Classification of grouplike categories*

ABSTRACT

In this paper we study grouplike monoids, defined as being monoids that contain a group to which we add an ordered set of idempotents. We classify finite categories with two objects having grouplike endomorphism monoids, by presenting a construction theorem for such categories, and proving that every grouplike category comes from this construction. Studying the algebraic properties of the endomorphism monoids allows us to gather extra information on the category itself, which in particular helps the counting problem because the nature of the monoids affects greatly the structure of the category. At the end of the paper, we give a count of certain categories with grouplike monoids, concluded from the properties of grouplike monoids that are studied in the paper.

KEYWORDS:

• finite categories
• grouplike monoids
• grouplike categories
• groupoids
• bimodules
• Mace4

1. Introduction

Finite categories are categories with a finite set of objects and a finite set of morphisms. The main purpose of our work in general (Ghannoum, 2022) thesis is the classification of finite categories. One associates to a finite category a square matrix such that the entries of the matrix are the numbers of morphisms between each pair of objects.

A more informative description of a finite category is obtained using the endomorphism monoids. We classify monoids of certain orders, then look at which categories have a given set of endomorphism monoids. For example, two monoids ${\mathsf{\text{C}}}_i,{\mathsf{\text{C}}}_j$ are called connected if there exists a category with two objects and non-empty morphism sets in both directions, where the monoids ${\mathsf{\text{C}}}_i$ and ${\mathsf{\text{C}}}_j$ are the endomorphism monoids of the two objects. Each object in the category is going to correspond to one of the monoids of endomorphisms that we specify. A groupoid is a category in which every morphism is invertible.

Starting from this idea, we can define matrices associated to categories in a different way. Instead of putting the number of morphisms between objects, we represent them in terms of the endomorphism monoids and bimodules of the category. For example a category $\mathsf{\text{C}}$ with two objects $X,Y$ will be represented as follows:

$\begin{array}{l}\left(\begin{array}{ll}A& L\\ R& B\end{array}\right)\end{array}$

where

$\begin{array}{l}\mathsf{\text{C}}(X,X)=A,\mathsf{\text{C}}(X,Y)=L,\mathsf{\text{C}}(Y,X)=R\text{and}\mathsf{\text{C}}(Y,Y)=B\end{array}$

such that A and B are monoids with a specific algebraic structure.

In this perspective the off-diagonal parts of the matrices (L and R above) are seen as bimodules over the corresponding endomorphism monoids, taking only the multiplication tables of $A\cdot L\cdot B$ and $B\cdot R\cdot A$. In some cases, the number of bimodules obtained specifies in a unique way the number of categories that could be obtained by two bimodules (we give some examples in section 5).

We study in this paper a specific type of finite categories called grouplike categories. To define grouplike categories, we need to define grouplike monoids first. A grouplike monoid is a monoid of the form $G\cup I$ where G is a group and $I=\{e_1,\dots ,e_k\}$ such that

(Ord) $(e_i\cdot e_j=e_i\text{and}{e}_j\cdot e_i=e_i)\text{if and only if}i\le j.$(Ord)

We denote it by $G^{\ast k}$. A grouplike category is a category whose endomorphism monoids are grouplike.

Counting associative structures has been a problem for years. We introduce some of the previous work in this area:

In 2009, A. Distler and T. Kelsey counted the monoids of orders eight, nine and ten. They weren’t able to achieve more than that as the number of semigroups of order 10 was unknown (Distler & Kelsey, 2009).

In 2012, the number of semigroups of order 10 was known by A. Distler, C. Jefferson, T. Kelsey, and L. Kotthoff (Distler et al., 2012).

In 2014, G. Cruttwell and R. Leblanc introduced the question: How many categories are there with n morphisms? It means with a total number of morphisms distributed between objects in an arbitrary way. They compared the numbers obtained with the number of monoids of order n, which lead to almost the same numbers up to order 10.

In 2017, S. Allouch and C. Simpson counted the categories whose number of morphisms between each pair of objects is 2. They were able to get an exact count up to order 3, and bounds for a general size (Allouch & Simpson, 2018).

This work is inspired by our results with W. Fussner, T. Jakl and C. Simpson in (Fussner et al., 2020). Using the program Mace4 (McCune, 2003), we gave a count of the categories associated to the matrix

$\begin{array}{l}\left(\begin{array}{ll}3& 3\\ 3& 3\end{array}\right).\end{array}$

We classified monoids of order 3 and presented the number of categories between each pair of monoids. The data obtained from the count showed that the nature of the endomorphism monoids affects the classification in a very important way, which led us to prove some properties about certain monoids in a category, and to propose classifying finite categories in terms of their endomorphism monoids. In particular, in this paper we study groups and monoids that contain a group. The classification of such monoids gives very interesting properties inside the category and gives us lots of additional information allowing us to compute the number of categories with such monoids in some cases.

We present a construction theorem for grouplike categories, and we prove that every grouplike category comes from this construction. We prove that having grouplike monoids as endomorphism sets affects the structure of the bimodules, and in this way, the category is almost determined with a strong part of its properties being known.

The notion of grouplike monoids can be generalized in the following way: instead of adding k identities to a group G, we add k different groups consecutively, leading to what could be then called an iterative grouplike monoid. This idea of generalization was suggested by Sam van Gool during a seminar at IRIF in Paris. Although we don’t pursue that here, it will furnish the opportunity for future work in the direction of the results of the present paper.

Using the notation $G^{\ast k}$ for grouplike monoids defined above, we now describe in more detail the results. The first definition encapsulates the condition that will appear in Theorem 4.11.

Definition 1.1.

Let L be a $(G_1^{\ast k_1},G_2^{\ast k_2})$-bimodule. We say that L is i-unigen if

$\begin{array}{l}L\cdot f_i=e_i\cdot L:=L_i\end{array}$

and $L_i\simeq G_1^{\ast i}$ as a left module and $L_i\simeq G_2^{\ast i}$ as a right module.

Remark 1.2.

The definition of L_i is given in (Equation L(L) $L_{ij}:=\{x\in L\mid e_i\cdot x=x\text{and}x\cdot f_j=x\}=e_i\cdot L\cdot f_j$(L) ), and the definition of $G^{\ast k}$ is given in Definition 2.6 and Definition 2.7.

Lemma 1.3.

Suppose L is an i-unigen $(G_1^{\ast k_1},G_2^{\ast k_2})$-bimodule with L_i as above, then there exists an isomorphism $G_1\simeq G_2$ such that $L_i=G_1^{\ast i}$ as $(G_1^{\ast k_1},G_2^{\ast k_2})$-bimodule. If i > 0 then the isomorphism is unique. If i = 0 then the isomorphism is well defined up to inner automorphisms.

Because of this lemma we can assume that $G_1=G_2=G$. In this case we say that a $(G^{\ast k_1},G^{\ast k_2})$-bimodule L is strongly i-unigen if it is i-unigen and the isomorphism of Lemma 1.3 can be taken as the identity.

Remark 1.4.

The proof of Lemma 1.3 is on page 19.

Definition 1.5.

Let

$\begin{array}{l}M=\left(\begin{array}{ll}{G}^{\ast k_1}& L\\ R& G^{\ast k_2}\end{array}\right)\end{array}$

be the algebraic matrix (Definition 2.3) of a grouplike category $\mathsf{\text{C}}$, such that $I_1=\{e_1,\dots ,e_{k_1}\}$ and $I_2=\{f_1,\dots ,f_{k_2}\}$. We define $i^{max}(\mathsf{\text{C}})$ to be the maximum index i such that there exist $x,y$ such that $x\cdot y=e_i$ and $y\cdot x=f_i$.

Let $\mathsf{\text{C}}$ be a category with one object $\{\alpha \}$ and endomorphism set $G^{\ast i}$. Let $x,y$ be two elements, and using the constant function

$\begin{array}{l}f:\{x,y\}\to \alpha \end{array}$

we obtain a new category ${\mathsf{\text{C}}}^{\prime }$ that could be denoted ${\mathsf{\text{C}}}^{\prime }=f^{\ast }(\mathsf{\text{C}})$, with $Ob({\mathsf{\text{C}}}^{\prime })=\{x,y\}$ and ${\mathsf{\text{C}}}^{\prime }(x,y)=\mathsf{\text{C}}(f(x),f(y))$.

If we apply this construction to a grouplike monoid $G^{\ast i}$, we get a category associated to the matrix

($M^{(1)}$) $\left(\begin{array}{ll}{G}^{\ast i}& G^{\ast i}\\ G^{\ast i}& G^{\ast i}\end{array}\right)$($M^{(1)}$)

of similar copies of $G^{\ast i}$. Such a category will be called groupoid-like.

Theorem 1.6.

Let G be a group and $G^{\ast i}$ a grouplike monoid of the form $G\cup I$ such that $I=\{e_1,\dots ,e_i\}$. Let $M^{(1)}$ be a matrix of the form

($M^{(1)}$) $\left(\begin{array}{ll}{G}^{\ast i}& G^{\ast i}\\ G^{\ast i}& G^{\ast i}\end{array}\right)$($M^{(1)}$)

of similar copies of $G^{\ast i}$, let ${\mathsf{\text{C}}}^{(1)}$ be the groupoid-like category associated to this matrix. Then we can extend the endomorphism sets $G^{\ast i}$ and we obtain a category ${\mathsf{\text{C}}}^{(2)}$ associated to the matrix

($M^{(2)}$) $\left(\begin{array}{ll}{G}^{\ast k_1}& G^{\ast i}\\ G^{\ast i}& G^{\ast k_2}\end{array}\right)$($M^{(2)}$)

such that for all $x\in G^{\ast i},y\in G^{\ast k_1}$, we have $x\cdot y=y\cdot x\in G^{\ast i}$. Same for $G^{\ast k_2}$. And there exists ${\mathsf{\text{C}}}^{(2)}$ a finite category associated to $M^{(2)}$.

