---
abstract: 'There are different categorizations of the definition of a [*ring*]{} such as [*Ann-category*]{} \[6\], [*ring category*]{} \[2\],... The main result of this paper is to prove that every axiom of the definition of a [*ring category*]{}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an [*Ann-category*]{}.'
author:
- |
    Nguyen Tien Quang, Nguyen Thu Thuy and Che Thi Kim Phung\
    [*Hanoi National University of Education*]{}
title: 'THE RELATION BETWEEN ANN-CATEGORIES AND RING CATEGORIES'
---

Introduction
============

Categories with monoidal structures $\oplus, \otimes$ (also called [*categories with distributivity constraints*]{}) were presented by Laplaza \[3\]. M. Kapranov and V.Voevodsky \[2\] omitted requirements of the axiomatics of Laplaza which are related to the commutativity constraints of the operation $\otimes$ and presented the name [*ring categories*]{} to indicate these categories.

To approach in an other way, monoidal categories can be “smoothed” to become a [*category with group structure,*]{} when they are added the definition of invertible objects (see Laplaza \[4\], Saavedra Rivano \[9\]). Now, if the back ground category is a [*groupoid*]{} (i.e., each morphism is an isomorphism) then we have [*monoidal category group-like*]{} (see A. Frölich and C. T. C. Wall \[1\], or a [*Gr-category*]{} (see H. X. Sinh \[11\]). These categories can be classified by $H^{3}(\Pi, A)$. Each Gr-category $\mathcal G$ is determined by 3 invariants: The group $\Pi$ of classes of congruence objects, $\Pi-$module $A$ of automorphisms of the unit $1$, and an element $\overline{h}\in H^{3}(\Pi, A),$ where $h$ is induced by the associativity constraint of $\mathcal G.$

In 1987, in \[6\], N. T. Quang presented the definition of an [*Ann-category*]{}, as a categorization of the definition of rings, when a symmetric Gr-category (also called Pic-category) is equiped with a monoidal structure $\otimes$. In \[8\], \[7\], Ann-categories and [*regular*]{} Ann-categories, developed from the ring extension problem, have been classified by, respectively, Mac Lane ring cohomology \[5\] and Shukla algebraic cohomology \[10\].

The aim of this paper is to show clearly the relation between the definition of an [*Ann-category*]{} and a [*ring category*]{}.

For convenience, let us recall the definitions. Moreover, let us denote $AB$ or $A.B$ instead of $A{\otimes }B.$

Fundamental definitions
=======================

[**The axiomatics of an Ann-category**]{}\
An Ann-category consists of:\
i) A groupoid ${\mathcal{A}}$ together with two bifunctors ${\oplus},{\otimes }:{\mathcal{A}}\times{\mathcal{A}}\longrightarrow {\mathcal{A}}.$\
ii) A fixed object $0\in {\mathcal{A}}$ together with naturality constraints $a^+,c,g,d$ such that $({\mathcal{A}},{\oplus},a^+,c,(0,g,d))$ is a Pic-category.\
iii) A fixed object $1\in{\mathcal{A}}$ together with naturality constraints $a,l,r$ such that $({\mathcal{A}},{\otimes },a,(1,l,r))$ is a monoidal $A$-category.\
iv) Natural isomorphisms ${\frak{L}},{\frak{R}}$ $${\frak{L}}_{A,X,Y}:A{\otimes }(X{\oplus}Y)\longrightarrow (A{\otimes }X){\oplus}(A{\otimes }Y)$$ $${\frak{R}}_{X,Y,A}:(X{\oplus}Y){\otimes }A\longrightarrow(X{\otimes }A){\oplus}(Y{\otimes }A)$$ such that the following conditions are satisfied:\
(Ann-1) For each $A\in {\mathcal{A}},$ the pairs $(L^A,\breve{L^A}),(R^A,\breve{R^A})$ determined by relations:\
$$\begin{aligned}
&L^A & = &A{\otimes }- \;\;\;\; &R^A&=&-{\otimes }A\\
&\breve{L^A}_{X,Y}& = &{\frak{L}}_{A, X, Y}\;\;\;\; &\breve{R^A}_{X,
Y}&=&\mathcal{R}_{X, Y, A}
\end{aligned}$$ are ${\oplus}$-functors which are compatible with $a^+$ and $c.$\
(Ann-2) For all $ A,B,X,Y\in {\mathcal{A}},$ the following diagrams:

$$\begin{diagram}
\node{(AB)(X{\oplus}Y)}\arrow{s,l}{\breve{L}^{AB}} \node{ A(B(X{\oplus}Y))}\arrow{w,t}{\quad a_{A, B, X{\oplus}Y}\quad}\arrow{e,t}{\quad
id_A{\otimes }\breve{L}^B\quad}\node{ A(BX{\oplus}BY)}\arrow{s,r}{\breve{L}^A}\\
\node{(AB)X{\oplus}(AB)Y}\node[2]{A(BX){\oplus}A(BY)}\arrow[2]{w,t}
{\quad\quad a_{A, B, X}{\oplus}a_{A, B, Y} \quad\quad}
\end{diagram}\tag{1.1}$$

