On the Axiomatics of Ann-Categories

ON THE AXIOMA TICS OF ANN-CA TEGORIES Nguy en Tien Quang, D. D. Hanh and N. T. Th uy Octob er 25 , 2018 Abstract In this paper, we ha ve studied the axiomatics of Ann-c ate gories and c ate goric al rings. These are the categories with d istributivit y constrain ts whose axiomatics are similar with those of ring structures. The main result w e ha ve achiev ed is p ro ving the indep endence of the axiomatics of Ann-category d efi nition. And then we hav e prov ed th at after adding an axiom into the definition of ca tegorical rings, w e obtain th e new axioma tics whic h is equiva lent to t h e one of Ann-categories. 1 In tro duction The definition of Ann-c ate gories was present ed b y Nguyen Tien Q uang in 1987 [4], which was re garded as a ca teg oricalization o f ring struc tur es. Each Ann- category is c haracter ed by 3 inv ariants: the ring R = Π 0 ( A ) of classes of inv er tible ob jects of A , the R -bimo dule Π 1 ( A ) = Aut(0) and the element f ∈ H 3 ( R, M ) in the third co homology group of the ring R with coefficients in the R -bimo dule M due to [5]. Recently , we hav e prov ed tha t this cohomology coincides with the one due to Maclane[3]. The cla ss of regular Ann-categories (whose commutativit y constraints sa tisfy the condition c X,X = id ) is classifi- cated by the coho mology group H 3 M ( R, M ) of the Z -a lg ebra due to Shukla[8]. In [1], M.Jibladze and T. Pirashvili pr esen ted the definition of c ate goric al rings a s a slightly mo dified version of the definition of Ann-catego ries and clas- sificated them with Maclane coho mology for r ings. In this pa p er, w e hav e made some comments on these tw o definitions. Fir st, we hav e prov ed that in the ax iomatics of Ann-categories , the compatibilit y of the functors ( L A , e L A ), ( R A , e R A ) with the commutativit y co ns train t c is dep enden t. So we have pr o ved tha t each Ann-categ ory is a categorical r ing due t o [1]. W e ha ve s een that, in order to prove the con verse, w e m us t add an a xiom int o the definition of categorical rings, that is the compatibilit y of the functor s ( L A , e L A ) , ( R A , e R A ) with t he unitivit y constraint. In [6 ], [7], thanks to t his axiom, we ma y construct the asso ciative bimo dule structure . There is a problem here: Is the new a dded ax iom indep endent o f the o thers in the definition of categoric al rings due to [1]? 1 2 On the axiomatics o f Ann-categ ories F or conv enience, in this pap er w e denote by AB the tensor pro duct of the t wo ob jects A and B , instead of A ⊗ B . 2 The axiomatics of An n-categories Definition 2.1 . An Ann-c ate gory c onsists of: i) A gr oup oid A to gether with two bifunctors ⊕ , ⊗ : A × A − → A . ii) A fi xe d obje ct 0 ∈ A to gether with natu r ality c onstr aints a + , c, g, d such that ( A , ⊕ , a + , c, (0 , g, d )) is a Pic -c ate gory. iii) A fi xe d obje ct 1 ∈ A to gether with natur ality c onstr aints a, l , r such that ( A , ⊗ , a, (1 , l , r )) is a monoidal A -c ate gory. iv) N atur al isomorph isms 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 