Now let $k_1,k_2\ge i$, suppose that we have the matrix

($M^{(3)}$) $\left(\begin{array}{ll}{G}^{\ast k_1}& L\\ R& G^{\ast k_2}\end{array}\right)$($M^{(3)}$)

such that L and R are strongly i-unigen bimodules. Then

$(M^{(1)})\subseteq (M^{(2)})\subseteq (M^{(3)}).$

And (Equation M⁽³⁾($M^{(3)}$) $\left(\begin{array}{ll}{G}^{\ast k_1}& L\\ R& G^{\ast k_2}\end{array}\right)$($M^{(3)}$) ) is a matrix of a unique grouplike category, denote it by ${\mathsf{\text{C}}}^{(3)}$, such that $i^{max}({\mathsf{\text{C}}}^{(3)})=i$. For all $x\in L,y\in R$, we have $x\cdot y=e_i\cdot x\cdot y\cdot e_i$.

We have

$\begin{array}{l}{\mathsf{\text{C}}}^{(1)}\subseteq {\mathsf{\text{C}}}^{(2)}\subseteq {\mathsf{\text{C}}}^{(3)}.\end{array}$

Theorem 1.7.

Every grouplike category comes from the construction described in Theorem 1.6.

Remark 1.8.

The proofs of Theorems 1.6 and 1.7 are on pages 19 and 20.

1.1. Organization of the paper

This paper is organized as follows:

1. In section 2, we give the definitions of grouplike monoids, bimodules and grouplike categories.

2. Since a grouplike monoid has two parts: a group G and a set I of idempotents, then we divide the discussion of each part into two sections:

   1. In section 3, we discuss the group part of the category and how the action of the group on the set of morphisms affects the category its structure.

   2. In section 4, we discuss the set of idempotents part which is formed of n elements, and we give the maximum element $i\in \{1,\dots ,n\}$ such that a grouplike category can be determined. This leads to the proofs of Theorems 1.6 and 1.7.

3. In section 5, we give some examples and we apply the results in the previous sections in order to obtain a count of the number of certain grouplike categories.

2. Preliminaries

In this section we introduce grouplike categories. These are groupoids (groups in the case of monoids) in which we add extra elements, having some specific properties, to their set of morphisms. The goal is to study the structure of these categories in order to make the classification problem easier and clearer.

Definition 2.1.

A monoid A is a set equipped with a binary operation $\cdot :A\times A\to A$ such that · is associative and there exists an identity element e such that for every element $a\in A$, the equations $e\cdot a=a$ and $a\cdot e=a$ hold.

Definition 2.2.

A bimodule is a set with actions on the left and the right of the respective monoids, such that the actions commute i.e. $(g\cdot x)\cdot h=g\cdot (x\cdot h)$. It can be seen as a category such that one of the sets of morphisms is empty, it’s called an upper triangulated category.

Definition 2.3.

(Carboni et al., 1987; Koslowski, 1997; Leinster, 1999a, 1999b, 2002). Let $\mathsf{\text{C}}$ be a finite category. Then we get a matrix where the diagonal entries are monoids and the off diagonals entries are bimodules. This matrix is called the algebraic matrix.

Definition 2.4.

A semicategory is a category without identity morphisms.

Lemma 2.5.

(Allouch & Simpson, 2014) Let $\mathsf{\text{C}}$ be a semicategory, let $u\in \mathsf{\text{C}}(x,y)$ with $x,y\in Ob(\mathsf{\text{C}})$, then we can add a morphism u^ʹ to $\mathsf{\text{C}}(x,y)$ such that $u^{\prime }\ne u$ and u^ʹ duplicates u for all composition operations. Then we get a new semicategory ${\mathsf{\text{C}}}^{\prime }$ with $Ob({\mathsf{\text{C}}}^{\prime })=Ob(\mathsf{\text{C}})$ and $Hom({\mathsf{\text{C}}}^{\prime })=Hom(\mathsf{\text{C}})\cup \{u^{\prime }\}$.

On the other hand, we can also add the missing identities in $Ob(\mathsf{\text{C}})$ to obtain a category $\mathsf{\text{B}}$.

The previous lemma shows that we can add morphisms consecutively to a category and obtain a new category (provided that we add identities too). In our case, we define a category whose objects have grouplike endomorphism monoids, we only add elements consecutively to the monoids. The elements are idempotents and identities to the elements of the monoids. This means that each time we add an identity element, the previous identity is no longer an identity.

We need the notion of bimodules in order to make the classification and counting problem easier.

Definition 2.6.

Let $\mathsf{\text{A}}$ be a semigroup. Whether or not $\mathsf{\text{A}}$ has a multiplicative identity element, we let e be a fresh element and

${\mathsf{\text{A}}}^{\ast }:=\mathsf{\text{A}}\cup \{e\}.$

Then ${\mathsf{\text{A}}}^{\ast }$ is a semigroup if the multiplication of $\mathsf{\text{A}}$ is extended by stipulating $xe=ex=x$ for all $x\in {\mathsf{\text{A}}}^{\ast }$. More generally, if $\mathsf{\text{A}}$ is a semigroup, we recursively define semigroups ${\mathsf{\text{A}}}^{\ast n}$ for $n\in \mathbb{N}$ by:

$\begin{array}{ll}{\mathsf{\text{A}}}^{\ast 0}=& \mathsf{\text{A}}\\ {\mathsf{\text{A}}}^{\ast (n+1)}=& ({\mathsf{\text{A}}}^{\ast n}{)}^{\ast }.\end{array}$

If $\mathsf{\text{A}}$ is a group, we say that the semigroups ${\mathsf{\text{A}}}^{\ast n}$, $n\in \mathbb{N}$, are grouplike.

Definition 2.7.

We say that a category is called a grouplike category with groups G_i if its endomorphism monoids are grouplike monoids of the form $G_i^{\ast k_i};k_i\in \mathbb{N}$.

Definition 2.8.

A band S is an idempotent semigroup, i.e. for all $x\in S$, $x^2=x$.

A semilattice is a commutative band.

Remark 2.9.

The set of idempotents in a grouplike monoid along with the group identity form a semilattice.

Proof.

Let $I=\{e_0,\dots ,e_k\}$ be the set of idempotents where e₀ is a group identity. I is a semilattice:

• commutative: for $i\le j$, $e_i\cdot e_j=e_j\cdot e_i=e_i$.

• idempotent: $e_i\cdot e_i=e_i$.

3. Group action and the orbits of the sets of morphisms

Let $\mathsf{\text{C}}$ be a category with n objects $X_1,\dots ,X_n$. For each object $X_i,X_j$ there exists a monoid $A_i=\mathsf{\text{C}}(X_i,X_i)$ and a monoid $A_j=\mathsf{\text{C}}(X_j,X_j)$ and two operations

$\begin{array}{l}l:A_i\times \mathsf{\text{C}}(X_i,X_j)\to \mathsf{\text{C}}(X_i,X_j)\end{array}$

and

$\begin{array}{l}r:\mathsf{\text{C}}(X_i,X_j)\times A_j\to \mathsf{\text{C}}(X_i,X_j)\end{array}$

which represent the left monoid action of A_i and the right monoid action of A_j on the set of morphisms from X_i to X_j. Note here that these are the entries of the matrix defined in Definition 2.3.

In our work here, we take monoids of the form ${\mathsf{\text{A}}}^{\ast n}$, specifically grouplike, which means that each monoid contains a subgroup. This subgroup does not act on the whole set of morphisms from X_i to X_j, but it acts on a subset of the previous set (it’s important to note here that when we say group action we mean that the identity condition holds). In the following, we introduce how the group action works, and what are exactly the subsets that the group acts on. We will be considering categories with two objects.

To avoid large paragraphs every time we have a grouplike category, we give a notation in which we specify its exact form and the structure of its grouplike monoids.

Notation 3.1.

We denote by $M(G_1^{\ast k_1},G_2^{\ast k_2},L,R)$ the matrix

$M=\left(\begin{array}{ll}{G}_1^{\ast k_1}& L\\ R& G_2^{\ast k_2}\end{array}\right)$

of monoids and bimodules such that $G_1^{\ast k_1}=G_1\cup I_1$ and $G_2^{\ast k_2}=G_2\cup I_2$, where G₁ and G₂ are groups and $I_1=\{e_1,\dots ,e_{k_1}\}$ and $I_2=\{f_1,\dots f_{k_2}\}$ such that the elements of I₁ and I₂ satisfy (Equation (Ord)(Ord) $(e_i\cdot e_j=e_i\text{and}{e}_j\cdot e_i=e_i)\text{if and only if}i\le j.$(Ord) ). We denote by e₀ and f₀ the identities of the groups G₁ and G₂. Let $Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ be the set of grouplike categories associated to M whose objects are are $X,Y$ such that the monoids and the bimodules are not empty. If $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ then $\mathsf{\text{C}}(X,X)=G_1^{\ast k_1}$, $\mathsf{\text{C}}(X,Y)=L$, $\mathsf{\text{C}}(Y,X)=R$ and $\mathsf{\text{C}}(Y,Y)=G_2^{\ast k_2}$.