$$\begin{diagram}
\node{(X{\oplus}Y)(BA)}\arrow{s,l}{\breve{R}^{BA}}\arrow{e,t}{\quad
a_{ X{\oplus}Y, B, A}\quad}\node{((X{\oplus}Y)B)A}\arrow{e,t}{\quad
\breve{R}^B{\otimes }id_A\quad}\node{ (XB{\oplus}YB)A}\arrow{s,r}{\breve{R}^A}\\
\node{X(BA){\oplus}Y(BA)}\arrow[2]{e,t} {\quad\quad a_{X, B, A}{\oplus}a_{Y, B, A} \quad\quad}\node[2]{(XB)A{\oplus}(YB)A}
\end{diagram}\tag{1.1'}$$ $$\begin{diagram}
\node{(A(X{\oplus}Y))B}\arrow{s,l}{\breve{L}^{A}{\otimes }id_B} \node{
A((X{\oplus}Y)B)}\arrow{w,t}{\quad a_{A, X{\oplus}Y,
B}\quad}\arrow{e,t}{\quad id_A{\otimes }\breve{R}^B\quad}\node{
A(XB{\oplus}YB)}\arrow{s,r}{\breve{L}^A}\\
\node{(AX{\oplus}AY)B}\arrow{e,t}{\quad \breve{R}^B
\quad}\node{(AX)B{\oplus}(AY)B}\node{A(XB){\oplus}A(YB)}\arrow{w,t}
{\quad a{\oplus}a\quad}
\end{diagram}\tag{1.2}$$ $$\begin{diagram}
\node{(A{\oplus}B)X{\oplus}(A{\oplus}B)Y}\arrow{s,r}{\breve{R}^{X}{\oplus}\breve{R}^Y} \node{(A{\oplus}B)(X{\oplus}Y)}\arrow{w,t}{\breve{L}^{A{\oplus}B}}\arrow{e,t}{
\breve{R}^{X{\oplus}Y}}\node{A(X{\oplus}Y){\oplus}B(X{\oplus}Y)}\arrow{s,l}{\breve{L}^A{\oplus}\breve{L}^B}\\
\node{(AX{\oplus}BX){\oplus}(AY{\oplus}BY)}\arrow[2]{e,t} {\quad\quad v
\quad\quad}\node[2]{(AX{\oplus}AY){\oplus}(BX{\oplus}BY)}
\end{diagram}\tag{1.3}$$

commute, where $v=v_{U,V,Z,T}:(U{\oplus}V){\oplus}(Z{\oplus}T)\longrightarrow(U{\oplus}Z){\oplus}(V{\oplus}T)$ is the unique functor built from $a^+,c,id$ in the monoidal symmetric category $({\mathcal{A}},{\oplus}).$\
(Ann-3) For the unity object $1\in {\mathcal{A}}$ of the operation ${\oplus},$ the following diagrams:

$$\begin{diagram}
\node{1(X\oplus Y)} \arrow[2]{e,t}{\breve{L}^1}
\arrow{se,b}{l_{X\oplus Y}}
\node[2]{1X\oplus 1Y} \arrow{sw,b}{l_X\oplus l_Y} \\
\node[2]{X\oplus Y}
\end{diagram}\tag{1.4}$$

$$\begin{diagram}
\node{(X\oplus Y)1} \arrow[2]{e,t}{\breve{R}^1}
\arrow{se,b}{r_{X\oplus Y}}
\node[2]{X1\oplus Y1} \arrow{sw,b}{r_X\oplus r_Y} \\
\node[2]{X\oplus Y}
\end{diagram}\tag{1.4'}$$

commute.

[**Remark.**]{} The commutative diagrams (1.1), (1.1’) and (1.2), respectively, mean that:\
$$\begin{aligned} (a_{A, B, -})\;:\;& L^A.L^B &\longrightarrow \;& L^{AB}\\
(a_{-,A,B})\;:\;&R^{AB}&\longrightarrow \;&R^A.R^B\\
(a_{A, - ,B})\;:\;&L^A.R^B&\longrightarrow \;&R^B.L^A
\end{aligned}$$ are ${\oplus}$-functors.\
The diagram (1.3) shows that the family $(\breve{L}^Z_{X,Y})_Z=(\mathcal{L}_{-,X,Y})$ is an ${\oplus}$-functor between the ${\oplus}$-functors $Z\mapsto Z(X{\oplus}Y)$ and $Z\mapsto ZX{\oplus}ZY$, and the family $(\breve{R}^C_{A,B})_C=(\mathcal{R}_{A, B,-})$ is an ${\oplus}$-functor between the functors $C\mapsto (A{\oplus}B)C$ and $C\mapsto AC{\oplus}BC.$\
The diagram (1.4) (resp. (1.4’)) shows that $l$ (resp. $r$) is an ${\oplus}$-functor from $L^1$ (resp. $R^1$) to the unitivity functor of the ${\oplus}$-category ${\mathcal{A}}$.

[**The axiomatics of a ring category**]{}

A [*ring category*]{} is a category ${\mathcal{R}}$ equiped with two monoidal structures ${\oplus},{\otimes }$ (which include corresponding associativity morphisms $a^{{\oplus}}_{A,B,C},a^{{\otimes }}_{A,B,C}$ and unit objects denoted 0, 1) together with natural isomorphisms $$u_{A,B}:A{\oplus}B\to B{\oplus}A,\qquad\qquad v_{A,B,C}:A{\otimes }(B{\oplus}C)\to (A{\otimes }B){\oplus}(A{\otimes }C)$$ $$w_{A,B,C}:(A{\oplus}B){\otimes }C\to (A{\otimes }C){\oplus}(B{\otimes }C),$$ $$x_A:A{\otimes }0\to 0,\qquad y_A:0{\otimes }A\to 0.$$ These isomorphisms are required to satisfy the following conditions.

