{l} \circ β_{B} \circ (ε_{A} \otimes 1_{B}) \circ f \\ = β_{A \otimes B} \circ (1_{e} \otimes δ^{l}) \circ (ε_{A} \otimes 1_{B}) \circ f \\ = β_{A \otimes B} \circ (ε_{A} \otimes (1_{A} \otimes 1_{B})) \circ (1_{A} \otimes δ^{l}) \circ f \\ = β_{A \otimes B} \circ (ε_{A} \otimes (1_{A} \otimes 1_{B})) \circ α_{A, A, B}^{- 1} \circ (δ_{A} \otimes 1_{B}) \circ f \\ = β_{A \otimes B} \circ α_{e, A, B}^{- 1} \circ (β_{A}^{- 1} \otimes 1_{B}) \circ f \\ = f, \end{array}

so $ϕ$ is a solution. Here we have used the identity $β_{A \otimes B} = (β_{A} \otimes 1_{B}) \circ α_{e, A, B}$ . By construction $ϕ$ is an arrow in $C$ , but we also have

\begin{array}{l} (1_{A} \otimes ϕ) \circ γ^{l} \\ = (1_{A} \otimes β_{B}) \circ (1_{A} \otimes (ε_{A} \otimes 1_{B})) \circ (1_{A} \otimes f) \circ γ^{l} \\ = (1_{A} \otimes β_{B}) \circ (1_{A} \otimes (ε_{A} \otimes 1_{B})) \circ (1_{A} \otimes δ^{l}) \circ f \\ = (1_{A} \otimes β_{B}) \circ (1_{A} \otimes (ε_{A} \otimes 1_{B})) \circ α_{A, A, B}^{- 1} \circ (δ_{A} \otimes 1_{B}) \circ f \\ = (1_{A} \otimes β_{B}) \circ α_{A, e, B}^{- 1} \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (δ_{A} \otimes 1_{B}) \circ f \\ = (γ_{A} \otimes 1_{B}) \circ (γ_{A}^{- 1} \otimes 1_{B}) \circ f \\ = f \\ = δ^{l} \circ ϕ, \end{array}

so $ϕ$ is a morphism in $S^{A} (C)$ . □

We have the following two corollaries to the previous proposition

Corollary 46. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ be external and let the underlying object for $δ$ be $B$ . If induced unit constraints exists they must be of the form

\begin{array}{l} l_{δ}^{A} & = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ π_{a, δ}^{A}, \\ r_{δ}^{A} & = γ_{B} \circ (1_{B} \otimes ε_{A}) \circ π_{δ, a}^{A} . \end{array}

Corollary 47. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ be external. Then the morphism $δ^{l} : δ \to a \otimes^{A} δ$ is a monomorphism.

Proof. Let $γ$ be any object in $S^{A} (C)$ and let $f, g : γ \to δ$ be any pair of morphisms. Assume that $δ^{l} \circ f = δ^{l} \circ g$ and deﬁne $h = δ^{l} \circ g$ . Then both $f$ and $h$ satisfy the equation

δ^{l} \circ ϕ = h .

By universality we can conclude that $f = g$ . □

Proposition 48. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ be external. Then induced unit and associativity constraints are unique if they exist.

Proof. By deﬁnition an external unit constraint $l_{δ}^{A}$ is a solution of the equation

δ^{l} \circ l_{δ}^{A} = π_{a, δ}^{A} .

But by universality this equation has a unique solution. The uniqueness of $r_{δ}^{A}$ is proved in a similar way. For $m_{r, s, t}^{A}$ we note that the morphism $t = (π_{δ, γ}^{A} \otimes 1_{ρ}) \circ π_{δ \otimes^{A} γ, ρ}^{A}$ is mono. Let $f$ and $g$ be two morphisms such that the third diagram in deﬁnition 44 commutes. Then we have

\begin{array}{l} t \circ f & = (π_{δ, γ}^{A} \otimes 1_{ρ}) \circ π_{δ \otimes^{A} γ, ρ}^{A} \circ f = α_{B, E, D} \circ (1_{δ} \otimes π_{γ, ρ}^{A}) \circ π_{δ, γ \otimes^{A} ρ}^{A}, \\ t \circ g & = (π_{γ, δ}^{A} \otimes 1_{ρ}) \circ π_{δ \otimes^{A} γ, ρ}^{A} \circ g = α_{B, E, D} \circ (1_{δ} \otimes π_{γ, ρ}^{A}) \circ π_{δ, γ \otimes^{A} ρ}^{A}, \end{array}

so $t \circ f = t \circ g$ . But $t$ is mono and therefore $f = g$ □

We can now give sufficient conditions for the existence of induced unit and associativity constraints.

Theorem 49. Let the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ be external and assume that for all $δ, γ$ and $ρ$ there exists a isomorphism $m_{δ, γ, ρ}^{A} : δ \otimes^{A} (γ \otimes^{A} ρ) \to (δ \otimes^{A} γ) \otimes^{A} ρ$ such that the following diagram commute.

MA
δ⊠A (γ⊠A ρ) --δ,γ,ρ (δ⊠A γ)⊠ρ
| |
1δ⊠A πAγ,ρ| πAδ,γ ⊗ 1ρ
| |
δ⊠A (γ⊗A ρ) (δ⊗A γ)⊠A ρ
| |
A | |A
πδ,γ⊗Aρ| πδ⊗Aγ,ρ

δ⊗A (γ⊗A ρ) mA--- (δ⊗A γ)⊗A ρ
δ,γ,ρ

Then a induced monoidal structure $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ on the category $S^{A} (C)$ exists.

Proof. Let us ﬁrst prove that $m_{δ, γ, ρ}^{A}$ are the components of a natural isomorphism. Let $δ^{'}, γ^{'}$ and $ρ^{'}$ be three other relations and let $f : δ \to δ^{^{'}}, g : γ \to γ^{'}$ and $h : ρ \to ρ^{'}$ be three morphisms. For any three relations $δ, γ$ and $ρ$ deﬁne $ϕ_{δ, γ, ρ} = (π_{δ, γ}^{A} \otimes 1_{ρ}) \circ π_{δ \otimes^{A} γ, ρ}^{A}$ and $ψ_{δ, γ, ρ} = (1_{δ} \otimes^{A} π_{γ, ρ}^{A}) \circ π_{δ, γ \otimes^{A} ρ}^{A}$ . Then we have

\begin{array}{l} ϕ_{δ^{'}, γ^{'}, ρ^{'}} \circ ((f \otimes^{A} g) \otimes^{A} h) \circ m_{δ, γ, ρ}^{A} \\ = ((f ⊠^{A} g) ⊠^{A} h) \circ ϕ_{δ, γ, ρ} \circ m_{δ, γ, ρ}^{A} \\ = ((f ⊠^{A} g) ⊠^{A} h) \circ M_{δ, γ, ρ}^{A} \circ ψ_{δ, γ, ρ} \\ = M_{δ^{'}, γ^{'}, ρ^{'}}^{A} \circ (f ⊠^{A} (g ⊠^{A} h)) \circ ψ_{δ, γ, ρ} \\ = M_{δ^{'}, γ^{'}, ρ^{'}}^{A} \circ ψ_{δ^{'}, γ^{'}, ρ^{'}} \circ (f \otimes^{A} (g \otimes^{A} h)) \\ = ϕ_{δ^{'}, γ^{'}, ρ^{'}} \circ m_{δ^{'}, γ^{'}, ρ^{'}}^{A} \circ (f \otimes^{A} (g \otimes^{A} h)) . \end{array}

From this the naturality of $m_{δ, γ, ρ}^{A}$ follows because $ϕ_{δ^{'}, γ^{'}, ρ^{'}}$ is mono. We have thus far proved that we have a natural isomorphism

m_{δ, γ, ρ}^{A} : δ \otimes^{A} (γ \otimes^{A} ρ) \to (δ \otimes^{A} γ) \otimes^{A} ρ .

A induced left unit constraint is a natural isomorphism in $S^{A} (C)$ that satisfy the equation

δ^{l} \circ ϕ = π_{a, δ}^{A} .

By deﬁnition $a \otimes^{A} δ$ is a universal cone on the diagram $P_{a, δ}^{A}$ . But $δ^{l}$ is also a universal cone on this diagram so there exists an isomorphism $ϕ : a \otimes^{A} δ \to δ$ such that $δ^{l} \circ ϕ = π_{a, δ}^{A}$ . We have seen that the only solution of this equation in $S^{A} (C)$ is given by $l_{δ}^{A} = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ π_{a, δ}^{A}$ where the underlying object for $δ$ is $B$ . We can therefore conclude that $l_{δ}^{A} : a \otimes^{A} δ \to δ$ is an isomorphism. This isomorphism is natural because if $δ$ and $δ^{'}$ are objects in $S^{A} (C)$ with underlying objects $B$ and $B^{'}$ and $f : δ \to δ^{'}$ is any morphism the naturality of $β$ and $π^{A}$ give

\begin{array}{l} f \circ l_{δ}^{A} \\ = f \circ β_{B} \circ (ε_{A} \otimes 1_{B}) \circ π_{a, δ}^{A} \\ = β_{B^{'}} \circ (1_{e} \otimes f) \circ (ε_{A} \otimes 1_{B}) \circ π_{a, δ}^{A} \\ = β_{B^{'}} \circ (ε_{A} \otimes f) \circ π_{a, δ}^{A} \\ = β_{B^{'}} \circ (ε_{A} \otimes 1_{B^{'}}) \circ (1_{A} \otimes f) \circ π_{a, δ}^{A} \\ = β_{B^{'}} \circ (ε_{A} \otimes 1_{B^{'}}) \circ π_{a, δ^{'}}^{A} \circ f \\ = l_{δ^{'}}^{A} \circ f . \end{array}

In a similar way we ﬁnd a natural isomorphism $r_{δ}^{A} = γ_{B} \circ (1_{B} \otimes ε_{A}) \circ π_{δ, a}^{A}$ . The proposition is proved if we can show that these three natural isomorphisms satisfy the MacLane coherence conditions for a monoidal category. We clearly have

a^{l} \circ l_{a}^{A} = π_{a, a}^{A} = a^{r} \circ r_{a}^{A} .

But $a^{l} = δ_{A} = a^{r}$ and $δ_{A}$ is a monomorphism so we have $l_{a}^{A} = r_{a}^{A}$ . This is the third MacLane condition. Let us now consider the second MacLane condition. Let $δ$ and $γ$ have underlying objects $B$ and $E$ . Using the formulas for $l_{δ}^{A}, r_{δ}^{A}$ and the deﬁnition of $m_{δ, γ, ρ}^{A}$ we ﬁnd

\begin{array}{l} π_{δ, γ}^{A} \circ (r_{δ}^{A} \otimes^{A} 1_{γ}) \circ m_{δ, a, γ}^{A} \\ = (r_{δ}^{A} \otimes 1_{E}) \circ π_{δ \otimes^{A} a, γ}^{A} \circ m_{δ, a, γ}^{A} \\ = (γ_{B} \circ (1_{B} \otimes ε_{A}) \circ π_{δ, a}^{A} \otimes 1_{E}) \circ π_{δ \otimes^{A} a, γ}^{A} \circ m_{δ, a, γ}^{A} \\ = (γ_{B} \otimes 1_{E}) \circ ((1_{B} \otimes ε_{A}) \otimes 1_{E}) \circ (π_{δ, a}^{A} ⊠^{A} 1_{γ}) \circ π_{δ \otimes^{A} a, γ}^{A} \circ m_{δ, a, γ}^{A} \\ = (γ_{B} \otimes 1_{E}) \circ ((1_{B} \otimes ε_{A}) \otimes 1_{E}) \circ α_{B, A, E} \circ (1_{δ} ⊠^{A} π_{a, γ}^{A}) \circ π_{δ, a \otimes^{A} γ}^{A} \end{array}

\begin{array}{l} = (γ_{B} \otimes 1_{E}) \circ α_{B, e, E} \circ (1_{B} \otimes (ε_{A} \otimes 1_{E})) \circ (1_{δ} ⊠^{A} π_{a, γ}^{A}) \circ π_{δ, a \otimes^{A} γ}^{A} \\ = (1_{B} \otimes β_{E}) \circ (1_{B} \otimes (ε_{A} \otimes 1_{E})) \circ (1_{B} \otimes π_{a, γ}^{A}) \circ π_{δ, a \otimes^{A} γ}^{A} \\ = (1_{B} \otimes β_{E} \circ (ε_{A} \otimes 1_{E}) \circ π_{a, γ}^{A}) \circ π_{δ, a \otimes^{A} γ}^{A} \\ = (1_{δ} ⊠^{A} l_{γ}^{A}) \circ π_{δ, a \otimes^{A} γ}^{A} \\ = π_{δ, γ}^{A} \circ (1_{δ} \otimes^{A} l_{γ}^{A}) \end{array}

and

π_{δ, γ}^{A}

is mono so we have

(r_{δ}^{A} \otimes^{A} 1_{γ}) \circ m_{δ, a, γ}^{A} = (1_{δ} \otimes^{A} l_{γ}^{A})

and this is the second MacLane condition. The ﬁrst MacLane condition follow from the assumptions in the Theorem and the fact that $⊠^{A}$ is a semimonoidal structure on $S^{A} (C)$ with associativity constraint $M^{A}$ . □

Since $S^{A} (C)$ is isomorphic to the category of relations a monoidal structure on $S^{A} (C)$ will induce one on the category of relations. Let the product in $ℛ^{A} (C)$ corresponding to $\otimes^{A}$ be $⊙^{A} : ℛ^{A} (C) \times ℛ^{A} (C) \to ℛ^{A} (C)$ . We thus have

⊙^{A} = Ψ \circ \otimes^{A} \circ (Φ \times Φ) .

We have the following explicit expression for the product

Proposition 50. For any pair of objects $r$ and $s$ in $ℛ^{A} (C)$ we have

r ⊙^{A} s = (γ_{A} \otimes β_{A}) \circ (1_{A} \otimes ε_{A} \otimes ε_{A} \otimes 1_{A}) \circ (r \otimes s) \circ π_{Φ (r), Φ (s)}^{A} .

Proof. We have a natural monomorphism

π^{A} : \otimes^{A} \to ⊠^{A} .

Since by deﬁnition $⊡^{A} = Ψ \circ ⊠^{A} \circ (Φ \times Φ)$ and $⊙^{A} = Ψ \circ \otimes^{A} \circ (Φ \times Φ)$ , horizontal composition of natural transformations give us a natural transformation

(1_{Ψ} \circ π^{A} \circ (1_{Φ} \times 1_{Φ})) : ⊙^{A} \to ⊡^{A} .