Remark 3.2.

More precisely if M is an algebraic matrix i.e. a matrix of monoids and bimodules, then Cat(M) is the set of pairs $(\mathsf{\text{C}},\beta )$ where $\mathsf{\text{C}}$ is a category and β is an isomorphism between the algebraic matrix of $\mathsf{\text{C}}$ and M. We usually don’t include this notation of β in our discussion.

Remark 3.3.

Denote by n_i the order of the group G_i. When we write $G^{\ast k_1}$ and $G^{{}^{\prime }\ast k_2}$, this means that the groups G and G^ʹ have the same order n.

Lemma 3.4.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Then:

1. G₁ acts on $G_1\cdot L$ and $R\cdot G_1$.

2. G₂ acts on $L\cdot G_2$ and $G_2\cdot R$.

Proof.

Let

$l:G_1\times (G_1\cdot L)\to G_1\cdot L.$

• For all $x\in L$ and $g\in G_1$ we have $e_0\cdot (g\cdot x)=g\cdot x$.

• For all $g_1,g_2,g_3\in G_1$ and $x\in L$ we have $(g_2\cdot g_3)\cdot (g_1\cdot x)=g_2\cdot (g_3\cdot g_1\cdot x)$.

The same holds for the other sets.

Moreover, the orbit of the set $G_1\cdot L$ is itself. Indeed, the orbit of $G_1\cdot L$ is a subset $G_1\cdot (G_1\cdot L)$. It remains to prove the other direction. Let $g\cdot x\in G_1\cdot L$, then

$g\cdot x=e_0\cdot g\cdot x\in G_1\cdot (G_1\cdot L).$

Similarly to the case where we have groups as objects, we can conclude that the group action on these sets is free.

Proposition 3.5.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. The actions of the group G₁ on $G_1\cdot L$ and $R\cdot G_1$ and the group G₂ on $L\cdot G_2$ and $G_2\cdot R$ are all free.

Proof.

Let $g_1,g_2,g\in G_1$ and $x\in L$. Suppose $g_1\cdot (g\cdot x)=g_2\cdot (g\cdot x)$. If we multiply both sides by $y\in R$, we obtain:

$g_1\cdot g\cdot x\cdot y=g_2\cdot g\cdot x\cdot y$

where $x\cdot y$ has 2 possibilities:

1. If $x\cdot y=e_i$ where $e_i\notin G_1$, then it’s sufficient to multiply by g⁻¹ on both sides to prove that $g_1=g_2$.

2. If $x\cdot y=g_3$ where $g_3\in G_1$, then if we multiply by $g_3^{-1}\cdot g^{-1}$ on both sides, we get $g_1=g_2$.

Corollary 3.6.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. The cardinal of each orbit in the set of morphisms is equal to the order of the group acting.

Proof.

Use Proposition 3.5.

This corollary could greatly help the enumeration problem here, because it tells us the number of possibilities in some blocks of a category, such as $G_1\cdot L$ and $L\cdot G_2$ for example, then we can compute how many times the multiplication of morphisms appear to obtain non isomorphic copies of blocks.

Lemma 3.7.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. For all $x:X\to Y$ there exists at least one $y:Y\to X$ such that $x\cdot y=i{d}_{G_1}$ and vice versa.

Proof.

Let $x\in \mathsf{\text{C}}(X,Y)$ and $y\in \mathsf{\text{C}}(Y,X)$. Suppose $x\cdot y\ne i{d}_{G_1}$:

1. If $x\cdot y=g\in G_1$, then

   $x\cdot y\cdot g^{-1}=i{d}_{G_1}.$

2. If $x\cdot y$ is equal to an idempotent e in $\mathsf{\text{C}}(X,X)$ then

   $\begin{array}{lll}x\cdot y& =& e\\ x\cdot y\cdot g& =& g;g\in G_1\\ x\cdot y\cdot g\cdot g^{-1}& =& i{d}_{G_1}.\end{array}$

Now since we have two group actions on the sets of morphisms, we want to understand the relation between these actions over the morphisms sets.

Lemma 3.8.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Then

$\begin{array}{l}{G}_1\cdot x\cdot G_2\subseteq G_1\cdot x\end{array}$

and

$\begin{array}{l}{G}_1\cdot x\cdot G_2\subseteq x\cdot G_2\end{array}$

for all $x\in \mathsf{\text{C}}(X,Y)$.

Proof.

Let $g\cdot x\cdot h\in G_1\cdot x\cdot G_2$,

$\begin{array}{lll}g\cdot x\cdot h& =& g\cdot x\cdot h\cdot e_2;e_2=i{d}_{G_2}\\ & =& g\cdot x\cdot h\cdot y\cdot x;y\in \mathsf{\text{C}}(Y,X)(\text{Lemma ,3.7})\\ & \in & G_1\cdot x.\end{array}$

Same for the second inequality.

Lemma 3.9.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Let $x\in L$, the orbit of x by the action of G₁ is the same orbit of x by the action of G₂, i.e. $G_1\cdot x=x\cdot G_2$.

Proof.

Suppose $n_1\le n_2$. From Lemma 3.4, there is a group action by G₁ on $G_1\cdot L$, and by Corollary 3.6

$|G_1\cdot x|=n_1.$

Similarly, there is a group action by G₂ on $G_1\cdot L\cdot G_2\subseteq G_1\cdot L$, then again by Corollary 3.6 we have

$\begin{array}{l}|g_1\cdot x\cdot G_2|=n_2\end{array}$

but

$\begin{array}{l}{g}_1\cdot x\cdot G_2\subseteq G_1\cdot x\cdot G_2\subseteq G_1\cdot x.\end{array}$

Therefore, $n_2\le n_1$, but we have $n_1\le n_2$, then we obtain that n₁ should be equal to n₂ and $G_1\cdot x\cdot G_2=G_1\cdot x$ and $G_1\cdot x\cdot G_2=x\cdot G_2$, hence

$G_1\cdot x=x\cdot G_2\text{for all}x\in L.$

Proposition 3.10.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. For all $x,y\in L$ we have

$G_1\cdot x=G_1\cdot y.$

Proof.

Let $g\cdot x\in G_1\cdot x$,

$\begin{array}{lll}g\cdot x& =& (g\cdot e_1)\cdot x;e_1=i{d}_{G_1}\\ & =& g\cdot (e_1\cdot x)\\ & =& g\cdot e_1\cdot x\cdot e_2(Lemma3.9)\\ & =& g\cdot e_1\cdot x\cdot z\cdot y\in G_1\cdot y(Lemma3.7)\end{array}$

and since $|G_1\cdot x|=|G_1\cdot y|$, hence equality.

Proposition 3.11.

Let $\mathsf{\text{C}}\in Cat(M(G^{\ast k_1},G^{{}^{\prime }\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Then the multiplication of the elements in the orbit of L and the elements in the orbit of R is the group G, i.e.

$\begin{array}{l}G\cdot \mathsf{\text{C}}(X,Y)\cdot \mathsf{\text{C}}(Y,X)\cdot G=G\end{array}$

and

$\begin{array}{l}{G}^{\prime }\cdot \mathsf{\text{C}}(Y,X)\cdot \mathsf{\text{C}}(X,Y)\cdot G^{\prime }=G^{\prime }.\end{array}$

Proof.

• $\subseteq$ : evident.

• $\supseteq$ : Let $g\in G$

  $g=g\cdot e_0\cdot e_0=g\cdot x\cdot y\cdot e_0\text{(Lemma, 3.7)}$

  where $x:X\to Y$ and $y:Y\to X$.

Definition 3.12.

Let $\mathsf{\text{C}}$ be a category with objects $\{X,Y\}$. Suppose that $g\cdot f$ is never equal to an identity for all $f,g\in Hom(\mathsf{\text{C}})$, then we can always reduce the category $\mathsf{\text{C}}$ to a new semi-category ${\mathsf{\text{C}}}^{\prime }$ such that $Ob({\mathsf{\text{C}}}^{\prime })=Ob(\mathsf{\text{C}})$ and $Hom({\mathsf{\text{C}}}^{\prime })=Hom(\mathsf{\text{C}})\setminus \{i{d}_{\mathsf{\text{C}}}\}$.

Similarly, we can eliminate morphisms that are not identities and still obtain a new semi-category.

The reason we want to eliminate some morphisms is because when we take a category whose objects have grouplike endomorphism monoids, then we could restrict the category to a smaller category with only groups as objects and the orbits in the sets of morphisms. Following this technique leads to proving some matrix properties about the coefficients on the off-diagonals. It will also clarify how such categories are built.

From the above lemmas and propositions, we conclude that we can divide each set of morphisms into two sets: the orbit of the set and the other elements that are not inside the orbit.

3.1. Groupoids

Definition 3.13.

In category theory, a groupoid generalizes the notion of group in several equivalent ways. A groupoid can be seen as a:

• group with a partial function replacing the binary operation;

• category in which every morphism is invertible. A groupoid with only one object is a usual group.

In the previous sections, we have discussed the structure of a grouplike category with two objects. From this category, we can extract a subcategory, which is exactly a groupoid, and we prove the following:

Theorem 3.14.

In every grouplike category with two objects, there is a sub-semicategory that is a groupoid with two objects, whose groups are the groups of the grouplike monoids.

Proof.

