The Relation between Ann-Categories and Ring Categories

THE RELA TION BETWEEN ANN -CA TEGORIES AND RING CA TEGORIES Nguy en Tien Quang, Nguy en Th u Th uy and Che Thi Kim Ph ung Hanoi Nationa l University of Educ ation No v em b er 20, 2018 Abstract There are different categorizatio ns of the defin ition of a ring such as A nn-c ate gory [6], ring c ate gory [2],.. . The main result of this pap er is to prov e that every axiom of the definition of a ring c ate gory , without the axiom x 0 = y 0 , can be dedu ced from the axiomatics of an Ann-c ate gory . 1 In tro duction Categorie s with monoidal structur es ⊕ , ⊗ (also called c ate gories with distributivity c on- str aints ) w e r e presented by Laplaza [3]. M. Kapra nov and V.V o evodsky [2] omitted require - men ts of the axio matics of Laplaza whic h are related to the commutativit y constraints of the op er a tion ⊗ a nd presented the name ring c ate gories to indica te these ca tegories. T o approach in an other wa y , mono idal categor ie s can b e “smo othed” to b ecome a c ate gory with gr oup structur e, when they a re added the definition of inv ertible ob jects (see Laplaza [4], Saavedra Riv ano [9]). No w , if the back ground ca tegory is a gr oup oid (i.e., each morphism is an isomorphism) then w e ha v e monoidal c ate gory gr oup-like (see A. F r¨ olich and C. T. C. W all [1], or a Gr-c ate gory (see H. X. Sinh [11]). These categories ca n be classified by H 3 (Π , A ). Each Gr -categor y G is determined by 3 inv a riants: The gro up Π of classes of co ngruence ob jects, Π − mo dule A of automo rphisms o f the unit 1, and an element h ∈ H 3 (Π , A ) , wher e h is induced by the a sso ciativity constr aint of G . In 1 9 87, in [6], N. T. Quang pr esented the definition of an Ann-c ate gory , a s a categoriz a - tion of the definition of rings, when a symmetric Gr-catego ry (also called Pic- categor y) is equip ed with a monoidal s tr ucture ⊗ . In [8], [7], Ann-categor ies and r e gular Ann-ca tegories , developed from the ring extension pro blem, have b een cla ssified b y , r esp ectively , Ma c La ne ring coho mology [5] a nd Shukla alg ebraic cohomo lo gy [10 ]. The aim o f this pape r is to show clearly the r elation b etw een the definition o f a n Ann- c ate gory and a ring c ate gory . F or co nv enience, let us r ecall the definitions. Moreo ver, let us denote AB or A.B instea d of A ⊗ B . 