If we evaluate this natural transformation at a pair of objects $r$ and $s$ in $ℛ^{A} (C)$ we get the following morphism in $ℛ^{A} (C)$

π_{Φ (r), Φ (s)}^{A} : r ⊙^{A} s \to r ⊡^{A} s .

But this means that

r ⊙^{A} s = (r ⊡^{A} s) \circ π_{Φ (r), Φ (s)}^{A}

and this is the formula in the proposition if we take into account the formula for $r ⊡^{A} s$ that we have derived earlier. □

We will now consider a few examples of the tensor product. Let us ﬁrst assume that the underlying category $C$ is $S e t s$ with its unique choice of natural $C$ -category $C$ . We have seen that all possible $A - A$ bicomodule structures $δ = {δ^{l}, δ^{r}}$ on a set $B$ are of the form $δ^{l} (x) = (f (x), x)$ and $δ^{r} (x) = (x, g (x))$ for some functions $f, g : B \to A$ . The relation on $A$ corresponding to $δ$ is clearly given by $r (x) = (f (x), g (x))$ . Let now $r (x) = (f (x), g (x))$ and $s (y) = (h (y), k (y))$ be two relations with domains $B$ and $E$ and let the corresponding $A - A$ bicomodules be $δ$ and $γ$ . The two maps $(δ^{r} \times 1_{E})$ and $(1_{B} \times γ^{l})$ are given by

\begin{array}{l} (δ^{r} \times 1_{E}) (x, y) & = (x, g (x), y), \\ (1_{B} \times γ^{l}) (x, y) & = (x, h (y), y) . \end{array}

In $S e t s$ the underlying object $X$ for the $A - A$ bicomodule $δ \otimes^{A} γ$ is the equalizer of the two given maps. We therefore ﬁnd that

X = {(x, y) ∣ g (x) = h (y)}

and

\begin{array}{l} {(δ \otimes^{A} γ)}^{l} (x, y) & = (f (x), x, y), \\ {(δ \otimes^{A} γ)}^{r} (x, y) & = (x, y, x, k (y)) . \end{array}

The map $π_{δ, γ}^{A} : X \to B \times E$ is the inclusion map. The relation $r ⊙^{A} s$ corresponding to $δ \otimes^{A} γ$ is then given by $(r ⊙^{A} s) (x, y) = (f (x), k (y))$ .

We have seen that each relation $r$ and $s$ is in fact a directed labelled graph. Each element in $B$ can be thought of as an arrow that has a source and a target in the vertex set $A$ for the graph and similarly for elements in $E$ . Let us deﬁne two arrows to be composable if the target of the ﬁrst is the same as the source of the second. The set $X$ then consists of all composable pairs of arrows from $B$ and $E$ . Two relations on $A$ , $B \subset A \times A$ and $E \subset A \times A$ , in the usual sense corresponds to relations $r$ and $s$ in our sense if we let $r (x, x^{'}) = (x, x^{'})$ and $s (y, y^{'}) = (y, y^{'})$ be the inclusion maps. If we use the same notation as above we ﬁnd $f (x, x^{'}) = x, g (x, x^{'}) = x^{'}, h (y, y^{'}) = y$ and $k (y, y^{'}) = y^{'}$ .

For this special case we ﬁnd

\begin{array}{l} X = {((x, y), (y, y^{'}))}, \\ (r ⊙^{A} s) ((x, y), (y, y^{'})) = (x, y^{'}) . \end{array}

We then observe that

(r ⊙^{A} s) (X) = B \circ E,

where $B \circ E$ is the usual composition of relations.

Let $t : D \to A \times A$ be a third relation with $t (z) = (p (z), q (z))$ and let $ρ$ be the $A - A$ bicomodule corresponding to $t$ . Let $X$ be the underlying object for $(δ \otimes^{A} γ) \otimes^{A} ρ$ and $Y$ the underlying object for $δ \otimes^{A} (γ \otimes^{A} ρ)$ . Direct calculation show that

\begin{array}{l} X & = {((x, y), z) ∣ g (x) = h (y), k (y) = p (z)}, \\ Y & = {(x, (y, z)) ∣ g (x) = h (y), k (y) = p (z)} . \end{array}

Deﬁne

\begin{array}{l} m_{δ, γ, ρ}^{A} (x, (y, z)) & = ((x, y), z), \\ l_{δ}^{A} (a, x) & = x, \\ r_{δ}^{A} (x, a) & = x . \end{array}

It is easy to see that $m_{δ, γ, ρ}^{A}$ is a morphism of relations. The underlying object for $a \otimes^{A} δ$ is easily seen to given by

Z = {(f (x), x) ∣ x \in B} .

Therefore $l_{δ}^{A}$ is clearly a isomorphism and

\begin{array}{l} (δ^{l} \circ l_{δ}^{A}) (f (x), x) \\ = δ^{l} (x) \\ = (f (x), x) \\ = π_{a, δ}^{A} (f (x), x) . \end{array}

In a similar way we show that $r_{δ}^{A}$ is a isomorphism that satisfy $δ^{r} \circ r_{δ}^{A} = π_{δ, a}^{A}$ . It is easy to see that $ω_{r, s, t}^{A}, β_{r}^{A}$ and $γ_{r}^{A}$ satisfy the MacLane coherence conditions. They are therefore the associativity and unit constraints for a monoidal structure $\otimes^{A}$ on $S^{A} (C)$ . A simple calculation show that they are induced.

In a similar way we can deﬁne products of any number of relations. It is evident that the product of $n$ relations consists of strings of composable arrows of length $n$ , one arrow from each relation. Note that $δ_{A}$ is a relation on $A$ . Let us assume that there exists a morphism of relations $f : δ_{A} \to r$ . This means that for each element $a \in A$ there exists a element $b_{a} = f (a)$ in $B$ such that $a$ is both the source and target of $b_{a}$ . If we now take all possible products of the relation $r$ we observe that the result is in fact the (internal) category generated by the graph deﬁned by the relation.

Let us next consider $V e c t_{k}$ with $\oplus$ as monoidal structure. Let $A$ be a linear space and let $r : B \to A \oplus A$ and $s : E \to A \oplus A$ be relations on $A$ . The domain for the product $r ⊙^{A} s$ is a linear subspace of $B \oplus E$

\begin{array}{l} V & = {(u, v) ∣ g (u) = h (v)}, \\ (r ⊙^{A} s) (u, v) & = (f (u), k (v)) . \end{array}

where $r (u) = (f (u), g (u))$ and $s (v) = (h (v), k (v))$ . Let $L : A \to A$ and $S : A \to A$ be two endomorphism of $A$ and let $B = E = A$ and $r : B \to A \oplus A$ $s : E \to A \oplus A$ be relations where $r (a) = (a, L (a))$ and $s (a) = (a, S (a))$ . We then have $f (a) = a, g (a) = L (a), h (a) = a$ and $k (a) = S (a)$ . Therefore the underlying object for $r ⊙^{A} s$ is $X = {(a, L (a))}$ and

(r ⊙^{A} s) (a, L (a)) = (a, (S \circ L) (a)),

so the image of $X$ in $A \oplus A$ is the graph of the composition of $L$ and $S$ . More generally let $L \subset A \oplus A$ and $S \subset A \oplus A$ be two linear subspaces and let $r : L \to A \oplus A$ and $s : S \to A \oplus A$ be the corresponding relations with $r$ and $s$ the inclusion maps. Then the image of the product relation of $L$ and $S$ in $A \oplus A$ is formed by selecting vectors in $u \in L$ , decomposing them as $u = a + b$ with $a, b \in A$ , selecting vectors $v \in S$ with decomposition $v = b + c$ and ﬁnally forming the vectors $w = a + c$ .

A monoid in the category of relations is a relation $r$ and two morphisms of relations

\begin{array}{l} μ_{r} & : r ⊙^{A} r \to r, \\ u_{r} & : δ_{A} \to r, \end{array}

such that the associativity and unit diagrams commute. Let us consider the case when the basic category is $S e t s$ . Then we have seen that a relation is a graph with vertex set $A$ and arrow set $B$ . The source and target for any arrow $x$ is given by $f (x), g (x) \in A$ where $r (x) = (f (x), g (x))$ . The domain $X$ for the relation $r ⊙^{A} r$ consists of all composable pairs of arrows from the graph $B$ ,

X = {(x, x^{'}) ∣ g (x) = f (x^{'})} .

The map $μ_{r}$ will deﬁne an associative rule of composition for composable pair of arrows in $B$ . Furthermore the unit map will provide for each vertex $a \in A$ an arrow $u (a)$ that has $a$ as both source and target and that acts as left and right unit for composition. The structure we have described is clearly a internal category in $S e t s$ with objects $A$ and arrows $B$ . Let $B \subset A \times A$ be a transitive and reﬂexive relation in the usual sense. If we deﬁne $r ((x, y)) = (x, y)$ then $r$ is clearly a relation in our sense. We have seen that the domain of the relation $r \otimes^{A} r$ is of the form

X = {((x, y), (y, z)) ∣ (x, y) \in B, (y, z) \in B},

and $(r \otimes^{A} r) (((x, y), (y, z))) = (x, z)$ . Deﬁne a map of sets $μ_{r} : X \to A \times A$ by $μ_{r} (((x, y), (y, z))) = (x, z)$ . But both $(x, y) \in B$ and $(y, z) \in B$ and since $B$ is transitive we have $(x, z) \in B$ and so we have in fact $μ_{r} : X \to B$ . But we also have

\begin{array}{l} (r \circ μ_{r}) (((x, y), (y, z))) \\ = r ((x, z)) \\ = (x, z) \\ = (r \otimes^{A} r) (((x, y), (y, z))), \end{array}

so $μ_{r} : r \otimes^{A} r \to r$ . The map $μ_{r}$ is clearly associative. Deﬁne a map of sets $u_{r} : A \to A \times A$ by $u_{r} (x) = (x, x)$ . Since the relation $B$ is reﬂexive we have $(x, x) \in B$ for all $x \in A$ and therefore we have $u_{r} : A \to B$ . This map is clearly a morphism of relations and acts as a left and right unit for the rule of composition $μ_{r}$ . We therefore have proved that the relation in our sense,corresponding to a reﬂexive and transitive relation in the usual sense, is in fact a monoid in the category of relations. We can thus think of monoids in $ℛ^{A} (C)$ as generalized reﬂexive and transitive relations or generalized categories.

3.8. Symmetries for the category of relations. From an algebraic point of view we know that commutative monoids is an important and interesting subclass of all monoids. From a categorical point of view the notion of commutativity can not be formulated unless there is a symmetry deﬁned on the category.

Let us therefore consider the notion of a symmetry for the category of relations. In $S e t s$ a relation $r$ with domain $B$ is a labelled and directed graph with vertex set $A$ and arrow set $B$ . We have seen that the tensor product of two relations $r$ and $s$ with domains $B$ and $E$ is a new graph on the vertex set $A$ where the set of arrows consists of all composable pairs of arrows from $B$ and $E$ . If $(x, y)$ with $x \in B$ and $y \in E$ is a composable pair of arrows it is clear that in general the pair $(y, x)$ is not a composable pair. It is thus evident that for relations the simple transposition $(x, y) \to (y, x)$ is not the right notion for a symmetry. We will develop our theory for the category $S^{A} (C)$ and use the isomorphism whenever we need the corresponding structures in the category of relations. Dual properties holds for the categories $S_{A} (C)$ and the category of $C$ -corelations.

Before we give the right deﬁnition of symmetry for the category of relations we need to introduce a new structure. Let $δ = {δ^{l}, δ^{r}}$ be an object in $S^{A} (C)$ . Let $r = Ψ (δ)$ be the corresponding relation and deﬁne $r^{*} = σ_{A, A} \circ r$ and

δ^{*} = Φ (r^{*}) .

An explicit expression for the new object $δ^{*}$ is given by the following.

Proposition 51. Let $δ$ be an object in $S^{A} (C)$ with underlying object $B$ . Then we have

\begin{array}{l} {(δ^{*})}^{l} & = σ_{B, A} \circ δ^{r}, \\ {(δ^{*})}^{r} & = σ_{A, B} \circ δ^{l} . \end{array}

Proof. Let $r = Ψ (δ)$ . Then we have

\begin{array}{l} {(δ^{*})}^{l} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (r^{*} \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{B}) \circ ((1_{A} \otimes ε_{A}) \otimes 1_{B}) \circ (σ_{A, A} \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (γ_{A} \otimes 1_{B}) \circ (σ_{e, A} \otimes 1_{B}) \circ ((ε_{A} \otimes 1_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ δ_{B} \\ = (β_{A} \otimes 1_{B}) \circ ((ε_{A} \otimes 1_{A}) \otimes 1_{B}) \circ (r \otimes 1_{B}) \circ σ_{B, B} \circ δ_{B} \\ = σ_{B, A} \circ (1_{B} \otimes β_{A}) \circ (1_{B} \otimes (ε_{A} \otimes 1_{A})) \circ (1_{B} \otimes r) \circ δ_{B} \\ = σ_{B, A} \circ δ^{r}, \end{array}

where we have used the commutativity of $δ_{B}$ . In a similar way we show that ${(δ^{*})}^{r} = σ_{A, B} \circ δ^{l}$ . □

Note that $δ$ and $δ^{*}$ both have the same underlying object. In order to extend the new operation $^{*}$ to morphisms we need the following lemma

Lemma 52. Let $δ$ and $γ$ be two objects in $S^{A} (C)$ with underlying objects $B$ and $E$ and let $f : δ \to γ$ be a morphism. Then the corresponding arrow in $C$ deﬁne a morphism in $S^{A} (C)$ with domain $δ^{*}$ and codomain $γ^{*}$ .

Proof. We have

\begin{array}{l} (1_{A} \otimes f) \circ {(δ^{*})}^{l} \\ = (1_{A} \otimes f) \circ σ_{B, A} \circ δ^{r} \\ = σ_{E, A} \circ (f \otimes 1_{A}) \circ δ^{r} \\ = σ_{E, A} \circ γ^{r} \circ f \\ = {(γ^{*})}^{l} \circ f, \end{array}

and in a similar way we prove that $(f \otimes 1_{A}) \circ {(δ^{*})}^{r} = {(γ^{*})}^{r} \circ f$ . □

We deﬁne $f^{*} : δ^{*} \to γ^{*}$ to be the morphism described in the previous lemma. It is evident that the map $T : S^{A} (C) \to S^{A} (C)$ deﬁned on objects and arrows by $T (δ) = δ^{*}$ and $T (f) = f^{*}$ is an endofunctor on $S^{A} (C)$ and since $σ$ is a symmetry we have $T \circ T = 1_{S^{A} (C)}$ . This show that the category $S^{A} (C)$ has a nontrivial action by the group $S_{2} = 〈 t ∣ t^{2} = 1 〉$ .