$K1 (\bullet{\oplus}\bullet)$ The isomorphisms $u_{A,B}$ define on ${\mathcal{R}}$ a structure of a symmetric monoidal category, i.e., they form a braiding and $u_{A,B}u_{B,A}=1.$

$K2 (\bullet{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C$ the diagram [$$\begin{diagram}
\node{A{\otimes }(B{\oplus}C)}\arrow{s,l}{A{\otimes }u_{B,C}} \arrow{e,t}{v_{A,B,C}}\node{(A{\otimes }B){\oplus}(A{\otimes }C)}\arrow{s,r}{u_{A{\otimes }B,A{\otimes }C}} \\
\node{A{\otimes }(C{\oplus}B)}\arrow{e,t}{v_{A,C,B}}\node{(A{\otimes }C){\oplus}(A{\otimes }B)}
\end{diagram}$$]{} is commutative.

$K3 ((\bullet{\oplus}\bullet){\otimes }\bullet)$ For any objects $A,B,C$ the diagram [$$\begin{diagram}
\node{(A{\oplus}B){\otimes }C}\arrow{s,l}{u_{A,B}{\otimes }C} \arrow{e,t}{w_{A,B,C}}\node{(A{\otimes }C){\oplus}(B{\otimes }C)}\arrow{s,r}{u_{A{\otimes }C,B{\otimes }C}} \\
\node{(B{\oplus}A){\otimes }C}\arrow{e,t}{w_{B,A,C}}\node{(B{\otimes }C){\oplus}(A{\otimes }C)}
\end{diagram}$$]{} is commutative.

$K4 ((\bullet{\oplus}\bullet{\oplus}\bullet){\otimes }\bullet)$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram}
\node{(A{\oplus}(B{\oplus}C)D)}\arrow{s,l}{a^{{\oplus}}_{A,B,C}{\otimes }D} \arrow{e,t}{w_{A,B{\oplus}C,D}}\node{AD{\oplus}((B{\oplus}C)D)}\arrow{e,t}{AD{\oplus}w_{B,C,D}}\node{AD{\oplus}(BD{\oplus}CD)}\arrow{s,r}{a^{{\oplus}}_{AD,BD,CD}} \\
\node{((A{\oplus}B){\oplus}C)D}\arrow{e,t}{w_{A{\oplus}B,C,D}}\node{(A{\oplus}B)D{\oplus}CD}\arrow{e,t}{w_{A,B,D}{\oplus}CD}\node{(AD{\oplus}BD){\oplus}CD}
\end{diagram}$$]{} is commutative.

$K5 (\bullet{\otimes }(\bullet{\oplus}\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram}
\node{A(B{\oplus}(C{\oplus}D))}\arrow{s,l}{A{\otimes }a^{{\oplus}}_{B,C,D}} \arrow{e,t}{v_{A,B,C{\oplus}D}}\node{AB{\oplus}A(C{\oplus}D)}\arrow{e,t}{AB{\oplus}v_{A,C,D}}\node{AB{\oplus}(AC{\oplus}AD)}\arrow{s,r}{a^{{\oplus}}_{AB,AC,AD}} \\
\node{A((B{\oplus}C){\oplus}D)}\arrow{e,t}{v_{A,B{\oplus}C,D}}\node{A(B{\oplus}C){\oplus}AD}\arrow{e,t}{v_{A,B,C}{\oplus}AD}\node{(AB{\oplus}AC){\oplus}AD}
\end{diagram}$$]{} is commutative.

$K6 (\bullet{\otimes }\bullet{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram [$$\begin{diagram}
\node{A(B(C{\oplus}D))}\arrow{s,l}{a^{{\otimes }}_{A,B,C{\oplus}D}} \arrow{e,t}{A{\otimes }v_{B,C,D}}\node{A(BC{\oplus}BD)}\arrow{e,t}{v_{A,BC,BD}}\node{A(BC){\oplus}A(BD)}\arrow{s,r}{a^{{\otimes }}_{A,B,C}{\oplus}a^{{\otimes }}_{A,B,D}} \\
\node{(AB)(C{\oplus}D)} \arrow[2]{e,t}{v_{AB,C,D}}\node[2]{(AB)C{\oplus}(AB)D}
\end{diagram}$$]{} is commutative.

$K7 ((\bullet{\oplus}\bullet){\otimes }\bullet{\otimes }\bullet)$ Similar to the above.

$K8 (\bullet{\otimes }(\bullet{\oplus}\bullet){\otimes }\bullet)$ Similar to the above.

