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} & = \end{array}