Let $\mathsf{\text{C}}$ be a grouplike category with objects $X,Y$. Let $\mathsf{\text{C}}(X,X)=G^{\ast k_1},\mathsf{\text{C}}(Y,Y)=G^{{}^{\prime }\ast k_2},\mathsf{\text{C}}(X,Y)=L$ and $\mathsf{\text{C}}(Y,X)=R$. $\mathsf{\text{G}}$ the sub-semicategory of $\mathsf{\text{C}}$ is of the form:

• $Ob(\mathsf{\text{G}})=\{G,G^{{}^{\prime }}\}$

• $Mor(\mathsf{\text{G}})=\{G,G^{{}^{\prime }},G\cdot L=L\cdot G^{{}^{\prime }},G^{{}^{\prime }}\cdot R=R\cdot G\}$

• Identities of $\mathsf{\text{G}}$ are 1_G and $1_{G^{{}^{\prime }}}$

• Let $x,y\in \mathsf{\text{G}}$, $x\circ y=x\cdot y$ with $\circ$ associative

Corollary 3.15.

Let $\mathsf{\text{C}}\in Cat(M(G^{\ast k_1},G^{{}^{\prime }\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. The category $\mathsf{\text{C}}$ determines an isomorphism between G and G^ʹ. The isomorphism is well defined up to inner automorphisms.

Proof.

Evident.

4. The sets of idempotents

In this section we study the role of the idempotent elements in the monoids, the idea is to interpret their action on the sets of morphisms.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Let:

$\begin{array}{l}{K}_1=\{e_0,e_1,\dots ,e_{k_1}\}\end{array}$

and

$\begin{array}{l}{K}_2=\{f_0,f_1,\dots ,f_{k_2}\}\end{array}$

be the sets of idempotents of $G_1^{\ast k_1}$ and $G_2^{\ast k_2}$ where $e_0=1_{G_1}$ and $f_0=1_{G_2}$. And let:

(L) $L_{ij}:=\{x\in L\mid e_i\cdot x=x\text{and}x\cdot f_j=x\}=e_i\cdot L\cdot f_j$(L)

and

(R) $R_{ji}:=\{y\in R\mid y\cdot e_i=y\text{and}{f}_j\cdot y=y\}=f_j\cdot R\cdot e_i$(R)

be the sets of morphisms that are fixed by e_i and f_j.

Notation 4.1.

$L_i:=L_{ii}$ and $R_j:=R_{jj}$.

In general we have

Lemma 4.2.

$L_{ij}=e_i\cdot L\cap L\cdot f_j$ and $R_{ji}=f_j\cdot R\cap R\cdot e_i$.

Proof.

We always have

$\begin{array}{l}{L}_{ij}\subseteq e_i\cdot L\text{and}{L}_{ij}\subseteq L\cdot f_j\end{array}$

then

$\begin{array}{l}{L}_{ij}\subseteq e_i\cdot L\cap L\cdot f_j.\end{array}$

Now let $x\in e_i\cdot L\cap L\cdot f_j$ then $e_i\cdot x=x$ and $x\cdot f_j=x$ hence $x\in L_{ij}$.

In the following lemma, we present some properties of the multiplication of the sets of morphisms by idempotent elements.

Lemma 4.3.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category, we will denote by $x,x^{\prime }$ elements of L and by $y,y^{\prime }$ elements of R.

1. $e_i\cdot x=x$ and $x\cdot f_j=x$ for all $e_i\in K_1$, $f_j\in K_2$ and $x\in L_0$.

2. $e_j\cdot (e_i\cdot x)=e_i\cdot x$ and $(x\cdot f_i)\cdot f_j=x\cdot f_i$ for all $x\in L$ and $i\le j$.

3. $x\cdot y\in G_1$ and $y\cdot x\in G_2$ for all $x\in L_0$ or $y\in R_0$.

4. If $x\cdot y\in K_1\setminus \{e_0\}$ then $y\cdot x\in K_2\setminus \{f_0\}$ and vice versa.

5. If $x\cdot y=e_0$ then $y\cdot x=f_0$. (This result is also proved in Lemma 4.4).

6. If $x\cdot y=e_i$ and $y\cdot x=f_j$ then $x\cdot f_j=e_i\cdot x$ and $y\cdot e_i=f_j\cdot y$.

   In this case we can always assume that $x\cdot f_j=e_i\cdot x=x$ and $y\cdot e_i=f_j\cdot y=y$ (because x and y could be replaced with $e_i\cdot x=x\cdot f_j$ and $y\cdot e_i=f_j\cdot y$ respectively).

Proof.

1. Let $x\in L_0$ and $e_i\in K_1$, we have:

   $e_i\cdot x=e_i\cdot (e_0\cdot x)=(e_i\cdot e_0)\cdot x=e_0\cdot x=x.$

   Same for $x\cdot f_j$.

2. $e_i\cdot e_j=e_j\cdot e_i=e_i$ for all $i\le j$ then $e_j\cdot (e_i\cdot x)=x$.

3. Let $g\in G_1$ such that $g\cdot x=g\cdot x^{\prime }$ then $g^{-1}\cdot g\cdot x=g^{-1}\cdot g\cdot x^{\prime }$ then $g^{\prime }\cdot x=g^{\prime }\cdot x^{\prime }$.

4. Suppose that $x\cdot y=e_i$ then $x\cdot y\cdot e_0=e_0$ then $x\cdot y=e_0$ (because $y\cdot e_0=y$) giving a contradiction.

5. Let $x\in L$ and $y\in R$ such that $x\cdot y=e_i\in K_1\setminus \{e_0\}$. Suppose that $y\cdot x=g\in G_2$. Then

   $\begin{array}{lll}(x\cdot y)\cdot (x\cdot y)& =& e_i\\ x\cdot g\cdot y& =& e_i\end{array}$

   but $x\cdot g\cdot y\in G_1$ and $e_i\in K_1\setminus \{e_0\}$, it means that $e_i\in G_1$, hence we get a contradiction. Therefore, $y\cdot x\in K_2\setminus \{f_0\}$.

6. From part (4), we can see that if $x\cdot y\in G_1$ then $y\cdot x\in G_2$. Suppose that $x\cdot y=e_0$ and $y\cdot x=g\in G_2$. Let us prove that $g=f_0$. We have

   $\begin{array}{ll}y\cdot x=& g\\ \Rightarrow x\cdot y\cdot x=& x\cdot g\\ \Rightarrow y\cdot e_0=& g\cdot y\\ \Rightarrow f_0\cdot (y\cdot e_0)=& g\cdot (y\cdot e_0)\\ & (\text{because}y\cdot e_0\text{and}g\cdot y\text{are in}{G}_2\cdot R).\end{array}$

   By the free action of G₂ on $G_2\cdot R$, we obtain that $g=f_0$.

7. $x\cdot y\cdot x=e_i\cdot x$ thus $x\cdot f_j=e_i\cdot x$

   $y\cdot x\cdot y=f_j\cdot y$ thus $y\cdot e_i=f_j\cdot y$.

   In addition, by part (b) $(x\cdot f_j)\cdot f_j=x\cdot f_j$ and $e_i\cdot (e_i\cdot x)=e_i\cdot x$, then we can assume that $x\cdot f_j=e_i\cdot x=x$.

From Lemma 4.3 (4) (5), we see that if one of the multiplications is an idempotent then the other way around should be an idempotent as well (we mean that it is not in the group part). That’s why in the following, we study the structure of the category whenever we have two elements such that their multiplications in both directions are idempotents.

Lemma 4.4.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Let i be the maximum index such that there exist $x\in L$ and $y\in R$; $x\cdot y=e_i$. And let j^ʹ be the maximum index such that there exist $x^{\prime }\in L$ and $y^{\prime }\in R$; $y^{\prime }\cdot x^{\prime }=f_{j^{\prime }}$. We can assume following Lemma 4.3 (6) that $e_i\cdot x=x$, $y\cdot e_i=y$, $x^{\prime }\cdot f_{j^{\prime }}=x^{\prime }$ and $f_{j^{\prime }}\cdot y^{\prime }=y^{\prime }$.

1. If i = 0 then $j^{\prime }=0$ and vice versa.

2. If $i,j^{\prime }\ge 1$ then $x\cdot y^{\prime }=e_i$ and $y^{\prime }\cdot x=f_{j^{\prime }}$. In addition, $e_i\cdot x=x\cdot f_{j^{\prime }}=x$ and $y^{\prime }\cdot e_i=f_{j^{\prime }}\cdot y^{\prime }=y^{\prime }$.

Proof.

For part (1), suppose i = 0, in this case by Lemma 4.3 (4) we have $x^{\prime }\cdot y^{\prime }$ is an idempotent, and by maximality of i it has to be e₀. Therefore

$\begin{array}{l}{f}_{j^{\prime }}=f_{j^{\prime }}^2=y^{\prime }\cdot x^{\prime }\cdot y^{\prime }\cdot x^{\prime }=y^{\prime }\cdot e_0\cdot x^{\prime }\in G_1.\end{array}$

Therefore, $j^{\prime }=0$.