The action have at least one ﬁxed-point

Proposition 53. Let $a = {δ_{A}, δ_{A}}$ be unit the object for the monoidal structure $\otimes^{A}$ on $S^{A} (C)$ Then we have $a^{*} = a .$

Proof. Recall that $δ_{A} : A \to A \otimes A$ deﬁnes a commutative coalgebra structure on $A$ . But then we have

\begin{array}{l} a^{*} \\ = {δ_{A}, δ_{A}}^{*} \\ = {σ_{A, A} \circ δ_{A}, σ_{A, A} \circ δ_{A}} \\ = {δ_{A}, δ_{A}} \\ = a . \end{array}

□

The nontrivial group of symmetries must be taken into account when the notion of a symmetry for the product structures $⊠^{A}$ and $\otimes^{A}$ in $S^{A} (C)$ are deﬁned. We have previously shown that for a symmetric monoidal category in the usual sense we have an interpretation of the Yang-Baxter equation and the unit symmetry conditions in terms of invariance with respect to the group $H = {T_{t} (σ), 1_{\otimes}}$ . This whole construction was based on a certain choice of action by the group $S_{2} = {1, t}$ on the category $C$ , $C^{2}$ and $C^{3}$ generated by the functors $T_{1} = 1_{C}$ , $T_{2} = τ$ and $T_{3} = (1_{C} \times τ) \circ (τ \times 1_{C}) \circ (1_{C} \times τ)$ . What is new for the category of relations is that we have a nontrivial action of $S_{2}$ . We will generalize this and consider monoidal categories $〈 C, \otimes, K_{e}, α, β, γ 〉$ where we have a nontrivial action of $S_{2}$ generated by a functor $T_{1} : C \to C$ . We use this action together with $τ$ to deﬁne actions of $S_{2}$ on the categories $C^{2}$ and $C^{3}$ generated by $T_{2} = τ \circ (T_{1} \times T_{1})$ and $T_{3} = (T_{1} \times T_{2}) \circ (T_{2} \times T_{1}) \circ (T_{1} \times T_{2})$ . It is easy to see that $T_{2} \circ T_{2} = 1_{C^{2}}$ and $T_{3} \circ T_{3} = 1_{C^{3}}$ so that these functors really deﬁnes an action of $S_{2}$ . Note that if $T_{1} = 1_{C}$ we get the action we discussed previously in the section on symmetries and group action. We now lift this action to the functor categories $[C^{2}, C]$ and $[C^{3}, C]$ in the usual way. From this point we proceed in a way that is exactly parallel to what we did in the section on symmetries and group action. In general one could imagine that the functor $T_{1}$ does not ﬁx the unit so that $T_{1} (e) \neq e$ . In this general situation we would assume the existence of a natural isomorphism $θ : K_{e} \to t K_{e}$ in addition to the isomorphism $σ : \otimes \to t \otimes$ . We would thus allow the constant functor $K_{e}$ to be ﬁxed only up to natural isomorphism. In this paper we will not consider such a possibility. This is because the unit is ﬁxed both for the usual case with trivial action and for the case of the category of relations $S^{A} (C)$ as proved in proposition 53. Allowing the unit to move would also make all formulas and derivations more complicated. With this out of the way we can now state that all results derived in the section on symmetries and group actions,up to and including corollary 12 ,also holds for the current situation if we substitute the $S_{2}$ action from this section in all statements. The proofs of these results are of course different since they must take into account the more general action considered in this section. We do not reproduce these proofs here since they are long and rather similar to the ones already given in the section on symmetries and group action.

We now reverse proposition 12 that characterized symmetric monoidal categories in terms of invariance and is lead to the following deﬁnition of symmetries for monoidal categories with a action of the group $S_{2}$ .

Deﬁnition 54. Let $C$ be a category where there is deﬁned an action of the group $S_{2}$ . Then $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ is a symmetric monoidal category if $〈 C, \otimes, K_{e}, α, β, γ 〉$ is a monoidal category and $σ : \otimes \to t \otimes$ is a natural isomorphism such that the following identities holds.

\begin{array}{l} σ \circ (1_{1_{C}} \times σ) & = (t α) \cdot (σ \circ (σ \times 1_{1_{C}})) \cdot α, \\ β & = (t γ) \cdot (σ \circ 1_{K_{e} \times 1_{C}}), \\ γ & = (t β) \cdot (σ \circ 1_{1_{C} \times K_{e}}), \\ t σ & = σ^{- 1} . \end{array}

We say that $σ$ is the symmetry for the monoidal category $〈 C, \otimes, K_{e}, α, β, γ 〉$ .

The ﬁrst condition is equivalent to the Yang-Baxter equation if we consider symmetric monoidal categories in the usual sense with trivial action of $S_{2}$ . We will in all cases call the ﬁrst condition for the Yang-Baxter equation. Note that even for the case of trivial action our notion of symmetric monoidal category is more general than the standard one. The standard deﬁnition of symmetry for a monoidal category implies that the Yang-Baxter equation holds but the fact that the Yang-Baxter equation holds for $σ$ does not necessarily imply that $σ$ is a symmetry in the usual sense.

We say that $〈 C, \otimes, α, σ 〉$ is a symmetric semimonoidal category if $〈 C, \otimes, α 〉$ is a semimonoidal category and the ﬁrst and last of the above conditions hold.

We will now apply the deﬁnition of symmetry for the case of the category of relations. For this case we denoted the map $T_{1}$ by $^{*}$ . Let us ﬁrst consider the structure $⊠^{A}$ . In terms of objects, the deﬁnition of a symmetry for the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ is as follows.

Deﬁnition 55. A symmetry for the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ is an isomorphism $S_{δ, γ}^{A} : δ ⊠^{A} γ \to {(γ^{*} ⊠^{A} δ^{*})}^{*}$ that is natural in $δ$ and $γ$ and such that the following identities are satisﬁed for all $δ, γ$ and $ρ$ .

\begin{array}{l} {(M_{ρ^{*}, γ^{*}, δ^{*}}^{A})}^{*} \circ {(1_{ρ}^{*} ⊠^{A} {(S_{δ, γ}^{A})}^{*})}^{*} \circ S_{δ ⊠^{A} γ, ρ}^{A} \circ M_{δ, γ, ρ}^{A} & = {({(S_{γ, ρ}^{A})}^{*} ⊠^{A} 1_{δ}^{*})}^{*} \circ S_{δ, γ ⊠^{A} ρ}^{A}, \\ {(S_{γ^{*}, δ^{*}}^{A})}^{*} \circ S_{δ, γ}^{A} & = 1_{δ ⊠^{A} γ} . \end{array}

In general many symmetric semimonoidal structures may exist for $S^{A} (C)$ with the product $⊠^{A}$ . We will now show there is always at least one.

Proposition 56. Let $δ, γ$ and $ρ$ be objects in $S^{A} (C)$ with underlying objects $B$ , $E$ and $D$ . Deﬁne

\begin{array}{l} M_{δ, γ, ρ}^{A} & = α_{B, E, D}, \\ S_{δ, γ}^{A} & = σ_{B, E} . \end{array}

where $α$ and $σ$ are the associativity constraint and symmetry for the category $C$ . Then $〈 S^{A} (C), ⊠^{A}, M^{A}, S^{A} 〉$ is a symmetric semimonoidal category.

Proof. We already know that $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ is a semimonoidal category. First we need to prove that $S_{δ, γ}^{A}$ is a morphism in $S^{A} (C)$ . Note that the underlying object for ${(γ^{*} ⊠^{A} δ^{*})}^{*}$ is $E \otimes B$ . If we use the fact that $σ$ is a symmetry in $C$ we have

\begin{array}{l} {({(γ^{*} ⊠^{A} δ^{*})}^{*})}^{l} \circ S_{δ, γ}^{A} \\ = σ_{E \otimes B, A} \circ {(γ^{*} ⊠^{A} δ^{*})}^{r} \circ σ_{B, E} \\ = σ_{E \otimes B, A} \circ α_{E, B, A} \circ (1_{E} \otimes {(δ^{*})}^{r}) \circ σ_{B, E} \\ = σ_{E \otimes B, A} \circ α_{E, B, A} \circ (1_{E} \otimes σ_{A, B} \circ δ^{l}) \circ σ_{B, E} \\ = σ_{E \otimes B, A} \circ α_{E, B, A} \circ (1_{E} \otimes σ_{A, B}) \circ (1_{E} \otimes δ^{l}) \circ σ_{B, E} \\ = σ_{E \otimes B, A} \circ α_{E, B, A} \circ (1_{E} \otimes σ_{A, B}) \circ σ_{A \otimes B, E} \circ (δ^{l} \otimes 1_{E}) \\ = (1_{A} \otimes σ_{B, E}) \circ α_{A, B, E}^{- 1} \circ (δ^{l} \otimes 1_{E}) \\ = (1_{A} \otimes S_{δ, γ}^{A}) \circ {(δ ⊠^{A} γ)}^{l} . \end{array}

and this proves that $S_{δ, γ}^{A}$ is a morphism in $S^{A} (C)$ . It is clearly an isomorphism and naturality is evident. The condition for $S^{A}$ to be a symmetry in $S^{A} (C)$ is satisﬁed since it turns into the condition for $σ$ being is a symmetry in the category $C$ . □

The previous proposition leads us to make the following deﬁnition.

Deﬁnition 57. A symmetric semimonoidal structure $〈 S^{A} (C), ⊠^{A}, M^{A}, S^{A} 〉$ on the category $S^{A} (C)$ is external if for all objects $δ, γ$ and $ρ$ we have

\begin{array}{l} M_{δ, γ, ρ}^{A} & = α_{B, E, D}, \\ S_{δ, γ}^{A} & = σ_{B, E} . \end{array}

where $B$ , $E$ and $D$ are the underlying objects for $δ, γ$ and $ρ$ and where $α$ and $σ$ are the associativity constraint and symmetry for the category $C$ .

The previous proposition then proves that an external symmetries on $S^{A} (C)$ with product $⊠^{A}$ always exists.

We now turn to the deﬁnition of symmetries for $S^{A} (C)$ with the product $\otimes^{A}$ . In terms of objects the general deﬁnition now takes the form

Deﬁnition 58. A symmetry for the monoidal category $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ is an isomorphism $s_{δ, γ}^{A} : δ \otimes^{A} γ \to {(γ^{*} \otimes^{A} δ^{*})}^{*}$ that is natural in $δ$ and $γ$ and such that the following identities are satisﬁed for all $δ, γ$ and $ρ$ .

\begin{array}{l} {(m_{ρ^{*}, γ^{*}, δ^{*}}^{A})}^{*} \circ {(1_{ρ}^{*} \otimes^{A} {(s_{δ, γ}^{A})}^{*})}^{*} \circ s_{δ \otimes^{A} γ, ρ}^{A} \circ m_{δ, γ, ρ}^{A} & = {({(s_{γ, ρ}^{A})}^{*} \otimes^{A} 1_{δ}^{*})}^{*} \circ s_{δ, γ \otimes^{A} ρ}^{A}, \\ {(l_{δ^{*}}^{A})}^{*} \circ s_{δ, a}^{A} & = r_{δ}^{A}, \\ {(r_{δ^{*}}^{A})}^{*} \circ s_{a, δ}^{A} & = l_{δ}^{A}, \\ {(s_{δ^{*}, γ^{*}}^{A})}^{*} \circ s_{δ, γ}^{A} & = 1_{δ \otimes^{A} γ} . \end{array}

Note that identity two and three are not independent. One can be derived from the other by using identity four and the fact that the neutral object $a$ is ﬁxed by the action of $S_{2}$ . There are several equivalent formulations of the ﬁrst symmetry condition

Proposition 59. Let $s^{A}$ be a natural isomorphism $s_{δ, γ}^{A} : δ \otimes^{A} γ \to (γ^{*} \otimes^{A} δ^{*})$ such that the following identities hold

\begin{array}{l} {(l_{δ^{*}}^{A})}^{*} \circ s_{δ, a}^{A} & = r_{δ}^{A}, \\ {(s_{δ^{*}, γ^{*}}^{A})}^{*} \circ s_{δ, γ}^{A} & = 1_{δ \otimes^{A} γ} . \end{array}

Then the following statements are equivalent:

$s^{A}$ is a symmetry.
${(m_{ρ^{*}, γ^{*}, δ^{*}}^{A})}^{*} \circ {(1_{ρ}^{*} \otimes^{A} {(s_{δ, γ}^{A})}^{*})}^{*} \circ s_{δ \otimes^{A} γ, ρ}^{A} \circ m_{δ, γ, ρ}^{A} = s_{δ, {(ρ^{*} \otimes^{A} γ^{*})}^{*}}^{A} \circ (1_{δ} \otimes^{A} s_{γ, ρ}^{A})$ .
${(m_{ρ^{*}, γ^{*}, δ^{*}}^{A})}^{*} \circ s_{{(γ^{*} \otimes^{A} δ^{*})}^{*}, ρ}^{A} \circ (s_{δ, γ}^{A} \otimes^{A} 1_{ρ}) \circ m_{δ, γ, ρ}^{A} = {({(s_{γ, ρ}^{A})}^{*} \otimes^{A} 1_{δ}^{*})}^{*} \circ s_{δ, γ \otimes^{A} ρ}^{A} .$
${(m_{ρ^{*}, γ^{*}, δ^{*}}^{A})}^{*} \circ s_{{(γ^{*} \otimes^{A} δ^{*})}^{*}, ρ}^{A} \circ (s_{δ, γ}^{A} \otimes^{A} 1_{ρ}) \circ m_{δ, γ, ρ}^{A} = s_{δ, {(ρ^{*} \otimes^{A} γ^{*})}^{*}}^{A} \circ (1_{δ} \otimes^{A} s_{γ, ρ}^{A})$ .

Proof. By naturality of $s^{A}$ we have the following two identities

\begin{array}{l} {({(s_{γ, ρ}^{A})}^{*} \otimes^{A} 1_{δ}^{*})}^{*} \circ s_{δ, γ \otimes^{A} ρ}^{A} & = s_{δ, {(ρ^{*} \otimes^{A} γ^{*})}^{*}}^{A} \circ (1_{δ} \otimes^{A} s_{γ, ρ}^{A}), \\ {(1_{ρ}^{*} \otimes^{A} {(s_{δ, γ}^{A})}^{*})}^{*} \circ s_{δ \otimes^{A} γ, ρ}^{A} & = s_{{(γ^{*} \otimes^{A} δ^{*})}^{*}, ρ}^{A} \circ (s_{δ, γ}^{A} \otimes^{A} 1_{ρ}) . \end{array}

The proposition now follows directly from these identities. □

Let us now consider the existence of symmetries.