$K9 ((\bullet{\oplus}\bullet){\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B,C,D$ the diagram

(12,4.5)

(0,0)[$((AC{\oplus}BC){\oplus}AD){\oplus}BD$]{} (0.1,1.5)[$(AC{\oplus}BC){\oplus}(AD{\oplus}BD)$]{} (0.3,3)[$(A{\oplus}B)C{\oplus}(A{\oplus}B)D$]{} (0.8,4.5)[$(A{\oplus}B)(C{\oplus}D)$]{}

(10,0)[$(AC{\oplus}(BC{\oplus}AD)){\oplus}BD$]{} (10,1.5)[$(AC{\oplus}(AD{\oplus}BC)){\oplus}BD$]{} (9.8,3)[$((AC{\oplus}AD){\oplus}BC){\oplus}BD$]{} (9.8,4.5)[$(AC{\oplus}AD){\oplus}(BC{\oplus}BD)$]{} (5.2,4.5)[$A(C{\oplus}D){\oplus}B(C{\oplus}D)$]{}

(1.8,4.3)[(0,-1)[0.8]{}]{} (1.8,2.8)[(0,-1)[0.8]{}]{} (1.8,1.3)[(0,-1)[0.8]{}]{}

(11.6,4.3)[(0,-1)[0.8]{}]{} (11.6,1.9)[(0,1)[0.8]{}]{} (11.6,1.3)[(0,-1)[0.8]{}]{}

(3.1,4.6)[(1,0)[1.9]{}]{} (8.3,4.6)[(1,0)[1.3]{}]{} (9.8,0.1)[(-1,0)[6.2]{}]{}

is commutative (the notation for arrows have been omitted, they are obvious).

$K10 (0{\otimes }0)$ The maps $x_0,y_0:0{\otimes }0\to 0$ coincide.

$K11 (0{\otimes }(\bullet{\oplus}\bullet))$ For any objects $A,B$ the diagram [$$\begin{diagram}
\node{0{\otimes }(A{\oplus}B)}\arrow{s,l}{y_{A{\oplus}B}} \arrow{e,t}{v_{0,A,B}}\node{(0{\otimes }A){\oplus}(0{\otimes }B)}\arrow{s,r}{y_a{\oplus}y_B}\\
\node{0}\node{0{\oplus}0}\arrow{w,t}{l^{{\oplus}}_0=r^{{\oplus}}_0}
\end{diagram}$$]{} is commutative.

$K12 ((\bullet{\oplus}\bullet){\otimes }0)$ Similar to the above.

$K13 (0{\otimes }1)$ The maps $y_1,r^{{\otimes }}_0:0{\otimes }1\to 0$ coincide.

$K14 (1{\otimes }0)$ Similar to the above.

$K15 (0{\otimes }\bullet{\otimes }\bullet)$ For any objects $A,B$ the diagram [$$\begin{diagram}
\node{0{\otimes }(A{\otimes }B)}\arrow{s,l}{y_{A{\otimes }B}} \arrow{e,t}{a^{{\otimes }}_{0,A,B}}\node{(0{\otimes }A){\otimes }B}\arrow{s,r}{y_A{\otimes }B}\\
\node{0}\node{0{\otimes }B}\arrow{w,t}{y_B}
\end{diagram}$$]{} is commutative.

$K16 (\bullet{\otimes }0{\otimes }\bullet),(\bullet{\otimes }\bullet{\otimes }0)$ For any objects $A,B$ the diagrams [$$\begin{diagram}
\node{A{\otimes }(0{\otimes }B)}\arrow{s,l}{A{\otimes }y_{B}} \arrow[2]{e,t}{a^{{\otimes }}_{A,0,B}}\node[2]{(A{\otimes }0){\otimes }B}\arrow{s,r}{x_A{\otimes }B}\\
\node{A{\otimes }0}\arrow{e,t}{x_A}\node{0}
\node{0{\otimes }B}\arrow{w,t}{y_B}
\end{diagram}$$]{} [$$\begin{diagram}
\node{A{\otimes }(B{\otimes }0)}\arrow{s,l}{A{\otimes }x_B}\arrow{e,t}{a^{{\otimes }}_{A,B,0}}\node{(A{\otimes }B){\otimes }0}\arrow{s,r}{x_{A{\otimes }B}}\\
\node{A{\otimes }0}\arrow{e,t}{x_A}\node{0}
\end{diagram}$$]{} are commutative.

$K17 (\bullet(0{\oplus}\bullet))$ For any objects $A,B$ the diagram [$$\begin{diagram}
\node{A{\otimes }(0{\oplus}B)}\arrow{s,l}{A{\otimes }l^{{\oplus}}_B} \arrow{e,t}{v_{A,0,B}}\node{(A{\otimes }0){\oplus}(A{\otimes }B)}\arrow{s,r}{x_A{\oplus}(A{\otimes }B)}\\
\node{A{\otimes }B}\node{0{\oplus}(A{\otimes }B)}\arrow{w,t}{l^{{\oplus}}_{A{\otimes }B}}
\end{diagram}$$]{} is commutative.

$K18 ((0{\oplus}\bullet){\otimes }\bullet),(\bullet{\otimes }(\bullet{\oplus}0)),((\bullet{\oplus}0){\otimes }\bullet)$ Similar to the above.

The relation between an Ann-category and a ring category
========================================================

In this section, we will prove that the axiomatics of a ring category, without K10, can be deduced from the one of an Ann-category. First, we can see that, the functor morphisms $a^{\oplus}, a^{\otimes}, u, l^{{\oplus}}, r^{{\oplus}}, v, w,$ in Definiton 2 are, respectively, the functor morphisms $a_{+}, a, c, g, d, {\frak{L}}, {\frak{R}}$ in Definition 1. Isomorphisms $x_A, y_A$ coincide with isomorphisms $\widehat{L}^A, \widehat{R}^A$ referred in Proposition 1.