For part (2), suppose $i^{\prime },j^{\prime }\ge 1$, then from Lemma 4.3 (4), suppose that $y\cdot x=f_j$ and $x^{\prime }\cdot y^{\prime }=e_{i^{\prime }}$. From our assumption following (Lemma 4.3 (6)), we get

$\begin{array}{l}{f}_j\cdot y=y\cdot e_i=y\text{and}x\cdot f_j=e_i\cdot x=x\end{array}$

and

$\begin{array}{l}{f}_{j^{\prime }}\cdot y^{\prime }=y^{\prime }\cdot e_{i^{\prime }}=y^{\prime }\text{and}{x}^{\prime }\cdot f_{j^{\prime }}=e_{i^{\prime }}\cdot x^{\prime }=x^{\prime }.\end{array}$

By maximality we have $i\ge i^{\prime }$ and $j^{\prime }\ge j$. We have

$x\cdot f_{j^{\prime }}=(x\cdot f_j)\cdot f_{j^{\prime }}=x\cdot (f_j\cdot f_{j^{\prime }})=x\cdot f_j=x.$

We obtain that

$x=x\cdot f_j=x\cdot f_{j^{\prime }}=x\cdot (y^{\prime }\cdot x^{\prime }).$

Then

$\begin{array}{l}{e}_i=x\cdot y=(x\cdot y^{\prime }\cdot x^{\prime })\cdot y=(x\cdot y^{\prime })\cdot (x^{\prime }\cdot y).\end{array}$

Therefore $x\cdot y^{\prime }\notin G_1$ since $i\ge 1$, then there exists $m\le i$ (by maximality of i) such that $x\cdot y^{\prime }=e_m$. Then $e_i=e_m\cdot (x^{\prime }\cdot y^{\prime })$.

Which implies that $e_m=e_m\cdot e_i=e_m^2\cdot (x^{\prime }\cdot y)=e_m\cdot (x^{\prime }\cdot y)=e_i$. Hence $e_m=e_i$ and $x\cdot y^{\prime }=e_i$.

Similarly we can prove that $y^{\prime }\cdot x=f_{j^{\prime }}$ and $y^{\prime }\cdot e_i=y^{\prime }$.

Corollary 4.5.

In the case of Lemma 4.4 (2), we have $x=x^{\prime }$ and $y=y^{\prime }$.

Proof.

$x=e_i\cdot x=x\cdot y^{\prime }\cdot x=x\cdot f_{j^{\prime }}\cdot x\cdot y^{\prime }\cdot x^{\prime }=e_i\cdot x^{\prime }$.

Where $x^{\prime }=e_{i^{\prime }}\cdot x^{\prime }=e_{i^{\prime }}\cdot e_i\cdot x^{\prime }=e_i\cdot e_{i^{\prime }}\cdot x^{\prime }=e_i\cdot x^{\prime }$. Then $x=x^{\prime }$.

Similarly we prove that $y=y^{\prime }$.

Remark 4.6.

As a conclusion of Lemma 4.4 and Corollary 4.5, There is a maximum element i and a maximum element j such that there exist $x\in L$ and $y\in R$ such that $x\cdot y=e_i$ and $y\cdot x=f_j$.

Proposition 4.7.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category. Let i and j be the maximum elements such that there exist $x\in L$ and $y\in R$; $x\cdot y=e_i$ and $y\cdot x=f_j$. Then

$\begin{array}{l}{L}_{ij}=e_i\cdot L=L\cdot f_j\end{array}$

and

$\begin{array}{l}{R}_{ji}=f_j\cdot R=R\cdot e_i.\end{array}$

Proof.

We want to prove that $e_i\cdot L\cdot f_j=e_i\cdot L=L\cdot f_j$.

We always have that $e_i\cdot L\cdot f_j\subseteq e_i\cdot L$ and $e_i\cdot L\cdot f_j\subseteq L\cdot f_j$. We prove the other direction.

Let $x^{\prime }\in L$

$\begin{array}{lll}{e}_i{x}^{\prime }& =& e_i{e}_i{x}^{\prime }\\ & =& e_i(xy)x^{\prime }\\ & =& e_{i}x(y{x}^{\prime })\\ & =& e_{i}x{f}_m\\ & =& e_i\underset{\in L}{\underbrace{x{f}_m}}{f}_j\text{(}{f}_m{f}_j=f_m\text{because}j\text{is the maximum)}\\ & \in & e_i\cdot L\cdot f_j.\end{array}$

Similarly we prove the others.

Theorem 4.8.

Let $\mathsf{\text{C}}\in Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ (Notation 3.1) be a grouplike category.

Suppose that i and j are the maximum elements (in the sense of Lemma 4.4) such that $x\cdot y=e_i$ and $y\cdot x=f_j$. By Lemma 4.3 (6) we can assume that

$f_j\cdot y=y\cdot e_i=y\text{and}x\cdot f_j=e_i\cdot x=x.$

Then we can construct a sub-semicategory ${\mathsf{\text{C}}}^{\prime }$ of the form

$\left(\begin{array}{ll}{G}_1^{\ast i}& L_{ij}\\ R_{ji}& G_2^{\ast j}\end{array}\right)$

where $L_{ij}=e_i\cdot L\cdot f_j$ and $R_{ji}=f_j\cdot R\cdot e_i$, such that ${\mathsf{\text{C}}}^{\prime }$ is the maximum sub-semicategory of this form.

Proof.

• ${\mathsf{\text{C}}}^{\prime }$ is a category:

• – Objects: X and Y

• – Morphisms: $G_1^{\ast i},L_{ij},R_{ji}$ and $G_2^{\ast j}$

• – Composition:

• * Let $z\in L_{ij}$ and $e^{\prime }\in K_i$ then we can write $z=e_i\cdot \tilde{z}\cdot f_j$, then

  $e^{\prime }\cdot z=e^{\prime }\cdot (e_i\cdot \tilde{z}\cdot f_j)=(e^{\prime }\cdot e_i)\cdot \tilde{z}\cdot f_j=e_i\cdot (e^{\prime }\cdot \tilde{z})\cdot f_j\in L_{ij}$

• * Let $z\in L_{ij}$ and $w\in R_{ji}$ then $z=e_i\cdot \tilde{z}\cdot f_j$ and $w=f_j\cdot \tilde{w}\cdot e_i$, then

  $\begin{array}{lll}z\cdot w& =& (e_i\cdot \tilde{z}\cdot f_j)\cdot (f_j\cdot \tilde{w}\cdot e_i)\\ & =& e_i\cdot (\tilde{z}\cdot f_j\cdot f_j\cdot \tilde{w})\cdot e_i\\ & =& e_i\cdot (\tilde{z}\cdot f_j\cdot \tilde{w})\cdot e_i\in G_1^{\ast i}\end{array}$

• – Identities: e_i and f_j

• ${\mathsf{\text{C}}}^{\prime }$ is the maximum category of this form:

  If we have two elements x^ʹ and y^ʹ outside of L_ij and R_ji such that $x^{\prime }\cdot y^{\prime }=e_{i^{\prime }}$ and $y^{\prime }\cdot x^{\prime }=f_{j^{\prime }}$ then by Lemma 4.4 we have $x\cdot y^{\prime }=e_i$ and $y^{\prime }\cdot x=f_{j^{\prime }}$. This is in contradiction with the maximality of i and j. Hence ${\mathsf{\text{C}}}^{\prime }$ is the maximum such category.

Corollary 4.9.

The monoids $G_1^{\ast i}$ and $G_2^{\ast j}$ are isomorphic. We call ${\mathsf{\text{C}}}^{\prime }$ a groupoid-like category.

Proof.

There exist $x,y$ such that $x\cdot y=e_i$ and $y\cdot x=f_j$ the identities. It follows that i = j and the groups G₁ and G₂ are isomorphic.

Proposition 4.10.

Let

$M=\left(\begin{array}{ll}{G}_1^{\ast k_1}& L\\ R& G_2^{\ast k_2}\end{array}\right)$

be a matrix of a grouplike category $\mathsf{\text{C}}$. By Theorem 4.8, we can construct a sub-semicategory ${\mathsf{\text{C}}}^{\prime }$ associated to the matrix

$M^{\prime }=\left(\begin{array}{ll}{G}_1^{\ast i}& L_i\\ R_i& G_2^{\ast i}\end{array}\right)$

of monoids and bimodules. For all $x\in L$ and all $y\in R$ we have

$\begin{array}{l}x\cdot y=(e_i\cdot x)\cdot (y\cdot e_i)\in G_1^{\ast i}\text{and}y\cdot x=(f_i\cdot y)\cdot (x\cdot f_i)\in G_2^{\ast i}.\end{array}$

.

Proof.

Suppose that $x\cdot y\in G_1^{\ast k_1}\setminus G_1^{\ast i}$ then there exists $e_k\notin G_1^{\ast i}$ such that $x\cdot y=e_k$. This is in contradiction with the maximality of i.

Conclusion: Let

$\begin{array}{l}M=\left(\begin{array}{ll}{G}_1^{\ast k_1}& L\\ R& G_2^{\ast k_2}\end{array}\right)\end{array}$

be a matrix of a grouplike category $\mathsf{\text{C}}$ whose objects X and Y. Then

1. $G_1\simeq G_2$.

2. There exists $i^{max}=max(i)$ such that there exist $x,y$ and $e_{i^{max}}=x\cdot y$.

   There exists $j^{max}=max(j)$ such that there exist $x,y$ and $f_{j^{max}}=y\cdot x$.