Deﬁnition 60. A symmetry for the monoidal category $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ is induced by a symmetry $S^{A}$ of the semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A} 〉$ if for all objects $δ, γ$ in $S^{A} (C)$ the following diagram commute

A SAδ,γ ∗ A ∗ ∗
δ⊠| γ------- (γ ⊠|δ )
| |
πAδ,γ| |(πAγ∗,δ∗)∗

δ⊗A γ--A---- (γ∗ ⊗Aδ∗)∗
sδ,γ

We will in the following show that an induced symmetry exists in the external case and is uniquely determined by the symmetry $S^{A}$ .

Recall that for any pair of objects $δ$ and $γ$ in $S^{A} (C)$ , the diagram $P_{δ, γ}^{A}$ was given by

$A A --MAδ,a,γ- A δ⊠(a⊠ γ) (δ ⊠ a)⊠ γ \ / 1δ⊠A γ\l / δr ⊠A1γ \ / δ⊠γ$

From the general theory of categories it is well known that isomorphisms of categories preserve universal cones. By deﬁnition $〈 γ^{*} \otimes^{A} δ^{*}, π_{γ^{*}, δ^{*}}^{A} 〉$ is a universal cone on the diagram $P_{γ^{*}, δ^{*}}^{A}$ and therefore $〈 {(γ^{*} \otimes^{A} δ^{*})}^{*}, {(π_{γ^{*}, δ^{*}}^{A})}^{*} 〉$ is a universal cone on the diagram ${(P_{γ^{*}, δ^{*}}^{A})}^{*}$ . But we have the following result

Lemma 61. Let the symmetric semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A}, S^{A} 〉$ be external, then $〈 δ \otimes^{A} γ, S_{δ, γ}^{A} \circ π_{δ, γ}^{A} 〉$ is a universal cone on ${(P_{γ^{*}, δ^{*}}^{A})}^{*}$ .

Proof. Let us ﬁrst prove that it is a cone. For this we must prove that the following identity

{(M_{γ^{*}, a, δ^{*}}^{A})}^{*} \circ {(1_{γ^{*}} ⊠^{A} {(δ^{*})}^{l})}^{*} \circ S_{δ, γ}^{A} \circ π_{δ, γ}^{A} = {({(γ^{*})}^{r} ⊠^{A} 1_{δ^{*}})}^{*},

holds in $S^{A} (C)$ . Since the semimonoidal structure on $S^{A} (C)$ is external, the previous identity is for the strict case equivalent to the following identity in $C$

α_{E, A, B} \circ (1_{E} \otimes {(δ^{*})}^{l}) \circ σ_{B, E} \circ π_{δ, γ}^{A} = ({(γ^{*})}^{r} \otimes 1_{B}) \circ σ_{B, E} \circ π_{δ, γ}^{A} .

But this identity follows from the Yang Baxter equation and the fact that $〈 δ \otimes^{A} γ, π_{δ, γ}^{A} 〉$ is a cone on $P_{δ, γ}^{A}$ .

\begin{array}{l} α_{E, A, B} \circ {(1_{γ^{*}} ⊠^{A} {(δ^{*})}^{l})}^{*} \circ σ_{B, E} \circ π_{δ, γ}^{A} \\ = α_{E, A, B} \circ (1_{E} \otimes σ_{B, A}) \circ (1_{E} \otimes δ^{r}) \circ σ_{B, E} \circ π_{δ, γ}^{A} \\ = α_{E, A, B} \circ (1_{E} \otimes σ_{B, A}) \circ σ_{B \otimes A, E} \circ (δ^{r} \otimes 1_{E}) \circ π_{δ, γ}^{A} \\ = α_{E, A, B} \circ (1_{E} \otimes σ_{B, A}) \circ σ_{B \otimes A, E} \circ α_{B, A, E} \circ (1_{B} \otimes γ^{l}) \circ π_{δ, γ}^{A} \\ = α_{E, A, B} \circ (1_{E} \otimes σ_{B, A}) \circ σ_{B \otimes A, E} \circ α_{B, A, E} \circ σ_{A \otimes E, B} \circ (γ^{l} \otimes 1_{B}) \circ σ_{B, E} \circ π_{δ, γ}^{A} \\ = α_{E, A, B} \circ (1_{E} \otimes σ_{B, A}) \circ σ_{B \otimes A, E} \circ α_{B, A, E} \circ σ_{A \otimes E, B} \circ (σ_{E, A} \otimes 1_{B}) \\ \circ ({(γ^{*})}^{r} \otimes 1_{B}) \circ σ_{B, E} \circ π_{δ, γ}^{A} \\ = ({(γ^{*})}^{r} \otimes 1_{B}) \circ σ_{B, E} \circ π_{δ, γ}^{A} . \end{array}

Let now $〈 θ, u 〉$ be any cone on ${(P_{γ^{*}, δ^{*}}^{A})}^{*}$ . The proposition is proved if we can show that the following equations has a unique solution $ϕ : θ \to δ \otimes^{A} γ$

S_{δ, γ}^{A} \circ π_{δ, γ}^{A} \circ ϕ = u .

The equation has at most one solution since $S_{δ, γ}^{A} \circ π_{δ, γ}^{A}$ is a monomorphism. In a calculation very similar to previous one we can prove that $〈 θ, {(S_{δ, γ}^{A})}^{- 1} \circ u 〉$ is a cone on $P_{δ, γ}^{A}$ . But $〈 δ \otimes^{A} γ, π_{δ, γ}^{A} 〉$ is a universal cone on $P_{δ, γ}^{A}$ and therefore there exists a morphism $h : θ \to δ \otimes^{A} γ$ in $S^{A} (C)$ such that $π_{δ, γ}^{A} \circ h = {(S_{δ, γ}^{A})}^{- 1} \circ u$ . Composing on both sides with $S_{δ, γ}^{A}$ show that $S_{δ, γ}^{A} \circ π_{δ, γ}^{A} \circ ϕ = u$ and the proposition is proved. □

We can now prove the existence of induced symmetries in the external case.

Theorem 62. Let the symmetric semimonoidal category $〈 S^{A} (C), ⊠^{A}, M^{A}, S^{A} 〉$ be external, then there exists a induced symmetry for the monoidal category $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ .

Proof. The previous lemma show that both $〈 δ \otimes^{A} γ, S_{δ, γ}^{A} \circ π_{δ, γ}^{A} 〉$ and $〈 {(γ^{*} \otimes^{A} δ^{*})}^{*}, {(π_{γ^{*}, δ^{*}}^{A})}^{*} 〉$ are universal cones on ${(P_{γ^{*}, δ^{*}}^{A})}^{*}$ . We can therefore conclude that there exists a unique morphism $s_{δ, γ}^{A} : δ \otimes^{A} γ \to {(γ^{*} \otimes^{A} δ^{*})}^{*}$ such that ${(π_{γ^{*}, δ^{*}}^{A})}^{*} \circ s_{δ, γ}^{A} = S_{δ, γ}^{A} \circ π_{δ, γ}^{A}$ . We will show that $s_{δ, γ}^{A}$ is a symmetry for the monoidal category $〈 S^{A} (C), \otimes^{A}, a, m^{A}, l^{A}, r^{A} 〉$ on $S^{A} (C)$ . The ﬁrst symmetry condition for $s^{A}$ follows from the ﬁrst symmetry condition for $S^{A}$ , the identity ${(π_{γ^{*}, δ^{*}}^{A})}^{*} \circ s_{δ, γ}^{A} = S_{δ, γ}^{A} \circ π_{δ, γ}^{A}$ and the fact that $m^{A}$ is induced by $M^{A}$ . For the second symmetry condition we have for the strict case

\begin{array}{l} {(l_{δ^{*}}^{A})}^{*} \circ s_{δ, a}^{A} \\ = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ π_{a, δ^{*}}^{A} \circ s_{δ, a}^{A} \\ = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ S_{δ, a}^{A} \circ π_{δ, a}^{A} \\ = β_{B} \circ (ε_{A} \otimes 1_{B}) \circ σ_{B, A} \circ π_{δ, a}^{A} \end{array}

\begin{array}{l} = β_{B} \circ σ_{B, e} \circ (1_{B} \otimes ε_{A}) \circ π_{δ, a}^{A} \\ = γ_{B} \circ (1_{B} \otimes ε_{A}) \circ π_{δ, a}^{A} \\ = r_{δ}^{A}, \end{array}

where we have used the fact that the symmetry $S^{A}$ is external. The last symmetry condition follows easily from the commutative diagram deﬁning $s^{A}$ in terms of $S^{A}$ and from the fact that $S^{A}$ is a symmetry. □

3.9. Commutative monoids in the category of relation. We will deﬁne the notion of a commutative monoid for categories with an action of $S_{2}$ and then apply this deﬁnition to the case of relations. Let now $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ be a symmetric monoidal category with an action of $S_{2}$ generated by the functor $T_{1} : C \to C$ . The conditions from deﬁnition 3 thus holds for $α, β, γ$ and $σ$ .

Our deﬁnition of a commutative monoid is a natural extension and categorization of the notion of a commutative monoids in algebra. Let $〈 M, \cdot, e 〉$ be a monoid in the usual algebraic sense, so that $M$ is a set and $\cdot$ is an associative product on $M$ with unit element $e$ . Deﬁne a new associative product on $M$ by $x * y = y \cdot x$ . Then $〈 M, *, e 〉$ is a new monoid on the same underlying set. The monoid $M$ is said to be commutative if $〈 M, *, e 〉$ is the same monoid as $〈 M, \cdot, e 〉$ and this is equivalent to the condition $x \cdot y = y \cdot x$ for all $x$ and $y$ in $M$ . The previous condition is really too strict since in algebra we consider isomorphic monoids to be essentially the same. Thus it would be more natural to require that the two monoids $〈 M, \cdot, e 〉$ and $〈 M, *, e 〉$ are isomorphic. From a categorical point of view the last condition is the only one that really makes sense since the relation of equality exists only between arrows and not between objects. If we now recall that the symmetry $σ$ is the categorization of the idea of changing order in the category $C$ we arrive at our deﬁnition of commutativity.

Let $X$ be a monoid in the category $C$ with product $μ : X \otimes X \to X$ and unit $u : e \to X$ . Deﬁne morphisms $μ^{σ} : T_{1} (X) \otimes T_{1} (X) \to T_{1} (X)$ and $u^{s} : e \to T_{1} (X)$ by

\begin{array}{l} μ^{σ} & = T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)}, \\ u^{s} & = T_{1} (u) . \end{array}

Proposition 63. $〈 T_{1} (X), μ^{σ}, u^{s} 〉$ is a monoid in $C .$

Proof. The Yang-Baxter equation and the naturality of $σ$ implies when evaluated on $(T_{1} (X), T_{1} (X), T_{1} (X))$ the following relation

\begin{array}{l} σ_{T_{1} (X), T_{1} (X \otimes X)} \circ (1_{T_{1} (X)} \otimes σ_{T_{1} (X), T_{1} (X)}) & = T_{1} (α_{X, X, X}) \circ σ_{T_{1} (X \otimes X), T_{1} (X)} \\ \circ (σ_{T_{1} (X), T_{1} (X)} \otimes 1_{T_{1} (X)}) . \end{array}

Using this relation we have

\begin{array}{l} μ^{σ} \circ (1_{T_{1} (X)} \otimes μ^{σ}) \\ = T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)} \circ (1_{T_{1} (X)} \otimes (T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)})) \\ = T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)} \circ (1_{T_{1} (X)} \otimes T_{1} (μ)) \circ (1_{T_{1} (X)} \otimes σ_{T_{1} (X), T_{1} (X)}) \\ = T_{1} (μ) \circ T_{1} (μ \otimes 1_{X}) \circ σ_{T_{1} (X), T_{1} (X \otimes X)} \circ (1_{T_{1} (X)} \otimes σ_{T_{1} (X), T_{1} (X)}) \\ = T_{1} (μ \circ (μ \otimes 1_{X})) \circ T_{1} (α_{X, X, X}) \circ σ_{T_{1} (X \otimes X), T_{1} (X)} \circ (σ_{T_{1} (X), T_{1} (X)} \otimes 1_{T_{1} (X)}) \\ = T_{1} (μ) \circ T_{1} (1_{X} \otimes μ) \circ σ_{T_{1} (X \otimes X), T_{1} (X)} \circ (σ_{T_{1} (X), T_{1} (X)} \otimes 1_{T_{1} (X)}) \\ = T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)} \circ ((T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)}) \otimes 1_{T_{1} (X)}) \\ = μ^{σ} \circ (μ^{σ} \otimes 1_{T_{1} (X)}), \end{array}

so the morphism $μ^{σ}$ is associative. The ﬁrst unit condition evaluated at the pair of objects $(e, T_{1} (X))$ given the identity

β_{e, T_{1} (X)} = T_{1} (γ_{X, e}) \circ σ_{e, T_{1} (X)} .

From the naturality of $σ$ and the fact that $〈 X, μ, u 〉$ is a monoid we have

\begin{array}{l} μ^{σ} \circ (u^{s} \otimes 1_{T_{1} (X)}) \\ = T_{1} (μ) \circ σ_{T_{1} (X), T_{1} (X)} \circ (T_{1} (u) \otimes 1_{T_{1} (X)}) \\ = T_{1} (μ) \circ T_{1} (1_{X} \otimes u) \circ σ_{e, T_{1} (X)} \\ = T_{1} (μ \circ (1_{X} \otimes u)) \circ σ_{e, T_{1} (X)} \\ = T_{1} (γ_{X, e}) \circ σ_{e, T_{1} (X)} \\ = β_{e, T_{1} (X)}, \end{array}

and this is the left condition on the unit. The proof for the right condition is similar. □

Recall that $ϕ : X \to Y$ is a morphism of monoids $ϕ : 〈 X, μ, u 〉 \to 〈 Y, μ^{'}, u^{'} 〉$ if the following two diagrams commute

φ⊗ φ
X⊗X ------------- Y ⊗Y
| |
μ |μ′
| |
X---------------- Y
φ

$X -------φ-------- Y \ / \u\ / /u′ e$

We are now ready to deﬁne the notion of a commutative monoid in the symmetric monoidal category $C$ .

Deﬁnition 64. Let $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ be a symmetric monoidal category. A monoid $〈 X, μ, u 〉$ in $C$ is commutative if there exists an isomorphism of monoids

ϕ : 〈 X, μ, u 〉 \to 〈 T_{1} (X), μ^{σ}, u^{s} 〉 .