whic h ar e c omp atible with a + and c . (Ann-2) F or all 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) N. T. Quang, D. D. Hanh a nd N. T . Th uy 3 ( 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 ommut e, wher e v = v U,V ,Z,T : ( U ⊕ V ) ⊕ ( Z ⊕ T ) − → ( U ⊕ Z ) ⊕ ( V ⊕ T ) is the unique functor built fr om a + , c, id in the monoidal symmetric c ate gory ( A , ⊕ ) . (Ann-3) F or the unity ob je 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) ( X ⊕ Y )1 X 1 ⊕ Y 1 X ⊕ Y ✲ ˘ R 1 ◗ ◗ ◗ ◗ s r X ⊕ Y ✑ ✑ ✑ ✑ ✰ r X ⊕ r Y (1.4’) c ommut e. Remark. The commut ative diagra ms (1.1), (1.1’) and (1.2), resp ectively , 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 dia gram (1.3 ) shows that the family ( ˘ L Z X,Y ) Z = ( L − ,X,Y ) is a ⊕ -functor betw een the ⊕ -functors Z 7→ Z ( X ⊕ Y ) a nd Z 7→ Z X ⊕ Z Y , and the family ( ˘ R C A,B ) C = ( R A,B , − ) is a ⊕ - functor b e t ween the functors C 7→ ( A ⊕ B ) C and C 7→ AC ⊕ B C. The diagram (1.4) (resp. (1.4’)) shows that l (resp. r ) is a ⊕ -functor from L 1 (resp. R 1 ) to the unitivity functor of the ⊕ -catego ry A . Prop osition 1. In the A nn-c ate gory A ther e exist u niquely isomorphisms: b L A : A ⊗ 0 − → 0 , b R A : 0 ⊗ A − → 0 such that the fol lowing diagr ams: AX L A ( g ) ← − − − − A (0 ⊕ X ) g x     y ˘ L A 0 ⊕ AX b L A ⊕ id ← − − − − A 0 ⊕ AX (1 . 5) AX L A ( d ) ← − − − − A ( X ⊕ 0) d x     y ˘ L A AX ⊕ 0 id ⊕ b L A ← − − − − AX ⊕ A 0 (1.5’) 4 On the axiomatics o f Ann-categ ories AX R A ( g ) ← − − − − (0 ⊕ X ) A g x     y ˘ R A 0 ⊕ AX b R A ⊕ id ← − − − − 0 A ⊕ X A (1 . 6) AX R A ( d ) ← − − − − ( X ⊕ 0) A d x     y ˘ R A AX ⊕ 0 id ⊕ b R A ← − − − − X A ⊕ 0 A (1.6’) c ommut e. Me aningly, L A and R A ar e functors which ar e c omp atible wi th the un itivity c onstr aint of t he op er ation ⊕ . Pr o of. Since the pair ( L A , ˘ L A ) is a ⊕ -functor which is c o mpatible with the asso ciativity constraint a + of the Pic a rd categor y ( A , ⊕ ) , it is also compatible with the unitivit y constraint (0 , g , d ) thanks to P rop.0.4.4 [6]. That means there exists uniquely the iso morphism b L A satisfying the diagrams (1.5) a nd (1 .5 ’). The pro of for b R A is completely similar . 3 A r emark on the axiomatics of A nn-categories The co mm utativity constraint c plays a q uite sp ecial role in the study of cate- gories with tensor pro duct. F o r example, in 198 1, Kasangia n Stefano and Ross i F abio [17] pr e sen ted the problem o f the r e search on the relationship b etw een some conditions for comm utativity co nstraint in symmetric monoidal categories . W e now consider the axiomatics of Ann-categories in another view. In the definition of a ring a s well as a mo dule, the a xiom ab out the commutation of the addition can b e omitted: It can b e implied from the other axio ms. Consider the axiomatics of an Ann-category , we can determine the commutativit y constraint c based o n the constraints L , R and a + thanks to the commutativ e diag ram (1.3). It leads us to consider the indep endence or dependence of the ax io ms r e lated to the commutativit y co nstraint c. That is the