We now prove that diagrams which commute in a ring category also do in an Ann-category.

K1 obviously follows from (ii) in the definition of an Ann-category.

The commutative diagrams $K2, K3, K4, K5$ are indeed the compatibility of functor isomorphisms $(L^A, \Breve L^A), (R^A, \Breve R^A)$ with the constraints $a_{+}, c$ (the axiom Ann-1).

The diagrams $K5-K9,$ respectively, are indeed the ones in (Ann-2). Particularly, K9 is indeed the decomposition of (1.3) where the morphism $v$ is replaced by its definition diagram: [$$\begin{diagram}
\node{(P{\oplus}Q){\oplus}(R{\oplus}S)}\arrow{s,l}{v} \arrow{e,t}{a_{+}}\node{((P{\oplus}Q){\oplus}R){\oplus}S}\node{(P{\oplus}(Q{\oplus}R)){\oplus}S}\arrow{w,t}{a_{+}{\oplus}S}\arrow{s,r}{(P{\oplus}c){\oplus}S}\\
\node{(P{\oplus}R){\oplus}(Q{\oplus}S)}\arrow{e,t}{a_{+}}\node{((P{\oplus}R){\oplus}Q){\oplus}S}\node{(P{\oplus}(R{\oplus}Q)){\oplus}S.}\arrow{w,t}{a_{+}{\oplus}S}
\end{diagram}$$]{}

[**The proof for K17, K18**]{}

Let $P,$ $P^{'}$ be Gr-categories, $(a_{+}, (0, g, d)), (a^{'}_{+}, (0^{'}, g^{'}, d^{'}))$ be respective constraints, and $(F, \Breve F):P\rightarrow P^{'}$ be $\oplus$-functor which is compatible with $(a_{+}, a^{'}_{+}).$ Then $(F, \Breve F)$ is compatible with the unitivity constraints $(0, g, d)), (0^{'}, g^{'}, d^{'})).$

First, the isomorphism $\widehat{F}:F0\to 0'$ is determined by the composition $$\begin{diagram}
\node{u=F0{\oplus}F0}\node{F(0{\oplus}0)}
\arrow {w,t}{\widetilde{F}}
\arrow{e,t}{F(g)}
\node{F0}\node{0'{\oplus}F0.}\arrow{w,t}{g'}
\end{diagram}$$ Since $F0$ is a regular object, there exists uniquely the isomorphism $\widehat{F}:F0\to 0'$ such that $\widehat{F}{\oplus}id_{F0}=u.$ Then, we may prove that $\widehat{F}$ satisfies the diagrams in the definition of the compatibility of the ${\oplus}$-functor $F$ with the unitivity constraints.

In an Ann-category ${\mathcal{A}},$ there exist uniquely isomorphisms $$\hat L^A: A{\otimes }0 \longrightarrow 0, \qquad \hat R^A:  0{\otimes }A \longrightarrow 0$$ such that the following diagrams [$$\begin{diagram}
\node{AX}\node{A(0{\oplus}X)}\arrow{w,t}{L^A(g)}\arrow{s,r}{\breve L^A\qquad(2.1)}
\node{AX}\node{A(X{\oplus}0)}\arrow{w,t}{L^A(d)}\arrow{s,r}{\breve L^A\qquad(2.1')}\\
\node{0{\oplus}AX}\arrow{n,l}{g}\node{A0{\oplus}AX}\arrow{w,t}{\hat L^A{\oplus}id}
\node{AX{\oplus}0}\arrow{n,l}{d}\node{AX{\oplus}A0}\arrow{w,t}{id{\oplus}\hat L^A}
\end{diagram}$$]{} [$$\begin{diagram}
\node{AX}\node{(0{\oplus}X)A}\arrow{w,t}{R^A(g)}\arrow{s,r}{\breve R^A\qquad(2.2)}
\node{AX}\node{(X{\oplus}0)A}\arrow{w,t}{R^A(d)}\arrow{s,r}{\breve R^A\qquad(2.2')}\\
\node{0{\oplus}AX}\arrow{n,l}{g}\node{0A{\oplus}XA}\arrow{w,t}{\hat R^A{\oplus}id}
\node{AX{\oplus}0}\arrow{n,l}{d}\node{XA{\oplus}0A}\arrow{w,t}{id{\oplus}\hat R^A}
\end{diagram}$$]{} commute, i.e., $L^A$ and $R^A$ are U-functors respect to the operation ${\oplus}$.

Since $(L^A, \breve L^A)$ are ${\oplus}$-functors which are compatible with the associativity constraint $a^{{\oplus}}$ of the Picard category $({\mathcal{A}},{\oplus}),$ it is also compatible with the unitivity constraint $(0,g,d)$ thanks to Lemma 1. That means there exists uniquely the isomorphism $\hat L^A$ satisfying the diagrams $(2.1)$ and $(2.1')$. The proof for $\hat R^A$ is similar. The diagrams commute in Proposition 1 are indeed K17, K18.