We will now apply these deﬁnitions to the $S e t s$ . For this case there is only one possible choice that makes $S e t s$ into a C-category. Let $r (x) = (f (x), g (x))$ and $s (y) = (h (y), k (y))$ be two relations with domains $B$ and $E$ . Then $r^{*} (x) = (g (x), f (x))$ and $s^{*} (y) = (k (y), h (y))$ . If $X$ and $Y$ are the underlying sets for $r \otimes^{A} s$ and ${(s^{*} \otimes^{A} r^{*})}^{*}$ we have

\begin{array}{l} X & = {(x, y) ∣ g (x) = h (y)}, \\ Y & = {(y, x) ∣ h (y) = g (x)}, \end{array}

and the relations are given by

\begin{array}{l} (r \otimes^{A} s) (x, y) & = (f (x), k (y)), \\ {(s^{*} \otimes^{A} r^{*})}^{*} (y, x) & = (f (x), k (y)) . \end{array}

Deﬁne $s_{r, s}^{A} (x, y) = (y, x)$ . Then clearly we have $s_{r, s}^{A} : X \to Y$ and also

\begin{array}{l} ({(s^{*} \otimes^{A} r^{*})}^{*} \circ s_{r, s}^{A}) (x, y) \\ = {(s^{*} \otimes^{A} r^{*})}^{*} (y, x) \\ = (f (x), k (y)) \\ = (r \otimes^{A} s) (x, y), \end{array}

so that we have a morphism in $s_{r, s}^{A} : r \otimes^{A} s \to {(s^{*} \otimes^{A} r^{*})}^{*}$ . It is straight forward to prove that $s^{A}$ is a symmetry on the category of relations. It is in fact induced by the symmetry of the external category $S e t s$ . Since a relation $r : B \to A \times A$ is a directed labelled graph it is clear that we get the relation $r^{*} : B \to A \times A$ by reversing all arrows in the relation $r$ . We have seen that $r$ is a monoid if there exists an associative rule of composition for composable arrows in $r$ such that for each object $x \in A$ there exists an arrow with source and target given by $x$ and that acts as right and left unit for the composition. Let $b$ and $b^{'}$ be two objects in $B$ . Then the rule of composition for the relation $r^{*}$ is deﬁned by ﬁrst reversing both arrows, then composing them as arrows in $r$ and then reversing the result to get an arrow in $r^{*}$ . Now an isomorphism $ϕ : r \to r^{*}$ is a bijective map with domain and codomain given by $B$ and such that

\begin{array}{l} g (ϕ (x)) & = f (x), \\ f (ϕ (x)) & = g (x), \end{array}

for all $x \in B$ . If $ϕ$ is also an isomorphism of the monoids $〈 r, μ, u 〉$ and $〈 r^{*}, μ^{s}, u^{s} 〉$ we must have

ϕ (μ ((x, y))) = μ ((ϕ (y), ϕ (x))),

for all objects $(x, y) \in A \times A$ such that $g (x) = f (y)$ . These conditions are in general impossible to satisfy for the identity map $ϕ = 1_{B}$ . Let $B \subset A \times A$ be a relation in the usual sense. Then $B$ corresponds to a relation in our sense if we deﬁne $r : B \to A \times A$ by $r ((x, y)) = (x, y)$ so that $f ((x, y)) = x$ and $g ((x, y)) = y$ . We know that if $r$ is a monoid in the category of relations then $B$ is a reﬂexive and transitive relation in the usual sense. Assume that $B$ is also symmetric so that it is in fact an equivalence relation. Thus we have $(x, y) \in B$ if and only if $(y, x) \in B$ . Then we can deﬁne a map $ϕ : B \to B$ by $ϕ ((x, y)) = (y, x)$ . For this map we have

\begin{array}{l} g (ϕ ((x, y))) & = g ((y, x)) = x = f ((x, y)), \\ f (ϕ ((x, y))) & = f ((y, x)) = y = g ((x, y)), \end{array}

so that $ϕ : r \to r^{*}$ . We have seen that the rule of composition and unit maps for $r$ are given by

\begin{array}{l} μ_{r} (((x, y), (y, z))) & = (x, z), \\ u_{r} (x) & = (x, x), \end{array}

but then the composition and unit maps for $r^{*}$ must be given by

\begin{array}{l} μ^{s} (((y, x), (z, y))) & = (z, x), \\ u^{s} (x) & = (x, x) . \end{array}

It is evident that $ϕ$ preserve that unit and the following computation show that $ϕ : 〈 r, μ, u 〉 \to 〈 r^{*}, μ^{s}, u^{s} 〉$ also preserve the product

\begin{array}{l} ϕ (μ (((x, y), (y, z)))) \\ = ϕ ((x, z)) \\ = (z, x) \\ = μ^{s} (((y, x), (z, y))) . \end{array}

We have thus proved the following result

Proposition 65. Let $B \subset A \times A$ be an equivalence relation. Deﬁne $r : B \to A \times A$ by $r ((x, y)) = (x, y)$ . Then $r$ is a commutative monoid in the category of relations with respect to the symmetry in $ℛ^{A} (C)$ induced by the symmetry $σ (x, y) = (y, x)$ in $S e t s$ .

Note that this result show that relations that are not equivalence relations in the usual sense might correspond to commutative monoids with respect to a different symmetry than the standard one used in the proposition. Such a class of relations would corresponds to an extension of the notion of equivalence that might be of interest.

4. Quantization of relations

In this section we apply our ideas of quantization as properties of functors in categories of representations of constraints. The constraints here are the system of functors and natural transformations deﬁning a symmetric monoidal category where we have an action of the group $S_{2}$ . Morphisms in this category of representations are what we call quantized functors. These are determined by a functor and a triple of natural isomorphisms that satisfy certain conditions that ensure that the functors behave in a natural way with respect to the representations. Properties of relations are coded in terms of commutative diagrams of arrows in the category of relations. Equivalence relations appears as commutative associative algebras with unit. In the last section we show how we can quantize relations by mapping them with quantized functors.

4.1. Quantized functors. Quantization has in our view its most natural formulation as a property of functors between categories. We will deﬁne quantization in the context of symmetric monoidal categories with an action of the group $S_{2}$ . The symmetries are supposed to be symmetries in our modiﬁes sense, they are natural isomorphisms that satisfy the conditions given in deﬁnition 54 .

Let now $〈 C_{i}, \otimes_{i}, P_{e_{i}}, α_{i}, β_{i}, γ_{i}, σ_{i} 〉$ for $i = 1, 2$ be two symmetric monoidal categories and let $F : C_{1} \to C_{2}$ be a functor.

Deﬁnition 66. A quantization of the functor $F$ is a triple of natural isomorphisms $〈 λ, μ, η 〉$

\begin{array}{l} λ & : \otimes_{2} \circ (F \times F) \to F \circ \otimes_{1}, \\ μ & : F \to t F, \\ η & : K_{e_{2}} \to F \circ K_{e_{1}}, \end{array}

such that the following relations hold

\begin{array}{l} α_{2} \circ 1_{F \times F \times F} & = (1_{\otimes_{2}} \circ (λ^{- 1} \times 1_{F})) \cdot (λ^{- 1} \circ 1_{\otimes_{1} \times 1_{C_{1}}}) \cdot (1_{F} \circ α_{1}) \\ \cdot (λ \circ 1_{1_{C_{1}} \times \otimes_{1}}) \cdot (1_{\otimes_{2}} \circ (1_{F} \times λ)), \\ β_{2} \circ 1_{F \times F} & = (1_{F} \circ β_{1}) \cdot (λ \circ 1_{K_{e_{1}} \times 1_{C_{1}}}) \cdot (1_{\otimes_{2}} \circ (η \times 1_{F})), \end{array}

\begin{array}{l} γ_{2} \circ 1_{F \times F} & = (1_{F} \circ γ_{1}) \cdot (λ \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{\otimes_{2}} \circ (1_{F} \times η)), \\ σ_{2} \circ 1_{F \times F} & = (1_{t \otimes_{2}} \circ (μ^{- 1} \times μ^{- 1})) \cdot (t λ^{- 1}) \cdot (μ \circ σ_{1}) \cdot λ, \\ t μ & = μ^{- 1} . \end{array}

The only true justiﬁcation of this deﬁnition, as for any mathematical deﬁnition, lies in the importance and depth of its consequences. We will now start investigating some of those consequences. We will ﬁrst show that quantized functors are composable.

Proposition 67. Let $F : C_{1} \to C_{2}$ and $G : C_{2} \to C_{3}$ be quantized functors with quantizations $〈 λ_{F}, μ_{F}, η_{F} 〉$ and $〈 λ_{G}, μ_{G}, η_{G} 〉$ . Then $G \circ F$ is a quantized functor with quantization $〈 λ_{G \circ F}, μ_{G \circ F}, η_{G \circ F} 〉$ where

\begin{array}{l} λ_{G \circ F} & = (1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F}), \\ μ_{G \circ F} & = μ_{G} \circ μ_{F}, \\ η_{G \circ F} & = (1_{G} \circ η_{F}) \cdot η_{G} . \end{array}

Proof. For the ﬁrst condition we have

\begin{array}{l} α_{3} \circ 1_{G \circ F \times G \circ F \times G \circ F} \\ = α_{3} \circ 1_{G \times G \times G} \circ 1_{F \times F \times F} \\ = [(1_{\otimes_{3}} \circ (λ_{G}^{- 1} \times 1_{G}) \circ 1_{F \times F \times F}) \cdot (λ_{G}^{- 1} \circ 1_{\otimes_{2} \times 1_{C_{2}}} \circ 1_{F \times F \times F}) \cdot (1_{G} \circ α_{2} \circ 1_{F \times F \times F}) \cdot (λ_{G} \circ 1_{C_{2} \times \otimes_{2} \circ 1_{F \times F \times F}}) \cdot (1_{\otimes_{3}} \circ (1_{G} \times λ_{G}) \circ 1_{F \times F \times F})] \\ = (1_{\otimes_{3}} \circ (λ_{G}^{- 1} \circ 1_{F \times F} \times 1_{G \circ F})) \cdot (λ_{G}^{- 1} \circ (1_{\otimes_{2} \circ (F \times F)} \times 1_{F})) \cdot (1_{G} \circ 1_{\otimes_{2}} \circ (λ_{F}^{- 1} \times 1_{F})) \cdot (1_{G} \circ λ_{F}^{- 1} \circ 1_{\otimes_{1} \times 1_{C_{1}}}) \cdot (1_{G} \circ 1_{F} \circ α_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{C_{1} \times \otimes_{1}}) \\ \cdot (1_{G} \circ 1_{\otimes_{2}} \circ (1_{F} \times λ_{F})) \cdot (λ_{G} \circ (1_{F} \times 1_{\otimes_{2} \circ (F \times F)})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times (λ_{G} \circ 1_{F \times F}))) \\ = (1_{\otimes_{3}} \circ ((λ_{G}^{- 1} \circ 1_{F \times F}) \times 1_{G \circ F})) \cdot (λ_{G}^{- 1} \circ (1_{\otimes_{2} \circ (F \times F)} \times 1_{F})) \cdot (1_{G \circ \otimes_{2}} \circ (λ_{F}^{- 1} \times 1_{F})) \cdot (1_{G} \circ λ_{F}^{- 1} \circ (1_{\otimes_{1}} \times 1_{1_{C_{1}}})) \cdot (1_{G \circ F} \circ α_{1}) \cdot (1_{G} \circ λ_{F} \circ (1_{1_{C_{1}}} \times 1_{\otimes_{1}})) \cdot (1_{G \circ \otimes_{2}} \circ (1_{F} \times λ_{F})) \cdot (λ_{G} \circ (1_{F} \times 1_{\otimes_{2} \circ (F \times F)})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times (λ_{G} \circ 1_{F \times F}))) \\ = (1_{\otimes_{3}} \circ ((λ_{G}^{- 1} \circ 1_{F \times F}) \times 1_{G \circ F})) \cdot (λ_{G}^{- 1} \circ (λ_{F}^{- 1} \times 1_{F})) \cdot (1_{G} \circ λ_{F}^{- 1} \circ (1_{\otimes_{1}} \times 1_{1_{C_{1}}})) \cdot (1_{G \circ F} \circ α_{1}) \cdot (1_{G} \circ λ_{F} \circ (1_{C_{1}} \times 1_{\otimes_{1}})) \cdot (λ_{G} \circ (1_{F} \times λ_{F})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times (λ_{G} \circ 1_{F \times F}))) \end{array}

\begin{array}{l} = (1_{\otimes_{3}} \circ ((λ_{G}^{- 1} \circ 1_{F \times F}) \times 1_{G \circ F})) \cdot (1_{\otimes_{3}} \circ 1_{G \times G} \circ (λ_{F}^{- 1} \times 1_{F})) \cdot (λ_{G}^{- 1} \circ (1_{F \circ \otimes_{1}} \times 1_{F})) \cdot (1_{G} \circ λ_{F}^{- 1} \circ (1_{\otimes_{1}} \times 1_{1_{C_{1}}})) \cdot (1_{G \circ F} \circ α_{1}) \cdot (1_{G} \circ λ_{F} \circ (1_{1_{C_{1}}} \times 1_{\otimes_{1}})) \cdot (λ_{G} \circ (1_{F} \times 1_{F \circ \otimes_{1}})) \cdot (1_{\otimes_{3}} \circ 1_{G \times G} \circ (1_{F} \times λ_{F})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times (λ_{G} \circ 1_{F \times F}))) \\ = (1_{\otimes_{3}} \circ ([(λ_{G}^{- 1} \circ 1_{F \times F}) \cdot (1_{G} \circ λ_{F}^{- 1})] \times 1_{G \circ F}) \cdot ([(λ_{G}^{- 1} \circ 1_{F \times F}) \cdot (1_{G} \circ λ_{F}^{- 1})] \circ (1_{\otimes_{1}} \times 1_{1_{C_{1}}})) \cdot (1_{G \circ F} \circ α_{1}) \cdot ([(1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F})] \circ (1_{1_{C_{1}}} \times 1_{\otimes_{1}}))) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times [(1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F})])) \\ = (1_{\otimes_{3}} \circ (λ_{G \circ F}^{- 1} \times 1_{G \circ F})) \cdot (λ_{G \circ F}^{- 1} \circ 1_{\otimes_{1} \times 1_{C_{1}}}) \cdot (1_{G \circ F} \circ α_{1}) \cdot (λ_{G \circ F} \circ 1_{1_{C_{1}} \times \otimes_{1}}) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times λ_{G \circ F})) . \end{array}