2 F undamen tal d efinitions Definition 2.1. The axiomatics of an Ann-category An Ann-c ate gory c onsists of: i) A gr oup oid A to gether with two bifunctors ⊕ , ⊗ : A × A − → A . ii) A fixe d obje ct 0 ∈ A t o gether with natur ality c onstr aints a + , c, g , d such that ( A , ⊕ , a + , c, (0 , g , d )) 1 N. T. Quang, N. T . Thuy a nd C. T. K. P hung 2 is a Pic-c ate gory. iii) A fixe d obje ct 1 ∈ A to gether with natur ality c onstr aints a, l , r s uch that ( A , ⊗ , a, (1 , l , r )) is a monoidal A -c ate gory. iv) Natur al isomorphisms L , R L A,X,Y : A ⊗ ( X ⊕ Y ) − → ( A ⊗ X ) ⊕ ( A ⊗ Y ) R X,Y ,A : ( X ⊕ Y ) ⊗ A − → ( X ⊗ A ) ⊕ ( Y ⊗ A ) such that the fol lowing c onditions ar e satisfie d: (Ann-1) F or e ach A ∈ A , the p airs ( L A , ˘ L A ) , ( R A , ˘ R A ) determine d by r elations: L A = A ⊗ − R A = − ⊗ A ˘ L A X,Y = L A,X,Y ˘ R A X,Y = R X,Y ,A ar e ⊕ - functors which ar e c omp atible with a + and c. (Ann-2) F or al l A, B , X , Y ∈ A , the fol lowing diagr ams: ( AB )( X ⊕ Y ) A ( B ( X ⊕ Y )) A ( B X ⊕ B Y ) ( AB ) X ⊕ ( AB ) Y A ( B X ) ⊕ A ( B Y ) ❄ ˘ L AB ✛ a A,B,X ⊕ Y ✲ id A ⊗ ˘ L B ❄ ˘ L A ✛ a A,B,X ⊕ a A,B,Y (1.1) ( X ⊕ Y )( B A ) (( X ⊕ Y ) B ) A ( X B ⊕ Y B ) A X ( B A ) ⊕ Y ( B A ) ( X B ) A ⊕ ( Y B ) A ❄ ˘ R BA ✲ a X ⊕ Y ,B ,A ✲ ˘ R B ⊗ id A ❄ ˘ R A ✲ a X,B,A ⊕ a Y ,B ,A (1.1’) ( A ( X ⊕ Y )) B A (( X ⊕ Y ) B ) A ( X B ⊕ Y B ) ( AX ⊕ AY ) B ( AX ) B ⊕ ( AY ) B A ( X B ) ⊕ A ( Y B ) ❄ ˘ L A ⊗ id B ✛ a A,X ⊕ Y ,B ✲ id A ⊗ ˘ R B ❄ ˘ L A ✲ ˘ R B ✛ a ⊕ a (1.2) ( A ⊕ B ) X ⊕ ( A ⊕ B ) Y ( A ⊕ B )( X ⊕ Y ) A ( X ⊕ Y ) ⊕ B ( X ⊕ Y ) ( AX ⊕ B X ) ⊕ ( AY ⊕ B Y ) ( AX ⊕ AY ) ⊕ ( B X ⊕ B Y ) ❄ ˘ R X ⊕ ˘ R Y ✛ ˘ L A ⊕ B ✲ ˘ R X ⊕ Y ❄ ˘ L A ⊕ ˘ L B ✲ v (1.3) c ommute, wher e v = v U,V ,Z,T : ( U ⊕ V ) ⊕ ( Z ⊕ T ) − → ( U ⊕ Z ) ⊕ ( V ⊕ T ) is the un ique functor built fr om a + , c, id in the monoidal symmetric c ate gory ( A , ⊕ ) . (Ann-3) F or the unity obje ct 1 ∈ A of the op er ation ⊕ , the fol lowing diagr ams: 1( X ⊕ Y ) 1 X ⊕ 1 Y X ⊕ Y ✲ ˘ L 1 ◗ ◗ ◗ ◗ s l X ⊕ Y ✑ ✑ ✑ ✑ ✰ l X ⊕ l Y (1.4) N. T. Quang, N. T . Thuy a nd C. T. K. P hung 3 ( X ⊕ Y )1 X 1 ⊕ Y 1 X ⊕ Y ✲ ˘ R 1 ◗ ◗ ◗ ◗ s r X ⊕ Y ✑ ✑ ✑ ✑ ✰ r X ⊕ r Y (1.4’) c ommute. Remark. The commutativ e diagr ams (1.1), (1.1’) and (1.2), res pec tively , mean that: ( a A,B , − ) : L A .L B − → L AB ( a − ,A,B ) : R AB − → R A .R B ( a A, − ,B ) : L A .R B − → R B .L A are ⊕ - functors. The diagra m (1.3) s hows that the family ( ˘ L Z X,Y ) Z = ( L − ,X,Y ) is an ⊕ -functor b etw een the ⊕ -functors Z 7→ Z ( X ⊕ Y ) and Z 7→ Z X ⊕ Z