[**The proof for 15, K16**]{}

Let $(F,\breve F), (G,\breve G)$ be ${\oplus}$-functors between ${\oplus}$-categories ${\mathcal{C}}, {\mathcal{C}}'$ which are compatible with the constraints $(0, g, d), (0', g', d')$ and $\widetilde F: F(0)\longrightarrow 0', \widetilde G: G(0)\longrightarrow 0'$ are respective isomorphisms. If $\alpha: F \longrightarrow G$ in an ${\oplus}$-morphism such that $\alpha_0$ is an isomorphism, then the diagram [$$\begin{diagram}
\node{F0}\arrow[2]{r,t}{\alpha_0}\arrow{se,b}{\hat F}\node[2]{G0}\arrow{sw,b}{\hat G}\\
\node[2]{0'}
\end{diagram}$$]{} commutes.

Let us consider the diagram

(9,3.8)

(-0.2,0.9)[$F0$]{} (2,0.9)[$F(0{\oplus}0)$]{} (4.8,0.9)[$G(0{\oplus}0)$]{} (7.8,0.9)[$G0$]{}

(-0.4,2.5)[$0'{\oplus}F0$]{} (2,2.5)[$F0{\oplus}F0$]{} (4.8,2.5)[$G0{\oplus}G0$]{} (7.6,2.5)[$0'{\oplus}G0$]{}

(3.8,0.1)[$u_0$]{} (3.6,3.7)[$id{\oplus}u_0$]{} (0.8,1.1)[$F(g)$]{} (3.6,1.1)[$u_{0{\oplus}0}$]{} (6.5,1.1)[$G(g)$]{} (0.9,2.7)[$\breve{F}{\oplus}id$]{} (3.5,2.7)[$u_0{\oplus}u_0$]{} (6.3,2.7)[$\breve{G}{\oplus}id$]{}

(-0.3,1.7)[$g'$]{} (2.3,1.7)[$\widetilde{F}$]{} (5.1,1.7)[$\widetilde{G}$]{} (7.7,1.7)[$g'$]{}

(0,0.8)[(0,-1)[0.8]{}]{} (0,0)[(1,0)[8]{}]{} (8,0)[(0,1)[0.8]{}]{}

(0,3.6)[(0,-1)[0.7]{}]{} (0,3.6)[(1,0)[8]{}]{} (8,3.6)[(0,-1)[0.7]{}]{}

(0,2.3)[(0,-1)[1]{}]{} (8,2.3)[(0,-1)[1]{}]{}

(2.6,1.3)[(0,1)[1]{}]{} (5.4,1.3)[(0,1)[1]{}]{}

(1.8,2.6)[(-1,0)[1]{}]{} (3.3,2.6)[(1,0)[1.3]{}]{} (6.1,2.6)[(1,0)[1.3]{}]{}

(0.3,1)[(1,0)[1.5]{}]{} (3.3,1)[(1,0)[1.3]{}]{} (6.1,1)[(1,0)[1.5]{}]{}

(3.8,3.2)[(I)]{} (1,1.7)[(II)]{} (3.7,1.7)[(III)]{} (6.4,1.7)[(IV)]{} (3.8,0.5)[(V)]{}

In this diagram, (II) and (IV) commute thanks to the compatibility of ${\oplus}$-functors $(F,\breve F), (G,\breve G)$ with the unitivity constraints; (III) commutes since $u$ is a ${\oplus}$-morphism; (V) commutes thanks to the naturality of $g'.$ Therefore, (I) commutes, i.e., $$\breve{G}\circ u_0{\oplus}u_0=\breve{F}{\oplus}u_0.$$ Since $F0$ is a regular object, $\breve{G}\circ u_0=\breve{F}.$

For any objects $X, Y\in \text{ob}{\mathcal{A}}$ the diagrams $$\begin{aligned}
{\scriptsize\begin{diagram}
\node{X{\otimes }(Y{\otimes }0)}\arrow{e,t}{id{\otimes }\widehat{L}^Y}\arrow{s,l}{a}\node{X{\otimes }0}
\arrow{s,r}{\widehat L^X \qquad(2.3)}
\node{0{\otimes }(X{\otimes }Y)}\arrow{e,t}{\widehat R^{XY}}\arrow{s,l}{a}\node{0}\\
\node{(X{\otimes }Y){\otimes }0}\arrow{e,t}{\widehat L^{XY}}\node{0}
\node{(0{\otimes }X){\otimes }Y}\arrow{e,t}{\widehat R^X{\otimes }id}\node{0{\otimes }Y}\arrow{n,r}{\widehat R^Y \qquad(2.3')}
\end{diagram}\nonumber}\end{aligned}$$ [$$\begin{diagram}
\node{X{\otimes }(0{\otimes }Y)}\arrow[2]{e,t}{a}\arrow{s,l}{id{\otimes }\hat R^Y}\node[2]{(X{\otimes }0){\otimes }Y}
\arrow{s,r}{\widehat L^X{\otimes }id\qquad(2.4)}\\
\node{X{\otimes }0}\arrow{e,t}{\widehat L^X}\node{0}
\node{0{\otimes }Y}\arrow{w,t}{\widehat R^Y}
\end{diagram}$$]{} commute.