The proof for the second and third conditions are the similar and we only show the proof for the third condition

\begin{array}{l} γ_{3} \circ 1_{G \circ F \times 1_{T}} \\ = γ_{3} \circ 1_{G \times 1_{T}} \circ 1_{F \times 1_{T}} \\ = [(1_{G} \circ γ_{2}) \cdot (λ_{G} \circ 1_{1_{C_{2}} \times K_{e_{2}}}) \cdot (1_{\otimes_{3}} \circ (1_{G} \times η_{G}))] \circ 1_{F \times 1_{T}} \\ = (1_{G} \circ γ_{2} \circ 1_{F \times 1_{T}}) \cdot (λ_{G} \circ 1_{1_{C_{2}} \times K_{e_{2}}} \circ 1_{F \times 1_{T}}) \cdot (1_{\otimes_{3}} \circ (1_{G} \times η_{G}) \circ 1_{F \times 1_{T}}) \\ = (1_{G} \circ [(1_{F} \circ γ_{1}) \cdot (λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{\otimes_{2}} \circ (1_{F} \times η_{F}))]) \cdot (λ_{G} \circ 1_{1_{C_{2}} \times K_{e_{2}}} \circ 1_{F \times 1_{T}}) \cdot (1_{\otimes_{3}} \circ (1_{G} \times η_{G}) \circ 1_{F \times 1_{T}}) \\ = (1_{G} \circ 1_{F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{G} \circ 1_{\otimes_{2}} \circ (1_{F} \times η_{F})) \cdot (λ_{G} \circ 1_{1_{C_{2}} \times K_{e_{2}}} \circ 1_{F \times 1_{T}}) \cdot (1_{\otimes_{3}} \circ (1_{G} \times η_{G}) \circ 1_{F \times 1_{T}}) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{G \circ \otimes_{2}} \circ (1_{F} \times η_{F})) \circ (λ_{G} \circ (1_{F} \times 1_{K_{e_{2}}})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G})) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (λ_{G} \circ (1_{F} \times η_{F})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G})) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (λ_{G} \circ (1_{F} \times η_{F})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times 1_{G \circ K_{e_{2}}})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G})) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (λ_{G} \circ (1_{F} \times η_{F})) \\ \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G})) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (λ_{G} \circ (1_{F} \times 1_{F \circ K_{e_{1}}})) \cdot (1_{\otimes_{3}} \circ (1_{G} \times 1_{G}) \circ (1_{F} \times η_{F})) \cdot \\ = (1_{G \circ F} \circ γ_{1}) \cdot (1_{G} \circ λ_{F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (λ_{G} \circ 1_{F \times F} \circ (1_{1_{C_{1}}} \times 1_{K_{e_{1}}})) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times (1_{G} \circ η_{F}))) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G})) & = (1_{G \circ F} \circ γ_{1}) \cdot ([(1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F})] \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times [(1_{G} \circ η_{F}) \cdot η_{G}])) \\ = (1_{G \circ F} \circ γ_{1}) \cdot (λ_{G \circ F} \circ 1_{1_{C_{1}} \times K_{e_{1}}}) \cdot (1_{\otimes_{3}} \circ (1_{G \circ F} \times η_{G \circ F})) . \end{array}

For the ﬁfth condition we have

\begin{array}{l} σ_{3} \circ 1_{G \circ F \times G \circ F} \\ = σ_{3} \circ 1_{G \times G} \circ 1_{F \times F} \\ = [(1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1})) \cdot (t λ_{G}^{- 1}) \cdot (μ_{G} \circ σ_{2}) \cdot λ_{G}] \circ 1_{F \times F} \\ = (1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1}) \circ 1_{F \times F}) \cdot (t λ_{G}^{- 1} \circ 1_{F \times F}) \cdot (μ_{G} \circ σ_{2} \circ 1_{F \times F}) \\ \cdot (λ_{G} \circ 1_{F \times F}) \\ = (1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1}) \circ 1_{F \times F}) \cdot (t λ_{G}^{- 1} \circ 1_{F \times F}) \\ \cdot (μ_{G} \circ [(1_{t \otimes_{2}} \circ (μ_{F}^{- 1} \times μ_{F}^{- 1})) \cdot (t λ_{F}^{- 1}) \cdot (μ_{F} \circ σ_{1}) \cdot λ_{F}]) \cdot (λ_{G} \circ 1_{F \times F}) \\ = (1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1}) \circ 1_{F \times F}) \cdot (t λ_{G}^{- 1} \circ 1_{F \times F}) \cdot (1_{t (G \circ \otimes_{2})} \circ (μ_{F}^{- 1} \times μ_{F}^{- 1})) \cdot (1_{t G} \circ (t λ_{F}^{- 1})) \cdot (μ_{G} \circ μ_{F} \circ σ_{1}) \cdot (1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F}) \end{array}

\begin{array}{l} = (1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1}) \circ 1_{F \times F}) \cdot ((t λ_{G}^{- 1}) \circ (μ_{F}^{- 1} \times μ_{F}^{- 1})) \cdot (1_{t G} \circ (t λ_{F}^{- 1})) \cdot (μ_{G} \circ μ_{F} \circ σ_{1}) \cdot (1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F}) \\ = (1_{t \otimes_{3}} \circ (μ_{G}^{- 1} \times μ_{G}^{- 1}) \circ 1_{F \times F}) \cdot (1_{t \otimes_{3}} \circ (1_{t G} \times 1_{t G}) \circ (μ_{F}^{- 1} \times μ_{F}^{- 1})) \cdot ((t λ_{G}^{- 1}) \circ (1_{t F} \times 1_{t F})) \cdot (1_{t G} \circ (t λ_{F}^{- 1})) \cdot (μ_{G} \circ μ_{F} \circ σ_{1}) \cdot (1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F}) \\ = (1_{t \otimes_{3}} \circ ({(μ_{G} \circ μ_{F})}^{- 1} \times {(μ_{G} \circ μ_{F})}^{- 1})) \cdot (t {[(1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F})]}^{- 1}) \cdot (μ_{G} \circ μ_{F} \circ σ_{1}) \cdot [(1_{G} \circ λ_{F}) \cdot (λ_{G} \circ 1_{F \times F})] \\ = (1_{t \otimes_{3}} \circ (μ_{G \circ F}^{- 1} \times μ_{G \circ F}^{- 1})) \cdot (t λ_{G \circ F}^{- 1}) \cdot (μ_{G \circ F} \circ σ_{1}) \circ λ_{G \circ F} . \end{array}

The last condition is clearly satisﬁed because action by $t$ pass through horizontal composition. □

As a consequence of this proposition the class of symmetric monoidal categories form a category where arrows are four tuples $〈 F, λ_{F}, μ_{F}, η_{F} 〉$ and where composition of four tuples is deﬁned using the previous proposition.

〈 G, λ_{G}, μ_{G}, η_{G} 〉 \circ 〈 F, λ_{F}, μ_{F}, η_{F} 〉 = 〈 G \circ F, λ_{G \circ F}, μ_{G \circ F}, η_{G \circ F} 〉 .

A given category $C$ with a product bifunctor $\otimes$ and unit functor $K_{e}$ is a symmetric monoidal category if the conditions on $α, β, γ, σ$ and $θ$ stated in deﬁnition 54 are satisﬁed. These conditions are equations that may have none or many solutions depending on the category $C$ and the choice of functors $\otimes$ and $K_{e}$ . We thus in general have a set of solutions. Let this set be denoted by $S$ . We will now show that there is a group acting on $S$ . The deﬁnition of this group action is derived from the formulas deﬁning a quantized functor. Let $G$ be the following group of natural isomorphisms

G = {(λ, μ, η) ∣ λ : \otimes \to \otimes, μ : 1_{C} \to 1_{C}, η : K_{e} \to K_{e}},

where the product is taken componentwise. The size of this group depends on the category $C$ and functors $\otimes$ and $P_{e}$ . Let now $(λ, μ, η)$ be any element of the group $G$ and deﬁne a mapping $F_{λ, μ, η}$ on $S$ by

F_{λ, μ, η} (α, β, γ, σ) = (\hat{α}, \hat{β}, \hat{γ}, \hat{σ}),

where

\begin{array}{l} \hat{α} & = (λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \cdot α \cdot (λ \circ (1_{1_{C}} \times λ)), \\ \hat{β} & = β \cdot (λ \circ (η \times 1_{1_{C}})), \\ \hat{γ} & = γ \cdot (λ \circ (1_{1_{C}} \times η)), \\ \hat{σ} & = (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (t λ^{- 1}) \cdot (μ \circ σ) \cdot λ . \end{array}

Let $H$ be the subgroup of $G$ deﬁned by the relations

\begin{array}{l} t μ & = μ^{- 1}, \\ μ \circ 1_{\otimes} \circ (μ^{- 1} \times 1_{K_{e}}) & = 1_{\otimes \circ (1_{C} \times K_{e})}, \\ μ \circ 1_{\otimes} \circ (1_{K_{e}} \times μ^{- 1}) & = 1_{\otimes \circ (K_{e} \times 1_{C})}, \\ t η & = (μ \circ 1_{K_{e}}) \cdot η . \end{array}

Then we have the following important result.

$F_{λ, μ, η} : S \to S$ and deﬁnes an action of the group $H$ on the set $S$ .

Proof. In order to prove that $(\hat{α}, \hat{β}, \hat{γ}, \hat{σ}) \in S$ we must show that $(\hat{α}, \hat{β}, \hat{γ}, \hat{σ})$ deﬁnes a symmetric monoidal structure. There are eight such conditions. For the ﬁrst condition we have (this is also a proof that the map $T_{1} (σ)$ from section 2.2 maps associativity constraints to associativity constraints)

\begin{array}{l} (\hat{α} \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (\hat{α} \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}}) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (α \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (λ \circ (1_{C} \times λ) \circ 1_{\otimes \times 1_{C} \times 1_{C}} \cdot (λ^{- 1} \circ (λ^{- 1} \times 1_{C})) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (λ \circ (1_{C} \times λ) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}}) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (λ \circ (1_{\otimes} \times λ)) \cdot (λ^{- 1} \circ (λ^{- 1} \times 1_{\otimes})) \cdot (α \circ 1_{C} \times 1_{C} \times \otimes) \cdot (λ \circ (1_{C} \times λ) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}}) \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (1_{\otimes} \circ (λ^{- 1} \times λ)) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (λ \circ (1_{C} \times λ) \circ 1_{1_{C} \times 1_{C} \times \otimes}) \\ = (λ^{- 1} \circ (λ^{- 1} \circ (1_{\otimes} \times 1_{1_{C}}) \times 1_{1_{C}})) \cdot (α \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (1_{\otimes} \circ (λ^{- 1} \times λ)) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (λ \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times 1_{\otimes})))) \\ = ([(λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \cdot α] \circ (1_{\otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot ((1_{\otimes} \circ (1_{1_{C}} \times 1_{\otimes})) \circ (λ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot ((1_{\otimes} \circ (1_{\otimes} \times 1_{1_{C}})) \circ (1_{1_{C}} \times 1_{1_{C}} \times λ)) \cdot ([α \cdot (λ \circ (1_{1_{C}} \times λ))] \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{\otimes})) \\ = ([(λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \cdot α] \circ (λ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot ([α \cdot (λ \circ (1_{1_{C}} \times λ))] \circ (1_{1_{C}} \times 1_{1_{C}} \times λ)) \\ = (λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}}) \circ (λ^{- 1} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (α \circ (1_{\otimes} \times 1_{1_{C}} \times 1_{1_{C}})) \cdot (α \circ (1_{1_{C}} \times 1_{1_{C}} \times 1_{\otimes})) \cdot (λ \circ (1_{1_{C}} \times λ) \circ (1_{1_{C}} \times 1_{1_{C}} \times λ)) \\ = (λ^{- 1} \circ ((λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (α \circ 1_{\otimes \times 1_{C} \times 1_{C}}) \cdot (α \circ 1_{1_{C} \times 1_{C} \times \otimes}) \cdot (λ \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times λ)))) \\ = (λ^{- 1} \circ ((λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (α \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α)) \cdot (λ \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times λ)))) \\ = (λ^{- 1} \circ ((λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (α \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{\otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times λ \times 1_{1_{C}})) \cdot (α \circ (1_{1_{C}} \times 1_{\otimes} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times 1_{\otimes}) \circ (1_{1_{C}} \times λ^{- 1} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α)) \cdot (λ \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times λ)))) \\ = (1_{\otimes} \circ ((λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (λ^{- 1} \circ ((1_{\otimes} \circ (1_{\otimes} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (α \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (1_{1_{C}} \times λ)) \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times (1_{\otimes} \circ (λ^{- 1} \times 1_{1_{C}})))) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α)) \cdot (1_{\otimes} \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times λ)))) \\ = (1_{\otimes} \circ ((λ^{- 1} \circ (λ^{- 1} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (α \times 1_{1_{C}})) \cdot (λ^{- 1} \circ (1_{\otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times λ \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times \otimes \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times 1_{\otimes}) \circ (1_{1_{C}} \times λ^{- 1} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times α)) \\ \cdot (1_{\otimes} \circ (1_{1_{C}} \times (λ \circ (1_{1_{C}} \times λ)))) \end{array}