Y , a nd the fa mily ( ˘ R C A,B ) C = ( R A,B , − ) is an ⊕ -functor b etw een the functors C 7→ ( A ⊕ B ) C and C 7→ AC ⊕ B C . The diagra m (1.4) (r e sp. (1.4’)) shows that l (r e sp. r ) is an ⊕ -functor from L 1 (resp. R 1 ) to the unitivity functor of the ⊕ -categ o ry A . Definition 2.2. The axiomatics of a ring category A ring c ate gory is a c at e gory R e qu ip e d with two monoidal str u ctur es ⊕ , ⊗ (which include c orr esp onding asso ciativity morphisms a ⊕ A,B ,C , a ⊗ A,B ,C and unit obje cts denote d 0, 1) to gether with natur al isomorphisms u A,B : A ⊕ B → B ⊕ A, v A,B ,C : A ⊗ ( B ⊕ C ) → ( A ⊗ B ) ⊕ ( A ⊗ C ) w A,B ,C : ( A ⊕ B ) ⊗ C → ( A ⊗ C ) ⊕ ( B ⊗ C ) , x A : A ⊗ 0 → 0 , y A : 0 ⊗ A → 0 . These isomorphisms ar e r e quir e d to satisfy the fol lowing c onditions. K 1( • ⊕ • ) The isomorphi sms u A,B define on R a struct ur e of a symm et ric monoi dal c ate gory, i.e. , they form a br aiding and u A,B u B ,A = 1 . K 2( • ⊗ ( • ⊕ • )) F or any obje cts A, B , C the diagr am A ⊗ ( B ⊕ C ) ( A ⊗ B ) ⊕ ( A ⊗ C ) A ⊗ ( C ⊕ B ) ( A ⊗ C ) ⊕ ( A ⊗ B ) ❄ A ⊗ u B,C ✲ v A,B,C ❄ u A ⊗ B,A ⊗ C ✲ v A,C,B is c ommu t ative. K 3(( • ⊕ • ) ⊗ • ) F or any obje cts A, B , C the diagr am ( A ⊕ B ) ⊗ C ( A ⊗ C ) ⊕ ( B ⊗ C ) ( B ⊕ A ) ⊗ C ( B ⊗ C ) ⊕ ( A ⊗ C ) ❄ u A,B ⊗ C ✲ w A,B,C ❄ u A ⊗ C,B ⊗ C ✲ w B,A,C is c ommu t ative. K 4(( • ⊕ • ⊕ • ) ⊗ • ) F or any obje ct s A, B , C, D the diagr am ( A ⊕ ( B ⊕ C ) D ) AD ⊕ (( B ⊕ C ) D ) AD ⊕ ( B D ⊕ C D ) (( A ⊕ B ) ⊕ C ) D ( A ⊕ B ) D ⊕ C D ( AD ⊕ B D ) ⊕ C D ❄ a ⊕ A,B,C ⊗ D ✲ w A,B ⊕ C,D ✲ AD ⊕ w B,C,D ❄ a ⊕ AD,B D,C D ✲ w A ⊕ B,C,D ✲ w A,B,D ⊕ C D N. T. Quang, N. T . Thuy a nd C. T. K. P hung 4 is c ommu t ative. K 5( • ⊗ ( • ⊕ • ⊕ • )) F or any obje ct s A, B , C, D the diagr am A ( B ⊕ ( C ⊕ D )) AB ⊕ A ( C ⊕ D ) AB ⊕ ( AC ⊕ AD ) A (( B ⊕ C ) ⊕ D ) A ( B ⊕ C ) ⊕ AD ( AB ⊕ AC ) ⊕ AD ❄ A ⊗ a ⊕ B,C,D ✲ v A,B,C ⊕ D ✲ AB ⊕ v A,C,D ❄ a ⊕ AB,AC,AD ✲ v A,B ⊕ C,D ✲ v A,B,C ⊕ AD is c ommu t ative. K 6( • ⊗ • ⊗ ( • ⊕ • )) F or any obje ct s A, B , C, D the diagr am A ( B ( C ⊕ D )) A ( B C ⊕ B D ) A ( B C ) ⊕ A ( B D ) ( AB )( C ⊕ D ) ( AB ) C ⊕ ( AB ) D ❄ a ⊗ A,B,C ⊕ D ✲ A ⊗ v B,C,D ✲ v A,BC,B D ❄ a ⊗ A,B,C ⊕ a ⊗ A,B,D ✲ v AB,C,D is c ommu t ative. K 7(( • ⊕ • ) ⊗ • ⊗ • ) Similar t o t he ab ove. K 8( • ⊗ ( • ⊕ • ) ⊗ • ) Similar t o t he ab ove. K 9(( • ⊕ • ) ⊗ ( • ⊕ • )) F or any obje cts A, B , C, D the dia gr am (( AC ⊕ B C ) ⊕ AD ) ⊕ B D ( AC ⊕ B C ) ⊕ ( AD ⊕ B D ) ( A ⊕ B ) C ⊕ ( A ⊕ B ) D ( A ⊕ B )( C ⊕ D ) ( AC ⊕ ( B C ⊕ AD )) ⊕ B D ( AC ⊕ ( AD ⊕ B C )) ⊕ B D (( AC ⊕ AD ) ⊕ B C ) ⊕ B D ( AC ⊕ AD ) ⊕ ( B C ⊕ B D ) A ( C ⊕ D ) ⊕ B ( C ⊕ D ) ❄ ❄ ❄ ❄ ✻ ❄ ✲ ✲ ✛ is c ommu t ative (the notation for arr ows have b e en omitte d, they ar e obvious). K 10 (0 ⊗ 0) The maps x 0 , y 0 : 0 ⊗ 0 → 0 c oincide. K 11 (0 ⊗ ( • ⊕ • )) F or any obje ct s A, B the diagr am 0 ⊗ ( A ⊕ B ) (0 ⊗ A ) ⊕ (0 ⊗ B ) 0 0 ⊕ 0 ❄ y A ⊕ B ✲ v 0 ,A,B ❄ y a ⊕ y B ✛ l ⊕ 0 = r ⊕ 0 is c ommu t ative. K 12 (( • ⊕ • ) ⊗ 0) Similar to the ab ove. K 13 (0 ⊗ 1) The maps y 1 , r ⊗ 0 : 0 ⊗ 1 → 0 c oincide . K 14 (1 ⊗ 0) Similar to the ab ove. K 15 (0 ⊗ • ⊗ • ) F or any obje cts A, B the dia gr am 0 ⊗ ( A ⊗ B ) (0 ⊗ A ) ⊗ B 0 0 ⊗ B ❄ y A ⊗ B ✲ a ⊗ 0 ,A,B ❄ y A ⊗ B ✛ y B is c ommu t ative. N. T. Quang, N. T . Thuy a nd C. T. K. P hung 5 K 16 ( • ⊗ 0 ⊗ • ) , ( • ⊗ • ⊗ 0) F or any obje cts A, B the diagr ams A ⊗ (0 ⊗ B ) ( A ⊗ 0) ⊗ B A ⊗ 0 0 0 ⊗ B ❄ A ⊗ y B ✲ a ⊗ A, 0 ,B ❄ x A ⊗ B ✲ x A ✛ y B A ⊗ ( B ⊗ 0) ( A ⊗ B ) ⊗ 0 A ⊗ 0 0 ❄ A ⊗ x B ✲ a ⊗ A,B, 0 ❄ x A ⊗ B ✲ x A ar e c ommutative. K 17 ( • (0 ⊕ • )) F or any obje cts A, B t he dia gr am A ⊗ (0 ⊕ B ) ( A ⊗ 0) ⊕ ( A ⊗ B ) A ⊗ B 0 ⊕ ( A ⊗ B ) ❄ A ⊗ l ⊕ B ✲ v A, 0 ,B ❄ x A ⊕ ( A ⊗ B ) ✛ l ⊕ A ⊗ B is c ommu t ative. K 18 ((0 ⊕ • ) ⊗ • ) , ( • ⊗ ( • ⊕ 0)) , (( • ⊕ 0) ⊗ • ) Similar to the ab ove. 3 The relatio n b et w een an Ann-category and a ring cat- egory In this section, we will prove that the axiomatics of a ring c ategory , without K10, can be deduced from the one of a n Ann-ca tegory . First, w e can see that, the functor morphisms a ⊕ , a ⊗ , u, l ⊕ , r ⊕ , v , w , in Definiton 2 are, resp ectively , the functor morphisms a + , a, c, g , d, L , R in Definition 1 . Isomo rphisms x A , y A coincide with isomor phis ms b L A , b R A referred in Pro p o - sition 1. W e no w pro ve that diagrams which commute in a ring category a lso do in an Ann- category . K1 obviously follows from (ii) in the definition of an Ann-category . The co mm utative dia grams K 2 , K 3 , K 4 , K 5 are indeed the compatibility of functor iso - morphisms ( L A , ˘ L A ) , ( R A , ˘ R A ) with the cons tr aints a + , c (the axiom Ann-1). The diag r ams K 5 − K 9 , res p ectively , are indeed the ones in (Ann-2 ). Particular ly , K9 is indeed the decompo s ition of (1.3) wher e the morphism v is repla ced b y its definition diagram: ( P ⊕ Q ) ⊕ ( R ⊕ S ) (( P ⊕ Q ) ⊕ R ) ⊕ S ( P ⊕ ( Q ⊕ R )) ⊕ S ( P ⊕ R ) ⊕ ( Q ⊕ S ) (( P ⊕ R ) ⊕ Q ) ⊕ S ( P ⊕ ( R ⊕ Q )) ⊕ S. ❄ v ✲ a + ✛ a + ⊕ S ❄ ( P ⊕ c ) ⊕ S ✲ a + ✛ a + ⊕ S The pro of for K17, K 18 Lemma 3.1. L et P, P ′ b e Gr-c ate gories, ( a + , (0 , g , d )) , ( a ′ + , (0 ′ , g ′ , d ′ )) b e r esp e ctive c on- str aints, and ( F, ˘ F ) : P → P ′ b e ⊕ -functor which is c omp atible with ( a + , a ′ + ) . Then ( F , ˘ F ) is c omp at ible with the unitivity c onstr aints (0 , g , d )) , (0 ′ , g ′ , d ′ )) . N. T. Quang, N. T . Thuy a nd C. T. K. P hung 6 First, the is omorphism b F : F 0 → 0 ′ is determined by the c o mp o sition u = F 0 ⊕ F 0 F ( 0 ⊕ 0 ) F 0 0 ′ ⊕ F 0 . ✛ e F ✲ F ( g ) ✛ g ′ Since F 0 is a reg ula r ob ject, there ex is ts uniquely the isomorphism b F : F 0 → 0 ′ such that b F ⊕ id F 0 = u. Then, we may prov e that b F satisfies the diagra ms in the definition of the compatibility of the ⊕ - functor F with the unitivity constraints. Prop ositi o n 1. In an Ann-c ate gory A , ther e exist uniquely isomorphisms ˆ L A : A ⊗ 0 − → 0 , ˆ R A : 0 ⊗ A − → 0 such that the fol lowing diagr ams AX A (0 ⊕ X ) AX A ( X ⊕ 0) 0 ⊕ AX A 0 ⊕ AX AX ⊕ 0 AX ⊕ A 0 ✛ L A ( g ) ❄ ˘ L A (2 . 1) ✛ L A ( d ) ❄ ˘ L A (2 . 1 ′ ) ✻ g ✛ ˆ L A ⊕ id ✻ d ✛ id ⊕ ˆ L A AX (0 ⊕ X ) A AX ( X ⊕ 0) A 0 ⊕ AX 0 A ⊕ X A AX ⊕ 0 X A ⊕ 0 A ✛ R A ( g ) ❄ ˘ R A (2 . 2) ✛ R A ( d ) ❄ ˘ R A (2 . 2 ′ ) ✻ g ✛ ˆ R A ⊕ id ✻ d ✛ id ⊕ ˆ R A c ommute, i.e., L A and R A ar e U- functors r esp e ct to t he op er ation ⊕ . Pr o of. Since ( L A , ˘ L A ) ar e ⊕ -functors which are compatible with the ass o ciativity constraint a ⊕ of the Picar d c a tegory ( A , ⊕ ) , it is also co mpatible with the unitivity constraint (0 , g , d ) thanks to Lemma 1. That means there exists uniquely the isomorphism ˆ L A satisfying the diagrams (2 . 1) and (2 . 1 ′ ). The pro o f for ˆ R A is similar. The dia g rams comm ute in Prop osition 1 are indeed K1 7, K18. The pro of for 15, K1 6 Lemma 3 . 2. L et ( F, ˘ F ) , ( G, ˘ G ) b e ⊕ -functors b et we en ⊕ -c ate gories C , C ′ which ar e c omp ati- ble with the c ons t r aints (0 , g , d ) , (0 ′ , g ′ , d ′ ) and e F : F (0) − → 0 ′ , e G : G (0) − → 0 ′ ar e r esp e ctive isomorphi sms. If α : F − → G in an ⊕ -morphism su ch that α 0 is an isomorphism, then the diagr am F 0 G 0 0 ′ ✲ α 0 ❅ ❅ ❘ ˆ F   ✠ ˆ G c ommutes. Pr o of. Le t us co ns ider the diagram F 0 F (0 ⊕ 0) G (0 ⊕ 0) G 0 0 ′ ⊕ F 0 F 0 ⊕ F 0 G 0 ⊕ G 0 0 ′ ⊕ G 0 u 0 id ⊕ u 0 F ( g ) u 0 ⊕ 0 G ( g ) ˘ F ⊕ id u 0 ⊕ u 0 ˘ G ⊕ id g ′ e F e G g ′ ✻ ❄ ❄ ❄ ✻ ✻ ✛ ✲ ✲ ✲ ✲ ✲ (I) (I I ) (I I I) (IV) (V) N. T. Quang, N. T . Thuy a nd C. T. K. P hung 7 In this dia gram, (I I) and (IV) comm ute tha nks to the compa tibility o f ⊕ -functors ( F, ˘ F ) , ( G, ˘ G ) with the unitivit y constraints; (II I) comm utes s ince u is a ⊕ -morphism; (V) commutes thank s to the na turality of g ′ . Therefore, (I) commutes, i.e ., ˘ G ◦ u 0 ⊕ u 0 = ˘ F ⊕ u 0 . Since F 0 is a regula r ob ject, ˘ G ◦ u 0 = ˘ F . Prop ositi o n 3.3 . F or any obje ct s X , Y ∈ ob A the diagr ams X ⊗ ( Y ⊗ 0) X ⊗ 0 0 ⊗ ( X ⊗ Y ) 0 ( X ⊗ Y ) ⊗ 0 0 (0 ⊗ X ) ⊗ Y 0 ⊗ Y ✲ id ⊗ b L Y ❄ a ❄ b L X (2 . 3) ✲ b R X Y ❄ a ✲ b L X Y ✲ b R X ⊗ id ✻ b R Y (2 . 3 ′ ) X ⊗ (0 ⊗ Y ) ( X ⊗ 0) ⊗ Y X ⊗ 0 0 0 ⊗ Y ✲ a ❄ id ⊗ ˆ R Y ❄ b L X ⊗ id (2 . 4) ✲ b L X ✛ b R Y c ommute. Pr o of. T o prove the first diagra m co mm utative, let us consider the diagr am X ⊗ ( Y ⊗ 0) X ⊗ 0 ( X ⊗ Y ) ⊗ 0 0 ✲ id ⊗ ˆ L Y ❄ a ◗ ◗ ◗ ◗ ◗ s \ L X ◦ L Y ❄ ˆ L X ✲ ˆ L X Y According to the axiom (1.1 ), ( a X,Y ,Z ) Z is an ⊕ -morphism from the functor L = L X ◦ L Y to the functor G = L X Y . Therefo r e, from Lemma 2, (I I) commutes. (I) comm utes thanks to the deter mination o f ˆ L of the comp os ition L = L ◦ L Y . So the o utside commutes. The second diagr am is prov ed similarly , thanks to the axiom (1.1’). T o prov e that the diagram (2.4) c ommut es, let us cons ider the diagram X ⊗ (0 ⊗ Y ) ( X ⊗ 0) ⊗ Y X ⊗ 0 0 0 ⊗ Y ✲ a ❄ id ⊗ d R Y ❅ ❅ ❅ ❅ ❘ ˆ H ❄ d L X ⊗ id     ✠ ˆ K ✲ d L X ✛ d R Y where H = L X ◦ R Y and K = R Y ◦ L X . Then (I I) and (I I I) commut e tha nks to the determination of the isomorphisms H and K . F rom the axiom (1.2), ( a X,Y ,Z ) Z is an ⊕ - morphism fro m the functor H to the functor K . So from Lemma 2, (I) c ommut es. T he r efore, the outside commut es. The dia grams in Pr op osition 2 a re indeed K15, K1 6 . Pro of for K11 Prop ositi o n 3.4 . In an Ann-c ate gory, the diagr am 0 ⊕ 0 0 (0 ⊗ X ) ⊕ (0 ⊗ Y ) 0 ⊗ ( X ⊕ Y ) ✲ g 0 = d 0 ✻ b R X ⊕ b R Y ✻ b R X Y (2 . 5) ✛ ˘ L 0 c ommutes. N. T. Quang, N. T . Thuy a nd C. T. K. P hung 8 Pr o of. Le t us co ns ider the diagram A ( B ⊕ C ) ⊕ 0( B ⊕ C ) A ( B ⊕ C ) ⊕ 0 ( AB ⊕ AC ) ⊕ (0 B ⊕ 0 C ) ( AB ⊕ AC ) ⊕ (0 ⊕ 0) ( AB ⊕ 0 B ) ⊕ ( AC ⊕ 0 C ) ( AB ⊕ 0) ⊕ ( AC ⊕ 0) ( A ⊕ 0) B ⊕ ( A ⊕ 0) C AB ⊕ AC ( A ⊕ 0)( B ⊕ C ) A ( B ⊕ C ) ˘ L A ⊕ ˘ L 0 v ˘ R B ⊕ ˘ R C ˘ L A ⊕ 0 ˘ L A ⊕ d − 1 0 v d AB ⊕ d AC ˘ L A ˘ R B ⊕ C d f ′ A ⊕ id ( id ⊕ id ) ⊕ ( b R B ⊕ b R C ) ( id ⊕ b R B ) ⊕ ( id ⊕ b R C ) ( d A ⊗ id ) ⊕ ( d A ⊗ id ) d A ⊗ id (I) (I I ) (I I I) (IV) (V) (VI) ✻ ❄ ❄ ❄ ✻ ❄ ✻ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✛ (2.6) In this diagr am, (V) co mm utes thank s to the axiom I(1.3), (I) comm utes tha nk s to the functorial prop erty of L ; the outside and (I I) co mm ute thank s to the co mpatibility of the functors R B ⊕ C , R B , R C with the unitivity c onstraint (0 , g , d ); (II I) co mmu tes thanks to the functorial pr o p erty v ; (VI) commutes thanks to the coher ence for the A CU-functor ( L A , ˘ L A ) . So (IV) c o mmut es. No te that A ( B ⊕ C ) is a regular ob ject resp ect to the op eration ⊕ , so the diag r am (2.5) commutes. W e hav e K11 . Similarly , we hav e K1 2. Pro of for K13, K14 Prop ositi o n 3.5 . In an Ann-c ate gory, we have b L 1 = l 0 , b R 1 = r 0 . Pr o of. W e will prove the first eq uation, the s econd o ne is pro ved similarly . Let us consider the diag ram (2.7). In this diagra m, the outside commutes thanks to the c o mpatibility of ⊕ -functor ( L 1 , ˘ L 1 ) with the unitivity constraint (0 , g , d ) res pe c t to the o pe ration ⊕ ; (I) commutes thanks to the functoria l prop erty of the is omorphism l ; (I I) c o mmut es thanks to the functorial prop erty of g ; (I I I) obviously commutes; (IV) commutes thanks to the axiom I(1.4). So (V) commutes, i.e., b L 1 ⊕ id 1 . 0 = l 0 ⊕ id 1 . 0 Since 1.0 is a regular ob ject r esp ect to the op er a tion ⊕ , b L 1 = l 0 . 0 ⊕ (1 . 0) (1 . 0) ⊕ (1 . 0) 0 ⊕ 0 0 ⊕ 0 0 ⊕ 0 0 1 . 0 1 . (0 ⊕ 0) ✛ ✛ ✻ ❄ ◗ ◗ ◗ s ✑ ✑ ✑ ✰    ✒ ❅ ❅ ❅ ■ ✲ ✻ ❄ ✛ b L 1 ⊕ id id ⊕ l 0 l 0 ⊕ l 0 ( V ) id g 1 . 0 ( I I ) g 0 ( I I I ) id ( I V ) ˘ L 1 (2 . 7) g 0 l 0 ( I ) l 0 ⊕ 0 L 1 ( g 0 ) = id ⊗ g 0 W e have K 14. Similarly , we hav e K1 3. N. T. Quang, N. T . Thuy a nd C. T. K. P hung 9 Definition 3.6. An Ann-c ate gory A is str ong if b L 0 = b R 0 . All the ab ov e r e s ults can b e stated as follows Prop ositi o n 3.7 . Each str ong Ann-c ate gory is a ring c ate gory. Remark 3.8. In our opinion, in the axiomatics of a ring c ate gory, t he c omp atibility of t he distributivity c onstr aint with t he unitivity c onstr aint (1 , l, r ) r esp e ct to the op er ation ⊗ is ne c ess ary, i.e., the diagr ams of (Ann-3) s hould b e adde d. Mor e over, if t he symmetric monoidal structu r e of the op er ation ⊕ is r eplac e d with the symmetric c ate goric al gr oup oid structure , then e ach ring c ate gory is an Ann-c ate gory. 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