3. $i^{max}=j^{max}$, and we obtain the following matrix

   $\begin{array}{l}\left(\begin{array}{ll}{G}_1^{\ast i^{max}}& L_{i^{max}{j}^{max}}\\ R_{j^{max}{i}^{max}}& G_2^{\ast j^{max}}\end{array}\right)\end{array}$

   of a sub-semicategory ${\mathsf{\text{C}}}^{\prime }$ of $\mathsf{\text{C}}$ whose objects X and Y are isomorphic. Thus

   $\begin{array}{l}{G}_1^{\ast i^{max}}\simeq G_2^{\ast j^{max}}\text{as monoids}\end{array}$

   and

   $\begin{array}{l}{G}_1^{\ast i^{max}}\simeq L_{i^{max}{j}^{max}}\simeq R_{j^{max}{i}^{max}}\text{as bimodules}.\end{array}$

4. The bimodule L has the property

   $\begin{array}{l}{L}_i=L\cdot f_i=e_i\cdot L\simeq G_1^{\ast i}\simeq G_2^{\ast i}\end{array}$

   i.e. L is i-unigen.

   and the bimodule R has the property

   $\begin{array}{l}{R}_i=f_i\cdot R=R\cdot e_i\simeq G_2^{\ast i}\simeq G_1^{\ast i}\end{array}$

   i.e. R is i-unigen. Where $i=i^{max}=j^{max}$.

5. For all $x\in L$, $e_i\cdot x=x\cdot f_i$.

6. The isomorphisms in 4 are inverses.

7. The multiplications of the elements of L by the elements of R are determined once we fix x and y.

Theorem 4.11.

Let $G_1,G_2$ be two groups, $k_1,k_2$, L, R be $(G_1^{\ast k_1},G_2^{\ast k_2})$-bimodules, and i; $i\le k_1$, $i\le k_2$ such that the following collection Equation P(i)(P(i)) $L_i=L\cdot f_i=e_i\cdot L\text{and}{R}_i=R\cdot e_i=f_i\cdot R$(P(i)) of properties holds:

(P(i)) $L_i=L\cdot f_i=e_i\cdot L\text{and}{R}_i=R\cdot e_i=f_i\cdot R$(P(i))

and such that there is $x_i\in L_i$ such that x_i determines an isomorphism

$\begin{array}{l}\begin{array}{lll}{G}_1^{\ast i}& \to & L_i\\ g& \mapsto & g{x}_i\end{array}\end{array}$

and an isomorphism

$\begin{array}{l}\begin{array}{lll}{G}_2^{\ast i}& \to & L_i\\ h& \mapsto & x_{i}h\end{array}.\end{array}$

Similarly for $y_i\in R_i$, determining an isomorphism $G_2^{\ast i}\simeq R_i\simeq G_1^{\ast i}$.

The isomorphisms $G_1^{\ast i}\simeq L_i\simeq G_2^{\ast i}$ and $G_2^{\ast i}\simeq R_i\simeq G_1^{\ast i}$ are assumed to be inverses for the property Equation P(i)(P(i)) $L_i=L\cdot f_i=e_i\cdot L\text{and}{R}_i=R\cdot e_i=f_i\cdot R$(P(i)) .

Then we get a category $\mathsf{\text{C}}$ with algebraic matrix

$\begin{array}{l}\left(\begin{array}{ll}{G}_1^{\ast k_1}& L\\ R& G_2^{\ast k_2}\end{array}\right)\end{array}$

such that $i=i^{max}=j^{max}$, $x_i\cdot y_i=e_i$ and $y_i\cdot x_i=f_i$ (L and R are i-unigen).

For $i\ge 1$, then the choice of x_i is unique and hence the category is unique.

For i = 0, then the choice of x₀ is not unique but the category is unique once x₀ and y₀ are fixed.

Proof.

The multiplications $G_1^{\ast k_1}\times L\to L$ and $L\times G_2^{\ast k_2}\to L$ are given by the bimodule structure of L. Similarly for R.

Let $x^{\prime }\in L$, $y^{\prime }\in R$,

$\begin{array}{l}{x}^{\prime }{y}^{\prime }:=\underset{\in L_i}{\underbrace{(e_i{x}^{\prime })}}\underset{\in R_i}{\underbrace{(y^{\prime }{e}_i)}}\in G_1^{\ast i}.\end{array}$

Similarly for $y^{\prime }{x}^{\prime }$. We can check that the multiplication is associative.

If we fix the matrix

$\begin{array}{l}M=\left(\begin{array}{ll}{G}_1^{\ast k_1}& L\\ R& G_2^{\ast k_2}\end{array}\right)\end{array}$

and we take $i^{\text{biggest}}=max\{i\mid \text{{P(i)} holds}\}$, then for the cases $i\ge 1$

$\begin{array}{l}Ca{t}_{i^{max}\ge 1}(M)=\{1\le i\le i^{\text{biggest}}\}.\end{array}$

For i = 0, the choices of x₀ and y₀ are not unique, see Remark 4.13.

Remark 4.12.

The condition Equation P(i)(P(i)) $L_i=L\cdot f_i=e_i\cdot L\text{and}{R}_i=R\cdot e_i=f_i\cdot R$(P(i)) is what we call i-unigen in Definition 1.1 (Thanks to Carlos Simpson for suggesting the terminology i-unigen).

Remark 4.13.

When i = 0, once we fix x₀, there are maybe several choices for y₀ that lead to inverse isomorphisms. The set of choices of y₀ is given by the center of the group. The cardinal of the set of pairs $(\mathsf{\text{C}},\beta )$ in $Cat(M(G_1^{\ast k_1},G_2^{\ast k_2},L,R))$ with $i^{max}=0$ is equal to the cardinal of the center of the group.

Proof of Lemma 1.3

Let L be an i-unigen bimodule. Let $x\in L_i$ such that $G_1^{\ast i}\cdot x=L_i$, where $G_1^{\ast i}\cdot x\simeq G_1^{\ast i}$. Then $x\cdot G_2^{\ast i}=L_i$:

Suppose $x\cdot g=x\cdot g^{\prime }$, $g\ne g^{\prime }$ in $G_2^{\ast i}$. Then since $G_1^{\ast i}\cdot x=L_i$, we get $y\cdot g=y\cdot g^{\prime }$ for all $y\in L_i$, then $g=g^{\prime }$. Then

$\begin{array}{l}\{x\}\times G_2^{\ast i}\hookrightarrow L_i\end{array}$

but they have the same cardinal, then

$\begin{array}{l}\{x\}\times G_2^{\ast i}\simeq L_i.\end{array}$

We define an isomorphism

$\begin{array}{l}\varphi :G_1^{\ast i}\to G_2^{\ast i}\end{array}$

as follows

If $g\in G_1^{\ast i}$ and $g\cdot x\in L_i$ then $g\cdot x=x\cdot h$; $h\in G_2^{\ast i}$. Set

$\varphi (g):=h\text{and}g\cdot x=x\cdot \varphi (g).$

Then

$\begin{array}{lll}(g{g}^{\prime })\cdot x& =& g\cdot (g^{\prime }\cdot x)\\ & =& g\cdot (x\cdot \varphi (g^{\prime }))\\ & =& (g\cdot x)\cdot \varphi (g^{\prime })\\ & =& (x\cdot \varphi (g))\cdot \varphi (g^{\prime })\\ & =& x\cdot (\varphi (g)\varphi (g^{\prime }))\\ & =& x\cdot (\varphi (g{g}^{\prime })).\end{array}$

By uniqueness of h, we have $\varphi (g{g}^{\prime })=\varphi (g)\varphi (g^{\prime })$, and Lemma 1.3 is proved.

Proof of Theorem 1.6

Using Theorem 4.11 we prove Theorem 1.6, as we have the same construction of a category. In Theorem 4.11 we start by choosing $i^{max}=j^{max}$ to get to the algebraic matrix (Equation M⁽³⁾($M^{(3)}$) $\left(\begin{array}{ll}{G}^{\ast k_1}& L\\ R& G^{\ast k_2}\end{array}\right)$($M^{(3)}$) ).

For the multiplication table of ${\mathsf{\text{C}}}^{(3)}$, the multiplication of $G^{\ast k_1}$ on L and R is given by the bimodule structure. It remains to find the maps

$L\times R\to G^{\ast k_1}\text{and}R\times L\to G^{\ast k_2}.$

Let $x\in L$ and $y\in R$,

$\begin{array}{l}x\cdot y=e_i\cdot x\cdot y\cdot e_i\end{array}$

where $e_i\cdot x$ and $y\cdot e_i$ are in L_i and R_i and these compositions are given by ${\mathsf{\text{C}}}^{(1)}$. And as in the conclusion part (7), we get the uniqueness of the category.

Proof of Theorem 1.7

From Theorem 4.8 and Proposition 4.10 we can prove Theorem 1.7. As in Remark 4.9 we prove that i and j should be the same, then the algebraic matrix obtained is

$M=\left(\begin{array}{ll}{G}^{\ast i}& L_i\\ R_i& G^{\ast i}\end{array}\right)$

where L_i and R_i are isomorphic to $G^{\ast i}$ (conclusion part (3)). Then M is the matrix (Equation M⁽¹⁾($M^{(1)}$) $\left(\begin{array}{ll}{G}^{\ast i}& G^{\ast i}\\ G^{\ast i}& G^{\ast i}\end{array}\right)$($M^{(1)}$) ).

We conclude that if we have a grouplike category then the two bimodules L and R are i-unigen and the resulting isomorphisms between these two bimodules are inverses. If we then identify the groups via these isomorphisms, we can say that L and R become strictly i-unigen. Then we get the structure described in Theorem 1.6, and Theorem 1.7 is proved.

Remark 4.14.

The condition that the isomorphisms should be inverses in Theorem 4.11 is very important to obtain the grouplike category. We give a counter example to show its importance (example 4.16), but first we define the category of a matrix that is a combination of two categories associated to matrices of bimodules.

Notation 4.15.