This proves the ﬁrst condition. For the second condition we have

\begin{array}{l} (1_{\otimes} \circ (\hat{γ} \times 1_{1_{C}})) \cdot (\hat{α} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ ([γ \cdot (λ \circ 1_{1_{C} \times K_{e}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times η))] \times 1_{1_{C}})) \cdot ([1_{\otimes} \circ (λ^{- 1} \times 1_{1_{C}}) \cdot (λ^{- 1} \circ 1_{\otimes \times 1_{C}}) \cdot α \cdot (λ \circ 1_{1_{C} \times \otimes}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ))] \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ ((γ \times 1_{1_{C}}) \cdot ((λ \circ 1_{1_{C} \times K_{e}}) \times 1_{1_{C}}) \cdot ((1_{\otimes} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}}))) \cdot (1_{\otimes} \circ (λ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ^{- 1} \circ 1_{\otimes \times 1_{C}} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ 1_{1_{C} \times \otimes} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((λ \circ 1_{1_{C} \times K_{e}}) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (λ^{- 1} \times 1_{1_{C}}) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ^{- 1} \circ 1_{\otimes \times 1_{C}} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ 1_{1_{C} \times \otimes} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((λ \circ (1_{1_{C}} \times 1_{K_{e}})) \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((λ^{- 1} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}})) \cdot (λ^{- 1} \circ 1_{\otimes \times 1_{C}} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ 1_{1_{C} \times \otimes} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}})) \cdot (λ^{- 1} \circ ((1_{\otimes} \circ (1_{1_{C}} \times 1_{K_{e}})) \times 1_{1_{C}})) \cdot (λ \circ 1_{1_{C} \times \otimes} \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ) \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \\ = (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}})) \cdot (λ^{- 1} \circ ((1_{\otimes} \circ (1_{1_{C}} \times 1_{K_{e}})) \times 1_{1_{C}})) \cdot (λ \circ (1_{1_{C}} \times (λ \circ 1_{K_{e} \times 1_{C}}))) \\ = (λ^{- 1} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (1_{1_{C}} \times η)) \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times (λ \circ 1_{K_{e} \times 1_{C}}))) \\ = (λ^{- 1} \circ (1_{G} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{\otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times (λ \circ 1_{K_{e} \times 1_{C}}))) \\ = (λ^{- 1} \circ (1_{G} \times 1_{1_{C}})) \circ (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{\otimes} \times 1_{1_{C}}) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \cdot (α \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}})) \cdot (λ \circ (1_{1_{C}} \times (λ \circ 1_{K_{e} \times 1_{C}}))) \\ = (λ^{- 1} \circ (1_{G} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (α \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}}) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \cdot (λ \circ (1_{1_{C}} \times (λ \circ 1_{K_{e}} c))) \\ = (λ^{- 1} \circ (1_{G} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}} \circ (1_{1_{C}} \times η 1_{1_{C}})) \cdot (λ \circ (1_{1_{C}} \times λ) \circ (1_{1_{C}} \times 1_{K_{e}} \times 1_{1_{C}})) \\ = (λ^{- 1} \circ (1_{G} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot ([(α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times λ))] \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \\ = (λ^{- 1} \circ (1_{1_{C}} \times 1_{B})) \cdot (1_{\otimes} \circ (γ \times 1_{1_{C}})) \cdot (α \circ 1_{1_{C} \times K_{e} \times 1_{C}}) \cdot ((λ \circ (1_{1_{C}} \times λ)) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \\ = (λ^{- 1} \circ (1_{1_{C}} \times 1_{B})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times β)) \cdot (λ \circ (1_{1_{C}} \times λ) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \\ = (λ^{- 1} \circ (1_{1_{C}} \times β)) \cdot (λ \circ (1_{1_{C}} \times λ) \circ (1_{1_{C}} \times η \times 1_{1_{C}})) \\ = (λ^{- 1} \circ (1_{1_{C}} \times β)) \cdot (λ \circ (1_{1_{C}} \times λ \circ (η \times 1_{1_{C}}))) \\ = (1_{\otimes} \circ (1_{1_{C}} \times (β \cdot (λ \circ (η \times 1_{1_{C}}))))) \\ = 1_{\otimes} \circ (1_{1_{C}} \times (β \cdot (λ \circ 1_{K_{e} \times 1_{C}}) \cdot (1_{\otimes} \circ (η \times 1_{1_{C}})))) \\ = 1_{\otimes} \circ (1_{1_{C}} \times \hat{β}) . \end{array}

This proves the second condition. For the third condition we have

\begin{array}{l} \hat{β} \circ 1_{1_{C} \times K_{e}} \\ = [β \cdot (λ \circ 1_{K_{e} \times 1_{C}} \cdot (1_{\otimes} \circ (η \times 1_{1_{C}})))] \circ 1_{1_{C} \times K} \\ = (β \circ 1_{1_{C} \times K}) \cdot (λ \circ (1_{K_{e}} \times 1_{1_{C}}) \circ 1_{1_{C} \times K}) \cdot (1_{\otimes} \circ (η \times 1_{1_{C}}) \circ 1_{1_{C} \times K}) \\ = (γ \circ 1_{K \times 1_{C}}) \cdot (λ \circ (1_{K_{e}} \times 1_{K_{e}})) \cdot (1_{\otimes} \circ (η \times 1_{K_{e}})) \\ = (γ \circ 1_{K \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times 1_{K_{e}}) \circ (1_{K_{e}} \times 1_{1_{C}})) \cdot (1_{\otimes} \circ (1_{K_{e}} \times η)) \\ = (γ \circ 1_{K \times 1_{C}}) \cdot (λ \circ (1_{1_{C}} \times 1_{K_{e}}) \circ 1_{K \times 1_{C}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times η) \circ 1_{K \times 1_{C}}) \\ = [(γ \cdot (λ \circ 1_{1_{C} \times K_{e}}) \cdot (1_{\otimes} \circ (1_{1_{C}} \times η)))] \circ 1_{K \times 1_{C}} \\ = \hat{γ} \circ 1_{K_{e} \times 1_{C}} . \end{array}

This proves the third condition. The proof of the fourth condition is very technical. In the proof we will use the following symbols

\begin{array}{l} L_{1} & = (1_{t \otimes} \circ (1_{1_{C}} \times t λ^{- 1})) \cdot (t λ^{- 1} \circ 1_{t \otimes \times 1_{C}}) \cdot (t α \cdot (t λ \circ 1_{t \otimes \times 1_{C}})), \\ L_{2} & = (1_{t \otimes} \circ (1_{1_{C}} \times t λ^{- 1})) \cdot (t λ^{- 1} \circ 1_{t \otimes \times 1_{C}}) \cdot (t α), \\ L_{3} & = (1_{t \otimes} \circ (1_{1_{C}} \times t λ^{- 1})) \cdot (t λ^{- 1} \circ 1_{t \otimes \times 1_{C}}), \\ L_{4} & = (1_{t \otimes} \circ (1_{1_{C}} \times t λ^{- 1})), \\ R & = ((t λ^{- 1}) \circ (t λ^{- 1} \times 1_{1_{C}})) \cdot ((μ \circ σ) \circ ((μ \circ σ) \times 1_{1_{C}})) \cdot (λ \circ (λ \times 1_{1_{C}})), \\ R_{1} & = ((μ \circ σ) \circ ((μ \circ σ) \times 1_{1_{C}})) \cdot (λ \circ (λ \times 1_{1_{C}})), \\ R_{2} & = (λ \circ (λ \times 1_{1_{C}})), \\ S_{1} & = α^{- 1} \cdot (λ \circ 1_{\otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (λ \times 1_{1_{C}})), \\ S_{2} & = (λ^{- 1} \circ 1_{1_{C} \times \otimes}) \cdot α^{- 1} \cdot (λ \circ 1_{\otimes \times 1_{C}}) \cdot (1_{\otimes} \circ (λ \times 1_{1_{C}})) . \end{array}

Using these symbols we have for the fourth condition

\begin{array}{l} (t \hat{α}) \cdot (\hat{σ} \circ (\hat{σ} \times 1_{1_{C}})) \\ = (t \hat{α}) \cdot ([(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (t λ^{- 1}) \cdot (μ \circ σ) \cdot λ] \circ ([(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (t λ^{- 1}) \cdot (μ \circ σ) \cdot λ] \times 1_{1_{C}})) \\ = L_{1} \cdot (1_{t \otimes} \circ (t λ \times 1_{1_{C}})) \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R \\ = L_{1} \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((t λ \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R \\ = L_{1} \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((t λ \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot (t λ^{- 1} \circ (t λ^{- 1} \times 1_{1_{C}})) \cdot R_{1} \\ = L_{1} \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R_{1} \\ = L_{2} \cdot (t λ \circ (1_{t \otimes} \times 1_{1_{C}})) \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R_{1} \\ = L_{2} \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R_{1} \\ = L_{2} \cdot ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot ((μ \circ σ) \circ ((μ \circ σ) \times 1_{1_{C}})) \cdot R_{2} \\ = L_{2} \cdot ((1_{1_{C}} \circ 1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \circ ((1_{1_{C}} \circ 1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot ((μ \circ σ \circ (1_{1_{C}} \times 1_{1_{C}})) \circ ((μ \circ σ \circ (1_{1_{C}} \times 1_{1_{C}})) \times 1_{1_{C}})) \cdot R_{2} \\ = L_{2} \cdot ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \circ ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R_{2} \\ = L_{2} \cdot ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \circ ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \times 1_{1_{C}})) \cdot R_{2} \\ = L_{2} \cdot ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \circ ((μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \times (μ \circ 1_{1_{C}} \circ μ^{- 1}))) \cdot R_{2} \\ = L_{2} \cdot ((1 μ \circ σ \circ (μ^{- 1} \times μ^{- 1})) \circ (μ \times μ) \circ (σ \times 1_{1_{C}}) \circ (μ^{- 1} \times μ^{- 1} \times μ^{- 1})) \cdot R_{2} \\ = L_{2} \cdot (μ \circ σ \circ ((σ \circ (μ^{- 1} \times μ^{- 1})) \times μ^{- 1})) \cdot R_{2} \\ = L_{3} \cdot (1_{1_{C}} \circ t α) \cdot (μ \circ σ \circ (σ \times 1_{1_{C}})) \cdot (1_{\otimes} \circ ((1_{\otimes} \circ (μ^{- 1} \times μ^{- 1})) \times μ^{- 1})) \cdot (λ \circ (λ \times 1_{1_{C}})) \\ = L_{3} \cdot (μ \circ (t α \cdot (σ \circ (σ \times 1_{1_{C}})))) \cdot (λ \circ (λ \circ (μ^{- 1} \times μ^{- 1})) \times μ^{- 1}) \\ = L_{3} \cdot (((μ \circ (σ \circ (1_{1_{C}} \times σ)) \cdot α^{- 1})) \cdot (λ \circ (λ \circ (μ^{- 1} \times μ^{- 1}))) \times μ^{- 1}) \\ = L_{3} \cdot (μ \circ ((σ \circ (1_{1_{C}} \times σ)) \cdot α^{- 1})) \cdot (λ \circ (λ \times 1_{1_{C}}) \circ (μ^{- 1} \times μ^{- 1} \times μ^{- 1})) \\ = L_{3} \cdot (μ \circ σ \circ (1_{1_{C}} \times σ)) \cdot α^{- 1} \cdot (λ \circ (λ \times 1_{1_{C}}) \circ (μ^{- 1} \times μ^{- 1} \times μ^{- 1})) \\ = L_{3} \cdot (μ \circ σ \circ (1_{1_{C}} \times σ)) \cdot ((α^{- 1} \cdot (λ \circ (λ \times 1_{1_{C}}))) \circ (μ^{- 1} \times μ^{- 1} \times μ^{- 1})) \\ = L_{3} \cdot (μ \circ σ \circ (1_{1_{C}} \times σ)) \cdot ((1_{\otimes} \circ (1_{1_{C}} \times 1_{\otimes})) \circ (μ^{- 1} \times μ^{- 1} \times μ^{- 1})) \cdot (α^{- 1} \cdot (λ \circ (λ \times 1_{1_{C}}))) \\ = L_{3} \cdot (μ \circ σ \circ (1_{1_{C}} \times σ)) \cdot (1_{\otimes} \circ (μ^{- 1} \times (1_{\otimes} \circ (μ^{- 1} \times μ^{- 1})))) \cdot S_{1} \\ = L_{3} \cdot (μ \circ σ \circ (μ^{- 1} \times (σ \circ (μ^{- 1} \times μ^{- 1})))) \cdot S_{1} \\ = L_{3} \cdot (((1_{1_{C}} \circ 1_{\otimes}) \cdot (μ \circ σ)) \circ ((μ^{- 1} \times (σ \circ (μ^{- 1} \times μ^{- 1})))) \cdot ((1_{1_{C}} \times (1_{t \otimes} \circ (1_{1_{C}} \times 1_{1_{C}}))))) \cdot S_{1} \\ = L_{3} \cdot ((1_{1_{C}} \circ 1_{t \otimes}) \circ (μ^{- 1} \times (σ \circ (μ^{- 1} \times μ^{- 1})))) \cdot ((μ \circ σ) \circ (1_{1_{C}} \times (1_{t \otimes} \circ (1_{1_{C}} \times 1_{1_{C}})))) \cdot S_{1} \\ = L_{3} \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times (μ \circ σ \circ (μ^{- 1} \times μ^{- 1})))) \cdot ((μ \circ σ) \circ (1_{1_{C}} \times (1_{t \otimes} \circ (1_{1_{C}} \times 1_{1_{C}})))) \cdot S_{1} \\ = L_{3} \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times (μ \circ σ \circ (μ^{- 1} \times μ^{- 1})))) \cdot ((μ \circ σ) \circ (1_{1_{C}} \times 1_{t \otimes})) \cdot S_{1} \\ = L_{3} \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ((μ \circ σ) \circ 1_{1_{C} \times \otimes}) \cdot S_{1} \\ = L_{4} \cdot (t λ^{- 1} \circ (1_{1_{C}} \times 1_{t \otimes})) \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ((μ \circ σ) \circ 1_{1_{C} \times \otimes}) \cdot S_{1} \\ = L_{4} \cdot (t λ^{- 1} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ((μ \circ σ) \circ 1_{1_{C} \times \otimes}) \cdot S_{1} \\ = L_{4} \cdot (t λ^{- 1} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = (1_{t \otimes} \circ (1_{1_{C}} \times t λ^{- 1})) \cdot (t λ^{- 1} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = (t λ^{- 1} \circ (μ^{- 1} \times μ^{- 1}) \circ (1_{1_{C}} \times [t λ^{- 1} \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = ([(1_{t \otimes} \cdot t λ^{- 1}) \circ ((μ^{- 1} \times μ^{- 1}) \cdot (1_{1_{C}} \times 1_{1_{C}}))] \circ (1_{1_{C}} \times [(t λ^{- 1} \circ (μ^{- 1} \times μ^{- 1})) \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = ([(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot t λ^{- 1}] \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot t λ^{- 1} \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = (\hat{σ} \circ (1_{1_{C}} \times [(1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot t λ^{- 1} \cdot (μ \circ σ)])) \cdot ([(μ \circ σ) \cdot λ] \circ 1_{1_{C} \times \otimes}) \cdot S_{2} \\ = (\hat{σ} \circ (1_{1_{C}} \times \hat{σ})) \cdot (1_{\otimes} \circ (1_{1_{C}} \times λ^{- 1})) \cdot S_{2} \\ = (\hat{σ} \circ (1_{1_{C}} \times \hat{σ})) \cdot {\hat{α}}^{- 1} . \end{array}

This proves the fourth condition. The ﬁfth and sixth condition is proved in a similar way and we only prove the sixth.