To prove the first diagram commutative, let us consider the diagram [$$\begin{diagram}
\node{X{\otimes }(Y{\otimes }0)}\arrow{e,t}{id{\otimes }\hat L^Y}\arrow{s,l}{a}\arrow{se,t}{\widehat{L^X\circ L^Y}}
\node{X{\otimes }0}\arrow{s,r}{\hat L^X}\\
\node{(X{\otimes }Y){\otimes }0}\arrow{e,t}{\hat L^{XY}}\node{0}
\end{diagram}$$]{} According to the axiom (1.1), $(a_{X, Y, Z})_Z$ is an ${\oplus}$-morphism from the functor $L = L^X\circ L^Y$ to the functor $G = L^{XY}$. Therefore, from Lemma 2, (II) commutes. (I) commutes thanks to the determination of $\hat L$ of the composition $L = L^\circ L^Y$. So the outside commutes.

The second diagram is proved similarly, thanks to the axiom (1.1’). To prove that the diagram (2.4) commutes, let us consider the diagram [$$\begin{diagram}
\node{X{\otimes }(0{\otimes }Y)}\arrow[2]{e,t}{a}\arrow{s,l}{id{\otimes }\widehat{R^Y}}\arrow{se,t}{\hat H}\node[2]{(X{\otimes }0){\otimes }Y}\arrow{s,r}{\widehat{L^X}{\otimes }id}\arrow{sw,t}{\hat K}\\
\node{X{\otimes }0}\arrow{e,b}{\widehat{L^X}}\node{0}\node{0{\otimes }Y}\arrow{w,b}{\widehat{R^Y}}
\end{diagram}$$]{} where $H = L^X\circ R^Y$ and $K = R^Y\circ L^X$. Then (II) and (III) commute thanks to the determination of the isomorphisms $H$ and $K$. From the axiom (1.2), $(a_{X,Y,Z})_Z$ is an ${\oplus}$-morphism from the functor $H$ to the functor $K$. So from Lemma 2, (I) commutes. Therefore, the outside commutes. The diagrams in Proposition 2 are indeed K15, K16.

[**Proof for K11**]{}

In an Ann-category, the diagram [$$\begin{diagram}
\node{0{\oplus}0}\arrow{e,t}{g_0=d_0}\node{0}\\
\node{(0{\otimes }X){\oplus}(0{\otimes }Y)}\arrow{n,l}{\widehat{R}^X{\oplus}\widehat{R}^Y}
\node{0{\otimes }(X{\oplus}Y)}\arrow{n,r}{\widehat{R}^{XY}\qquad(2.5)}\arrow{w,t}{\breve{L}^0}
\end{diagram}$$]{} commutes.

Let us consider the diagram

(14,7.3)

(1.2,0)[$A(B{\oplus}C){\oplus}0(B{\oplus}C)$]{} (8.1,0)[$A(B{\oplus}C){\oplus}0$]{} (1,1.7)[$(AB{\oplus}AC){\oplus}(0B{\oplus}0C)$]{} (7.9,1.7)[$(AB{\oplus}AC){\oplus}(0{\oplus}0)$]{} (1.1,3.5)[$(AB{\oplus}0B){\oplus}(AC{\oplus}0C)$]{} (8.2,3.5)[$(AB{\oplus}0){\oplus}(AC{\oplus}0)$]{} (1.3,5.2)[$(A{\oplus}0)B{\oplus}(A{\oplus}0)C$]{} (9,5.2)[$AB{\oplus}AC$]{} (1.7,7)[$(A{\oplus}0)(B{\oplus}C)$]{} (8.9,7)[$A(B{\oplus}C)$]{}

(2.9,0.8)[$\breve{L}^A{\oplus}\breve{L}^0$]{} (2.9,2.9)[$v$]{} (2.9,4.5)[$\breve{R}^B{\oplus}\breve{R}^C$]{} (2.9,6.2)[$\breve{L}^{A{\oplus}0}$]{}

(9.8,0.8)[$\breve{L}^A{\oplus}d_0^{-1}$]{} (9.8,2.9)[$v$]{} (9.8,4.5)[$d_{AB}{\oplus}d_{AC}$]{} (9.8,6.2)[$\breve{L}^A$]{}

(0.2,2.6)[$\breve{R}^{B{\oplus}C}$]{} (11.7,2.6)[$d$]{}

(5.5,0.3)[$f'_A{\oplus}id$]{} (4.6,2)[$(id{\oplus}id){\oplus}(\widehat{R}^B{\oplus}\widehat{R}^C)$]{} (4.7,3.8)[$(id{\oplus}\widehat{R}^B){\oplus}(id{\oplus}\widehat{R}^C)$]{} (4.9,5.5)[$(d_A{\otimes }id){\oplus}(d_A{\otimes }id)$]{} (5.6,7.3)[$d_A{\otimes }id$]{}

(5.5,6.3)[(I)]{} (5.5,4.5)[(II)]{} (5.5,2.7)[(III)]{} (5.5,1)[(IV)]{} (0.5,4.3)[(V)]{} (11.2,4)[(VI)]{}

(2.7,0.5)[(0,1)[0.9]{}]{} (2.7,3.3)[(0,-1)[0.9]{}]{} (2.7,5)[(0,-1)[0.9]{}]{} (2.7,6.8)[(0,-1)[1.1]{}]{}