Let

$\begin{array}{l}{N}^{(1)}=\left(\begin{array}{ll}A& L\\ \mathrm{\varnothing }& B\end{array}\right)\end{array}$

be a matrix of bimodule L, and let ${\mathsf{\text{C}}}^{(1)}$ be a category associated to $N^{(1)}$.

Similarly let

$\begin{array}{l}{N}^{(2)}=\left(\begin{array}{ll}A& \mathrm{\varnothing }\\ R& B\end{array}\right)\end{array}$

be a matrix of bimodule R, and let ${\mathsf{\text{C}}}^{(2)}$ be a category associated to $N^{(2)}$.

We denote by $N=N^{(1)}\cup N^{(2)}$ the matrix of the form

$\begin{array}{l}\left(\begin{array}{ll}A& L\\ R& B\end{array}\right)\end{array}$

of monoids and bimodules L and R in ${\mathsf{\text{C}}}^{(1)}$ and ${\mathsf{\text{C}}}^{(2)}$. We denote by $\mathsf{\text{C}}$ a category associated to N.

Example 4.16.

Consider the matrix

$\begin{array}{l}{N}^{(1)}=\left(\begin{array}{ll}{\mathbb{Z}}_3& L\\ \mathrm{\varnothing }& {\mathbb{Z}}_3\end{array}\right)\end{array}$

where

$\begin{array}{l}\begin{array}{lll}{\mathbb{Z}}_3=\{1,2,3\}& L=\{4,5,6\}& {\mathbb{Z}}_3=\{7,8,9\}\end{array}\end{array}$

such that

$\begin{array}{l}{\mathbb{Z}}_3\cdot L=L\cdot {\mathbb{Z}}_3=\left(\begin{array}{lll}4& 5& 6\\ 5& 6& 4\\ 6& 4& 5\end{array}\right)\end{array}$

then $N^{(1)}$ is the matrix of the bimodule L. Let ${\mathsf{\text{C}}}^{(1)}$ be a bimodule category associated to $N^{(1)}$ with the above table of multiplication.

Consider the matrix

$\begin{array}{l}{N}^{(2)}=\left(\begin{array}{ll}{\mathbb{Z}}_3& \mathrm{\varnothing }\\ R& {\mathbb{Z}}_3\end{array}\right)\end{array}$

where

$\begin{array}{l}\begin{array}{lll}{\mathbb{Z}}_3=\{1,2,3\}& R=\{4,5,6\}& {\mathbb{Z}}_3=\{7,8,9\}\end{array}\end{array}$

such that

$\begin{array}{l}{\mathbb{Z}}_3\cdot R=\left(\begin{array}{lll}4& 5& 6\\ 5& 6& 4\\ 6& 4& 5\end{array}\right)\end{array}$

and

$\begin{array}{l}R\cdot {\mathbb{Z}}_3=\left(\begin{array}{lll}4& 6& 5\\ 5& 4& 6\\ 6& 5& 4\end{array}\right).\end{array}$

Similarly, $N^{(2)}$ is a matrix of the bimodule R. Notice that we changed the multiplication table of $R\cdot {\mathbb{Z}}_3$ by the involution of the group ${\mathbb{Z}}_3$. Let ${\mathsf{\text{C}}}^{(2)}$ be a bimodule category associated to $N^{(2)}$ with the above table of multiplication.

But the matrix

$\begin{array}{l}N=\left(\begin{array}{ll}{\mathbb{Z}}_3& L\\ R& {\mathbb{Z}}_3\end{array}\right)\end{array}$

with the above bimodules ${\mathsf{\text{C}}}^{(1)}$ and ${\mathsf{\text{C}}}^{(2)}$ doesn’t admit a grouplike category. This was shown by calculating using Mace4. It also follows from Conclusion part (6) since the isomorphisms given by L and R are not inverses.

We can conclude now the general structure of grouplike categories with only 2 objects $X_1,X_2$.

$\begin{array}{lllllllllllll}& & X_1\to X_1& & & X_1\to X_2& & & X_2\to X_1& & & X_2\to X_2& \\ X_1& & & & & & & & & & & & \\ \downarrow & & G^{\ast k_1}& & & L& & & & & & & \\ X_1& & & & & & & & & & & & \\ X_1& & & & & & & & & & & & \\ \downarrow & & & & & & & & G^{\ast i^{max}}& & & L& \\ X_2& & & & & & & & & & & & \\ X_2& & & & & & & & & & & & \\ \downarrow & & R& & & G^{\ast i^{max}}& & & & & & & \\ X_1& & & & & & & & & & & & \\ X_2& & & & & & & & & & & & \\ \downarrow & & & & & & & & R& & & G^{\ast k_2}& \\ X_2& & & & & & & & & & & & \end{array}$

5. Applications

Remark 5.1.

The number of monoids is known until order 10 (Jipsen, OEIS, peterjipsen, OEIS). It is not easy to get a count of higher order using Mace4 as it is limited to very small orders. Some AI and machine learning techniques are now applied to classify some semigroups as in (Simpson, 2021). In our applications below, we need the list of monoids of order 3 as we are going to give a count of some categories having 3 morphisms in each morphism set.

List all the 7 monoids of size 3:

(1) $\begin{array}{lllllllll}comp& & {\mathsf{\text{C}}}_1& {\mathsf{\text{C}}}_2& {\mathsf{\text{C}}}_3& {\mathsf{\text{C}}}_4& {\mathsf{\text{C}}}_5& {\mathsf{\text{C}}}_6& {\mathsf{\text{C}}}_7\\ 2^2& =& 1& 2& 2& 2& 2& 2& 3\\ 3^2& =& 3& 2& 3& 3& 2& 3& 2\\ 2\circ 3& =& 3& 2& 2& 3& 3& 2& 1\\ 3\circ 2& =& 3& 2& 3& 2& 3& 2& 1\end{array}$(1)

We combine monoids of three elements together in one category, two monoids ${\mathsf{\text{C}}}_i,{\mathsf{\text{C}}}_j$ are called connected if there exists a category where the monoids ${\mathsf{\text{C}}}_i$ and ${\mathsf{\text{C}}}_j$ are the endomorphism monoids of the two objects. Viewing them as objects, each object is one of the monoids of endomorphisms listed before, and this graph of a category is associated to the matrix:

$\left(\begin{array}{ll}3& 3\\ 3& 3\end{array}\right).$

From Equation Table 1(1) $\begin{array}{lllllllll}comp& & {\mathsf{\text{C}}}_1& {\mathsf{\text{C}}}_2& {\mathsf{\text{C}}}_3& {\mathsf{\text{C}}}_4& {\mathsf{\text{C}}}_5& {\mathsf{\text{C}}}_6& {\mathsf{\text{C}}}_7\\ 2^2& =& 1& 2& 2& 2& 2& 2& 3\\ 3^2& =& 3& 2& 3& 3& 2& 3& 2\\ 2\circ 3& =& 3& 2& 2& 3& 3& 2& 1\\ 3\circ 2& =& 3& 2& 3& 2& 3& 2& 1\end{array}$(1) , we can remark that ${\mathsf{\text{C}}}_5$ and ${\mathsf{\text{C}}}_6$ are grouplike monoids. Where ${\mathsf{\text{C}}}_5$ is of the form $G^{\ast 1}$ such that $G={\mathbb{Z}}_2$, and ${\mathsf{\text{C}}}_6$ is of the form $G^{\ast 2}$ such that $G=\{2\}$ is the trivial group.

Example 5.2.

The number of categories between ${\mathsf{\text{C}}}_6$ and itself Let $\mathsf{\text{B}}$ be a category with two objects X and Y associated to the matrix

$\begin{array}{l}{M}_1=\left(\begin{array}{ll}3& 3\\ 0& 3\end{array}\right)\end{array}$

such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(X,Y)=L$ and $\mathsf{\text{C}}(Y,X)=\mathrm{\varnothing }$. Fix

$\begin{array}{l}{\mathsf{\text{C}}}_6=\{1,2,3\}L=\{4,5,6\}{\mathsf{\text{C}}}_6=\{7,8,9\}\end{array}$

(these are two distinct copies of ${\mathsf{\text{C}}}_6$).

1. $\mathbf{\text{i = 0:}}$ the number of categories associated to M₁ is 64 (calculated by Mace4 [14]). But Since ${\mathsf{\text{C}}}_6$ is a grouplike category containing the trivial group $\{2\}$, then the orbit of L has size 1 and it’s unique. This means that $2\ast x$ is unique for all $x\in L$. Suppose it’s equal to 4. This reduces the number of possibilities to 15.

   Similarly, let $\mathsf{\text{D}}$ be a category with two objects X and Y associated to the matrix

   $\begin{array}{l}{M}_2=\left(\begin{array}{ll}3& 0\\ 3& 3\end{array}\right)\end{array}$

   such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(X,Y)=\mathrm{\varnothing }$ and $\mathsf{\text{C}}(Y,X)=R$.

   For the same reason, the number of possible categories associated to M₂ with the orbit condition is 15.

   Then a category $\mathsf{\text{C}}$ associated to the matrix

   $\begin{array}{l}M=\left(\begin{array}{ll}3& 3\\ 3& 3\end{array}\right)\end{array}$

   such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_6$, $\mathsf{\text{C}}(X,Y)=L$ and $\mathsf{\text{C}}(Y,X)=R$, has 15 left bimodules $A_1,\dots ,A_{15}$ and 15 right bimodules $B_1,\dots ,B_{15}$.