\begin{array}{l} (t \hat{β}) \cdot (\hat{σ} \circ 1_{1_{C} \times K_{e}}) \\ = t β \cdot (t λ \circ (1_{1_{C}} \times t η)) \cdot (((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot (t λ^{- 1}) \cdot (μ \circ σ) \cdot λ) \circ 1_{1_{C} \times K_{e}}) \\ = γ \cdot (t σ \circ 1_{1_{C} \times K_{e}}) \cdot (t λ \circ (1_{1_{C}} \times t η)) \cdot (1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1}) \circ 1_{1_{C} \times K_{e}}) \cdot (t λ^{- 1} \circ 1_{1_{C} \times K_{e}}) \cdot (μ \circ σ \circ 1_{1_{C} \times K_{e}}) \cdot (λ \circ 1_{1_{C} \times K_{e}}) \\ = γ \cdot (μ \circ λ \circ (μ^{- 1} \times (μ^{- 1} \circ t η))) \\ = γ \cdot (μ \circ λ \circ (μ^{- 1} \times η)) \\ = γ \cdot (λ \circ (1_{1_{C}} \times η)) \cdot (μ \circ 1_{\otimes} \circ (μ^{- 1} \times 1_{K_{e}})) \\ = γ \cdot (λ \circ (1_{1_{C}} \times η)) \\ = \hat{γ} . \end{array}

For the seventh condition we have

\begin{array}{l} t \hat{σ} \\ = t ((1_{t \otimes} \circ (μ^{- 1} \times μ^{- 1})) \cdot t λ^{- 1} \cdot (μ \circ σ) \cdot λ) \\ = (1_{\otimes} \circ (μ \times μ)) \cdot λ^{- 1} \cdot (μ^{- 1} \circ σ^{- 1}) \cdot t λ \\ = (1_{\otimes} \circ (μ \times μ)) \cdot (λ^{- 1} \circ (1_{1_{C}} \times 1_{1_{C}})) \cdot ((μ^{- 1} \circ σ^{- 1}) \circ (1_{1_{C}} \times 1_{1_{C}})) \end{array}

\begin{array}{l} \cdot (t λ \circ (1_{1_{C}} \times 1_{1_{C}})) \\ = (λ^{- 1} \cdot (μ^{- 1} \circ σ^{- 1}) \cdot t λ \cdot 1_{t \otimes}) \circ (1_{1_{C} \times 1_{C}} \cdot (μ \times μ)) \\ = λ^{- 1} \cdot (μ^{- 1} \circ σ^{- 1}) \cdot t λ \cdot (1_{t \otimes} \circ (μ \times μ)) \\ = {\hat{σ}}^{- 1} . \end{array}

□

From this point of view the quantizations of the identity functor on a symmetric monoidal category $〈 C, \otimes, K_{e}, α, β, γ, σ 〉$ is exactly equal to the subgroup of $H$ that ﬁx the point $(α, β, γ, σ)$ .

4.2. Quantization of algebraic structures . Let $〈 C_{i}, \otimes_{i}, P_{e_{i}}, α_{i}, β_{i}, γ_{i}, σ_{i} 〉$ be symmetric monoidal categories for $i = 1, 2$ and let $F : C_{1} \to C_{2}$ be a quantized functor with quantization $(λ, μ, η)$ . Let the $S_{2}$ action on $C_{1}$ and $C_{2}$ be generated by the functors $T_{1} : C_{1} \to C_{1}$ and $T_{2} : C_{2} \to C_{2}$ . In this section we will work with objects and need the object formulation of the conditions deﬁning a symmetric monoidal category and quantized functors. We collect these conditions in the following proposition whose proof consists of applying the deﬁnition of vertical composition and horizontal composition.

Proposition 68.

\begin{array}{l} {(α_{2})}_{F (X), F (Y), F (Z)} & = (λ_{X, Y}^{- 1} \otimes_{2} 1_{F (Z)}) \circ λ_{X \otimes_{1} Y, Z}^{- 1} \circ F ({(α_{1})}_{X, Y, Z}) \circ λ_{X, Y \otimes_{1} Z} \circ (1_{F (X)} \otimes_{2} λ_{Y, Z}), {(β_{2})}_{F (X), F (Y)} \\ = F (β_{X; Y}) \circ λ_{e_{1}, Y} \circ (η_{X} \otimes_{2} 1_{F (Y)}), \\ {(γ_{2})}_{F (X), F (Y)} & = F ({(γ_{1})}_{X, Y}) \circ λ_{X, e_{1}} \circ (1_{F (X)} \otimes_{2} η_{Y}), {(σ_{2})}_{F (X), F (Y)} \\ = T_{2} (T_{2} (μ_{Y}^{- 1}) \otimes_{2} T_{2} (μ_{X}^{- 1})) \circ T_{2} (λ_{T_{1} (Y), T_{1} (X)}^{- 1}) \circ μ_{T_{1} (T_{1} (Y) \otimes_{1} T_{1} (X))} \circ F (σ_{X; Y}) \circ λ_{X, Y} . \end{array}

Quantized functors preserve algebraic structures. Let $〈 X, ν, u 〉$ be a monoid in the symmetric monoidal category $C_{1}$ and Deﬁne arrows in $C_{2}$

\begin{array}{l} ν^{λ} & : F (X) \otimes_{2} F (X) \to F (X), \\ u^{η} & : e_{2} \to F (X), \end{array}

by $ν^{λ} = F (ν) \circ λ_{X, X}$ and $u^{η} = F (u) \circ η$ . In

Proposition 69. $〈 F (X), ν^{λ}, u^{η} 〉$ is a monoid in $C_{2}$ .

Proof. Since $〈 X, ν, u 〉$ is a monoid in $C_{1}$ we have the identities

\begin{array}{l} ν \circ (1_{X} \otimes_{1} ν) & = ν \circ (ν \otimes_{1} 1_{X}) \circ {(α_{1})}_{X, X, X}, \\ ν \circ (u \otimes_{1} 1_{X}) & = {(β_{1})}_{X, X}, \\ ν \circ (1_{X} \otimes_{1} ν) & = {(γ_{1})}_{X, X} . \end{array}

If we use these identities and the relations from proposition 68 we have

\begin{array}{l} ν^{λ} \circ (1_{F (X)} \otimes_{2} ν^{λ}) \\ = F (ν) \circ λ_{X, X} \circ (1_{F (X)} \otimes_{2} F (ν)) \circ (1_{F (X)} \otimes_{2} λ_{X, X}) \\ = F (ν) \circ F (1_{X} \otimes_{1} ν) \circ λ_{X, X \otimes_{1} X} \circ (1_{F (X)} \otimes_{2} λ_{X, X}) \\ = F (ν \circ (1_{X} \otimes_{1} ν)) \circ λ_{X, X \otimes_{1} X} \circ (1_{F (X)} \otimes_{2} λ_{X, X}) \\ = F (ν) \circ F (ν \otimes_{1} 1_{X}) \circ F ({(α_{1})}_{X, X, X}) \circ λ_{X, X \otimes_{1} X} \circ (1_{F (X)} \otimes_{2} λ_{X, X}) \\ = F (ν) \circ F (ν \otimes_{1} 1_{X}) \circ λ_{X \otimes_{1} X, X} \circ (λ_{X, X} \otimes_{2} 1_{F (X)}) \circ {(α_{2})}_{F (X), F (X), F (X)} \\ = F (ν) \circ λ_{X, X} \circ (F (ν) \otimes_{2} 1_{F (X)}) \circ (λ_{X, X} \otimes_{2} 1_{F (X)}) \circ {(α_{2})}_{F (X), F (X), F (X)} \\ = ν^{λ} \circ (ν^{λ} \otimes_{2} 1_{F (X)}) \circ {(α_{2})}_{F (X), F (X), F (X)}, \end{array}

and

\begin{array}{l} ν^{λ} \circ (u^{λ} \otimes_{2} 1_{F (X)}) \\ = F (ν) \circ λ_{X, X} \circ (F (u) \otimes_{2} F (1_{X})) \circ (η_{X} \otimes_{2} 1_{F (X)}) \\ = F (ν) \circ F (u \otimes_{1} 1_{X}) \circ λ_{e_{1}, X} \circ (η_{X} \otimes_{2} 1_{F (X)}) \\ = F (ν \circ (u \otimes 1_{X})) \circ λ_{e_{1}, X} \circ (η_{X} \otimes_{2} 1_{F (X)}) \\ = F ({(β_{1})}_{X, X}) \circ λ_{e_{1}, X} \circ (η_{X} \otimes_{2} 1_{F (X)}) \\ = {(β_{2})}_{F (X), F (X)} . \end{array}

□

We call the monoid $〈 F (X), ν^{λ}, u^{η} 〉$ a quantization of the monoid $〈 X, ν, u 〉$ in $C_{1}$ . Quantization of comonoids is deﬁned by duality. Let us assume that the monoid $〈 X, ν, u 〉$ is commutative. This property is preserved by quantization.

Proposition 70. Let $〈 X, ν, u 〉$ be a commutative monoid in $C_{1}$ . Then

$〈 F (X), ν^{λ}, u^{η} 〉$ is a commutative monoid in $C_{2}$ .

Proof. Using the exchange identity for horizontal and vertical composition of natural transformations, the two last conditions in the deﬁnition of quantized functors 66 and the symmetry conditions $t σ_{i} = σ_{i}^{- 1}, i = 1, 2$ we get the following identity

t (1_{F} \circ σ_{1}) \cdot (t λ) \cdot (1_{t \otimes_{2}} \circ (μ \times μ)) = (μ \circ 1_{\otimes}) \cdot λ \cdot (t σ_{2} \circ 1_{F \times F}) .

The $(X, X, X)$ component of this identity is gives after application of the functor $T_{2}$ the following relation

\begin{array}{l} F ({(σ_{1})}_{T_{1} (X), T_{1} (X)}) \circ λ_{T_{1} (X), T_{1} (X)} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = T_{2} (μ_{X \otimes_{1} X}) \circ T_{2} (λ_{X, X}) \circ {(σ_{2})}_{T_{2} (F (X)), T_{2} (F (X))} . \end{array}

But then we have

\begin{array}{l} T_{2} (μ_{X}) \circ {(ν^{λ})}^{σ_{2}} \\ = T_{2} (μ_{X}) \circ T_{2} (F (ν)) \circ T_{2} (λ_{X, X}) \circ {(σ_{2})}_{T_{2} (F (X)), T_{2} (F (X))} \\ = F (T_{1} (ν)) \circ T_{2} (μ_{X \otimes_{1} X}) \circ T_{2} (λ_{X, X}) \circ {(σ_{2})}_{T_{2} (F (X)), T_{2} (F (X))} \\ = F (T_{1} (ν)) \circ F ({(σ_{1})}_{T_{1} (X), T_{1} (X)}) \circ λ_{T_{1} (X), T_{1} (X)} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = {(ν^{σ_{1}})}^{λ} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) . \end{array}

Since $〈 X, ν, u 〉$ is commutative in $C_{1}$ there exists an isomorphism $ϕ : T_{1} (X) \to X$ such that the following identity holds

ϕ \circ ν^{σ_{1}} = ν \circ (ϕ \otimes_{1} ϕ) .

Let the isomorphism $\hat{ϕ} : T_{2} (F (X)) \to F (X)$ be deﬁned by $\hat{ϕ} = F (ϕ) \circ T_{2} (μ_{X})$ . For this isomorphism in $C_{2}$ we have

\begin{array}{l} \hat{ϕ} \circ {(ν^{λ})}^{σ_{2}} \\ = F (ϕ) \circ T_{2} (μ_{X}) \circ {(ν^{λ})}^{σ_{2}} \\ = F (ϕ) \circ {(ν^{σ_{1}})}^{λ} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = F (ϕ) \circ F (ν^{σ_{1}}) \circ λ_{T_{1} (X), T_{1} (X)} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = F (ν) \circ F (ϕ \otimes_{1} ϕ) \circ λ_{T_{1} (X), T_{1} (X)} \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = F (ν) \circ λ_{X, X} \circ (F (ϕ) \otimes_{2} F (ϕ)) \circ (T_{2} (μ_{X}) \otimes_{2} T_{2} (μ_{X})) \\ = ν^{l} \circ (\hat{ϕ} \otimes_{2} \hat{ϕ}), \end{array}

and this proves that $〈 F (X), ν^{λ}, u^{λ} 〉$ is a commutative monoid. □

Commutative comonoids will by duality also be preserved by quantization. Similar results holds for other algebraic structures like modules and comodules. As a special case of the above constructions let $F = I_{C}$ and let $〈 X, ρ, u 〉$ be a commutative monoid in $C$ . Then any element $(λ, μ, η)$ in the group $H$ described in the previous section deﬁnes a quantization $〈 X, ρ^{λ}, u^{η} 〉$ of the given monoid. We thus get a whole family of quantized product and unit structures on the object $X$ . Each such quantized product and unit does not deﬁne a commutative monoid with respect to the original structure $〈 α, β, γ, σ 〉$ , but with respect to the structure $〈 \hat{α}, \hat{β}, \hat{γ}, \hat{σ} 〉$ .

References

[1] Fronsdal C. Lichnerowicz A. Bayen F., Flato M. and Sternheimer D. Deformation theory and quantization , II. Ann. Phys. (NY), 111:61–110,111–151, 1978.

[2] Louise De Broglie. Ondes et quanta. Comptes rendus de l’Academie des Sciences, 177:507–510, September 1923.

[3] Louise De Broglie. Recherches sur la Theorie Des Quanta. PhD thesis, University of Paris, 1924.

[4] Louise De Broglie. Phase wave of louise deBroglie. A. J. Phys., 40(9):1315–1320, September 1972.

[5] Albert Einstein. Zur theorie der lichterzeugung and lichtabsorption. Annalen der Physik, page 1, 1906.

[6] W. Heisenberg. Uber quantentheoretische umdentung kinematischer und mechanischer beziehungen. Zeitschrift fur physik, 33:879–893, Juli 1925.

[7] Woodhouse N. M. J. Geometric Quantization. Oxford Mathematical Monographs. Oxford Science Publications, 2 edition, 1992.

[8] Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. springer, 1998.

[9] V. V. Lychagin. Colour calculus and colour quantizations. geometric and algebraic structures in differential equations. Acta Appl. Math., 41(1-3):193 – 226, 1995.

[10] V. V. Lychagin. Calculus and quantizations over hopf algebras. Acta Appl. Math., 51(3):303–352, 1998.

[11] V. V. Lychagin. Quantum mechanics on manifolds. geometrical aspects of nonlineardifferential equations. Acta Appl. Math., 56(2-3):231–251, 1999.

[12] M. Planck. Zur theorie des gesetzes der energieverteilung im normalspectrum. Verhandl. Dtsch. phys. Ges., 2:237, 1900.

[13] Jakub Rembielinski, editor. Deformation Quantization: Twenty Years After. Amer. Inst. Phys., Woodbury, Ny., 1998.

[14] B. L. Van Der Waerden. Sources of Quantum Mechanics. Dover, 1967.

Faculty of science, University of Tromsø, Tromsø 9037, Norway

E-mail address: perj@math.uit.no

E-mail address: Valentin.Lychagin@matnat.uit.no

Received November 10, 2004