(9.6,0.4)[(0,1)[1]{}]{} (9.6,3.2)[(0,-1)[0.9]{}]{} (9.6,4)[(0,1)[1]{}]{} (9.6,6.8)[(0,-1)[1.1]{}]{}

(4.2,0.1)[(1,0)[3.7]{}]{} (4.4,1.8)[(1,0)[3.3]{}]{} (4.4,3.6)[(1,0)[3.6]{}]{} (4.2,5.3)[(1,0)[4.6]{}]{} (3.9,7.1)[(1,0)[4.7]{}]{}

(0,0.1)[(1,0)[1]{}]{} (12,7.1)[(-1,0)[1.7]{}]{}

(0,0.1)[(0,1)[7]{}]{} (0,7.1)[(1,0)[1.5]{}]{} (12,0.1)[(-1,0)[1.9]{}]{} (12,0.1)[(0,1)[7]{}]{}

(13,4)[(2.6)]{}

In this diagram, (V) commutes thanks to the axiom I(1.3), (I) commutes thanks to the functorial property of ${\frak{L}};$ the outside and (II) commute thanks to the compatibility of the functors $R^{B{\oplus}C},R^B,R^C$ with the unitivity constraint $(0,g,d);$ (III) commutes thanks to the functorial property $v;$ (VI) commutes thanks to the coherence for the ACU-functor $(L^A,\breve{L}^A).$ So (IV) commutes. Note that $A(B{\oplus}C)$ is a regular object respect to the operation ${\oplus},$ so the diagram (2.5) commutes. We have K11.

Similarly, we have K12.

[**Proof for K13, K14**]{}

In an Ann-category, we have $$\widehat{L}^1=l_0,\widehat{R}^1=r_0.$$

We will prove the first equation, the second one is proved similarly. Let us consider the diagram (2.7). In this diagram, the outside commutes thanks to the compatibility of ${\oplus}$-functor $(L^1,\breve{L}^1)$ with the unitivity constraint $(0,g,d)$ respect to the operation ${\oplus};$ (I) commutes thanks to the functorial property of the isomorphism $l;$ (II) commutes thanks to the functorial property of $g;$ (III) obviously commutes; (IV) commutes thanks to the axiom I(1.4). So (V) commutes, i.e., $$\widehat{L}^1{\oplus}id_{1.0}=l_0{\oplus}id_{1.0}$$ Since 1.0 is a regular object respect to the operation ${\oplus},$ $\widehat{L}^1=l_0.$

(5.5,4)

(0,0)[$0{\oplus}(1.0)$]{} (4.5,0)[$(1.0){\oplus}(1.0)$]{} (1.5,1.3)[$0{\oplus}0$]{} (3.3,1.3)[$0{\oplus}0$]{} (3.3,2.5)[$0{\oplus}0$]{} (1.8,2.5)[$0$]{}

(0.4,3.5)[$1.0$]{} (4.7,3.5)[$1.(0{\oplus}0)$]{}

(4.3,0.1)[(-1,0)[3.1]{}]{} (4.5,3.6)[(-1,0)[3.6]{}]{} (0.5,0.3)[(0,1)[3]{}]{} (5.3,3.4)[(0,-1)[3]{}]{} (0.7,3.3)[(3,-2)[1]{}]{} (5,3.3)[(-3,-2)[1]{}]{} (0.7,0.3)[(1,1)[0.9]{}]{} (4.8,0.3)[(-1,1)[0.9]{}]{} (2.3,1.4)[(1,0)[0.8]{}]{} (3.6,1.6)[(0,1)[0.8]{}]{} (1.9,2.4)[(0,-1)[0.8]{}]{} (3.1,2.6)[(-1,0)[1]{}]{}

(2.3,-0.3)[$\widehat{L}^1{\oplus}id$]{} (1.3,0.6)[$id{\oplus}l_0$]{} (3.5,0.6)[$l_0{\oplus}l_0$]{} (2.5,0.6)[$(V)$]{} (2.5,1.1)[$id$]{} (-0.1,2)[$g_{1.0}$]{} (0.8,2)[$(II)$]{} (1.6,2)[$g_0$]{} (2.3,2)[$(III)$]{} (3.3,2)[$id$]{} (4.2,2)[$(IV)$]{} (5.4,2)[$\breve{L}^1$]{} (7.5,2)[$(2.7)$]{} (2.5,2.7)[$g_0$]{} (1.2,3.1)[$l_0$]{} (2.5,3.1)[$(I)$]{} (3.9,3.1)[$l_{0{\oplus}0}$]{} (1.6,3.7)[$L^1(g_0)=id{\otimes }g_0$]{}

We have K14.

Similarly, we have K13.

An Ann-category ${\mathcal{A}}$ is [*strong*]{} if $\widehat{L}^0=\widehat{R}^0.$

All the above results can be stated as follows

Each strong Ann-category is a ring category.

In our opinion, in the axiomatics of a [*ring category,*]{} the compatibility of the distributivity constraint with the unitivity constraint $(1, l, r)$ respect to the operation $\otimes$ is necessary, i.e., the diagrams of (Ann-3) should be added.

Moreover, if the symmetric monoidal structure of the operation $\oplus$ is replaced with the symmetric categorical groupoid structure, then each ring category is an Ann-category.

An open question: May the equation $\widehat{L}^0=\widehat{R}^0$ be proved to be independent in an Ann-category?

[99]{}

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