We have two cases:

(a)	The bimodules are the same $\begin{array}{l}\left(\begin{array}{ll}{G}^{\ast 2}& A_i\\ B_i& G^{\ast 2}\end{array}\right).\end{array}$ By Theorem 4.11, there are exactly 15 possibilities for this matrix (up to isomorphism).
(b)	The bimodules are different $\begin{array}{l}\left(\begin{array}{ll}{G}^{\ast 2}& A_i\\ B_j& G^{\ast 2}\end{array}\right)\end{array}$ such that $1\le i<j\le 15$. $\begin{array}{ll}& \backslash \frac{15\times 14}{2}\end{array}$ There are $\frac{15\times 14}{2}=105$ possibilities.

Hence for i = 0, we have 120 categories.

2.	$\mathbf{\text{i = 1:}}$ there are 2 left bimodules $A_1,A_2$ and 2 right bimodules $B_1,B_2$. Then the possibilities are:

(a)	The bimodules are the same $\begin{array}{l}\left(\begin{array}{ll}{G}^{\ast 2}& A_i\\ B_i& G^{\ast 2}\end{array}\right)\end{array}$ and we have 2 categories that could be associated to this matrix.
(b)	The bimodules are different $\begin{array}{l}\left(\begin{array}{ll}{G}^{\ast 2}& A_i\\ B_j& G^{\ast 2}\end{array}\right)\end{array}$ and we have 1 category (up to isomorphism).

Hence in total there are $120+3=123$ categories between ${\mathsf{\text{C}}}_6$ and itself.

Example 5.3.

The number of categories between ${\mathsf{\text{C}}}_5$ and itself

The number of categories between ${\mathsf{\text{C}}}_5$ and itself can also be found bimodules. Let $\mathsf{\text{B}}$ be a category with two objects X and Y associated to the matrix

$\begin{array}{l}{M}_1=\left(\begin{array}{ll}3& 3\\ 0& 3\end{array}\right)\end{array}$

such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(X,Y)=L$ and $\mathsf{\text{C}}(Y,X)=\mathrm{\varnothing }$.

And similarly let $\mathsf{\text{D}}$ be a category with two objects X and Y associated to the matrix

$\begin{array}{l}{M}_2=\left(\begin{array}{ll}3& 0\\ 3& 3\end{array}\right)\end{array}$

such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(X,Y)=\mathrm{\varnothing }$ and $\mathsf{\text{C}}(Y,X)=R$.

Then there exists a category $\mathsf{\text{C}}$ with two objects associated to the matrix

$\begin{array}{l}M=\left(\begin{array}{ll}3& 3\\ 3& 3\end{array}\right)\end{array}$

such that $\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_5$, $\mathsf{\text{C}}(X,Y)=L$ and $\mathsf{\text{C}}(Y,X)=R$. Where $\mathsf{\text{B}}$ is a left bimodule and $\mathsf{\text{D}}$ is a right bimodule.

Since there is 1 left and 1 right bimodule (also calculated by Mace4) and since $i^{max}=0$, then the number of possible y₀ is equal to the order of the center of the group ${\mathbb{Z}}_2$ which is 2 (Theorem 4.11 and Remark 4.13). Hence there are 2 categories between ${\mathsf{\text{C}}}_5$ and itself.

Definition 5.4.

A category $\mathsf{\text{C}}$ is called reduced if there do not exist two distinct isomorphic objects in $\mathsf{\text{C}}$.

Lemma 5.5.

Let $\mathsf{\text{C}}$ be a reduced category associated to the matrix

$\begin{array}{l}\left(\begin{array}{ll}A& L\\ R& B\end{array}\right).\end{array}$

If there exists $x\in L$ and $y\in R$ such that $y\cdot x=1_B$, then $|B|<|A|$ and B is a sub-monoid of A disjoint from $\{1_A\}$.

Proof.

Let

$\begin{array}{lllll}\varphi & :& A& \to & B\\ & & b& \mapsto & x\cdot f\cdot y\end{array}$

We have

$\begin{array}{ll}\varphi (f{f}^{\prime })=& x\cdot f{f}^{\prime }\cdot y=x\cdot f\cdot 1_B\cdot f^{\prime }\cdot y\\ =& (x\cdot f\cdot y)\cdot (x\cdot f^{\prime }\cdot y)=\varphi (f)\varphi (f^{\prime }).\end{array}$

ϕ is injective, indeed, suppose that $\varphi (f)=1_A$, then

$\begin{array}{lll}x\cdot f\cdot y& =& 1_A\\ \Rightarrow x\cdot f\cdot y\cdot x& =& x\\ \Rightarrow x\cdot f& =& x\\ \Rightarrow y\cdot x\cdot f& =& y\cdot x\\ \Rightarrow f& =& 1_B.\end{array}$

Remark 5.6.

We note that if $|A|=|B|$ then $x\cdot y=1_A$ and $\mathsf{\text{C}}$ is not reduced.

Proposition 5.7.

Let $M=\left(\begin{array}{ll}3& a\\ b& 3\end{array}\right)$ and $\mathsf{\text{C}}$ be a category associated to M whose objects are X and Y. Then

$\begin{array}{l}\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_5⟺\mathsf{\text{C}}(X,X)={\mathsf{\text{C}}}_5.\end{array}$

Proof.

Suppose that $\mathsf{\text{C}}(X,X)=A,\mathsf{\text{C}}(X,Y)=L,\mathsf{\text{C}}(Y,X)=R$ and $\mathsf{\text{C}}(Y,Y)={\mathsf{\text{C}}}_5={\mathbb{Z}}_2\cup \{1\}$. Consider the algebraic matrix of $\mathsf{\text{C}}$

$\begin{array}{l}{M}^{alg}=\left(\begin{array}{ll}A& L\\ R& {\mathsf{\text{C}}}_5\end{array}\right)\end{array}$

such that $|A|=3,|L|=a,|R|=b$ and $|{\mathsf{\text{C}}}_5|=3$.

Let ${\mathsf{\text{C}}}^{\prime }$ be a subcategory of $\mathsf{\text{C}}$ defined in the following way

1. (Objects of ${\mathsf{\text{C}}}^{\prime }$): X and Y.

2. (Morphisms of ${\mathsf{\text{C}}}^{\prime }$): $A,L\cdot {\mathbb{Z}}_2,{\mathbb{Z}}_2\cdot R$ and ${\mathbb{Z}}_2$. (It means that we take the orbits of L and R by the action of ${\mathbb{Z}}_2$).

3. (Composition): by Proposition 3.11 we have that the composition of orbits goes into the group ${\mathbb{Z}}_2$.

4. (Identities): 1_A and $1_{{\mathbb{Z}}_2}$.

We obtain that ${\mathsf{\text{C}}}^{\prime }$ is associated to the following algebraic matrix

$\begin{array}{l}{M}^{\prime alg}=\left(\begin{array}{ll}A& L\cdot {\mathbb{Z}}_2\\ {\mathbb{Z}}_2\cdot R& {\mathbb{Z}}_2\end{array}\right).\end{array}$

Since there exist $x\in L\cdot {\mathbb{Z}}_2$ and $y\in {\mathbb{Z}}_2\cdot R$ such that $x\cdot y=1_{{\mathbb{Z}}_2}$, then from Lemma 5.5 we have that ${\mathbb{Z}}_2$ is a sub-monoid of A disjoint from $\{1_A\}$. Therefore, $A={\mathbb{Z}}_2\cup \{1\}={\mathsf{\text{C}}}_5$.

Theorem 5.8.

Let $M=\left(\begin{array}{llll}3& a_{12}& \dots & a_{1n}\\ a_{21}& 3& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& \dots & \dots & 3\end{array}\right)$ be a strictly positive matrix and let $\mathsf{\text{C}}$ be a reduced category associated to M with $Ob(\mathsf{\text{C}})=\{1,\dots ,n\}$.

If there exists $i\in Ob(\mathsf{\text{C}})$ such that $\mathsf{\text{C}}(i,i)={\mathsf{\text{C}}}_5$ then $\mathsf{\text{C}}(j,j)={\mathsf{\text{C}}}_5\mathrm{\forall }j\in Ob(\mathsf{\text{C}})$.

Proof.

If $\mathsf{\text{C}}$ is a category associated to M then every regular sub-matrix of M of size 2 is associated to a full sub-category of $\mathsf{\text{C}}$ [2] with two objects, then we apply Proposition 5.7 to get the result.

The classification of finite categories depends mostly on the algebraic structure of the endomorphism monoids. Monoids that are groups or contain a group impose some restrictions on the cardinality of the sets of morphisms. The properties obtained by studying such monoids make the classification and the counting problem easier. Counting finite structures is not easy in general, but optimizing the number of categories that could be obtained is a good start. We are now studying other types of monoids hoping to get classification theorems similar to the ones we have seen in this paper.

Acknowledgements

I would like to thank Wesley Fussner for he is the one that suggested working on grouplike monoids and categories, and Angel Toledo for the reference to Leinster’s paper. A very big thanks to my advisor Carlos Simpson for his constant help during my thesis.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

Supported by the Agence Nationale de la Recherche program 3ia Côte d’Azur ANR-19- P3IA-0002, and European Research Council Horizons 2020 grant 670624 (Mai Gehrke’s DuaLL project).

Additional information

Funding

Supported by the Agence Nationale de la Recherche program 3ia Côte d’Azur ANR-19- P3IA-0002, and European Research Council Horizons 2020 grant 670624 (Mai Gehrke’s DuaLL project).