compatibility of c, the co mpatibilit y of c with a + and the compatibility of the functors L A = A ⊗ − , R A = − ⊗ A with c . In this section, w e will prov e the indepe ndence of the last requir emen t. Prop osition 2. In the Ann-c ate gory A , the c omp atibility of the functors ( L A , ˘ L A ) , ( R A , ˘ R A ) with the c ommutativity c onst r aint c an b e de duc e d fr om the other ax- ioms r efer e d in Definition 1, without the c omp atibility of c and the c omp atibility of c with a + . Pr o of. First, w e prov e that the diagr am: X ( A 1 ⊕ B 1) L − − − − → X ( A 1) ⊕ X ( B 1) id X ⊗ c   y   y c X ( B 1 ⊕ A 1) L − − − − → X ( B 1) ⊕ X ( A 1) (2.1) commute in step b y step: 1. Consider the diagram (2.2), w e can see that: N. T. Quang, D. D. Hanh a nd N. T . Th uy 5 The regio ns (I), (IV) co mm ute thanks to the naturalit y of the iso morphism L , the reg ions (I I), (VI II), (IX) commute thanks to the axiom (1.1); the r egions (II I), (VI I) comm ute thanks to the axio m (1.2); the reg ions (V), (X) comm ute thanks to the axiom (1.3); the regio n (VI) commutes tha nks to the naturality of the isomorphism v . Ther efore, the outside region commutes. 2. Consider the diagram (2.3 ) in whic h the region (I I) is exactly the outside region of the dia gram (2.2) whose c o mm utation was prov ed right ab ov e. The religions (I) and (I I I) comm ute thanks to the coherence for ⊗ -functor ( L X , ˘ L X ); t ( L X , ˘ L X ); the regions (IV) a nd (V) co mm ute thanks to the axio m (1.3) and the definition of the isomo rphism v . Therefore, the outside re g ion co mm utes. 3. Now w e co nsider the dia g ram (2.4) whos e outside regio n is the one of the diagram (2.3). In this diagr am, the r eligions (I), (I I) commute thanks to the naturality of L , so the r egion (II I) co mm utes. Hence, from the re g ular prope r t y of the ob ject X ( A 1) and X ( B 1) for the addition ⊕ , we c a n deduce that the diagram (2.1) commutes. 4. Finally , we prove that the diagr am X ( A ⊕ B ) L − − − − → X A ⊕ X B id ⊗ c   y c   y X ( B ⊕ A ) L − − − − → X B ⊕ X A commutes by em b edding it into the dia gram: X ( A 1 ⊕ B 1) X ( A 1) ⊕ X ( B 1 ) X ( A ⊕ B ) X A ⊕ X B X ( B ⊕ A ) X B ⊕ X A X ( B 1 ⊕ A 1) X ( B 1) ⊕ X ( A 1) ◗ ◗ ◗ ◗ ◗ ◗ s id ⊗ ( r ⊕ r ) (I) ❄ id ⊗ c (IV) ✲ L ✑ ✑ ✑ ✑ ✑ ✑ ✰ id ⊗ r ⊕ id ⊗ r ❄ c ❄ id ⊗ c (I I) ✲ L ❄ c (V) ✲ L ✑ ✑ ✑ ✑ ✑ ✑ ✸ id ⊗ ( r ⊕ r ) (I I I) ✲ L ◗ ◗ ◗ ◗ ◗ ◗ ❦ id ⊗ r ⊕ id ⊗ r In this diagram, the o utside region is exac tly the one of the comm utative dia- gram (2.1), the regio ns (I) and (I II) co mm ute thanks to the naturality of the isomorphism L ; the regions (IV) and (V) comm ute thanks to the naturality of the isomorphis m c . So the r egion (II) c omm utes. 6 On the axiomatics o f Ann-categ ories Because o f the symmetr y , we can deduce the co mpatibilit y of the functor ( R A , ˘ R A ) with c . This completes the pr oo f. X ( A 1 ⊕ A 1) ⊕ X ( B 1 ⊕ B 1) ✏ ✏ ✏ ✏ ✏ ✏ ✶ L ⊕ L X (( A 1 ⊕ A 1) ⊕ ( B 1 ⊕ B 1)) P P P P P P q L ( X ( A 1) ⊕ X ( A 1)) ⊕ ( X ( B 1) ⊕ X ( B 1)) ❅ ❅ ■ ( a ⊕ a ) ⊕ ( a ⊕ a ) (( X A )1 ⊕ ( X A )1) ⊕ (( X B )1 ⊕ ( X B )1) ( X A )(1 ⊕ 1) ⊕ ( X B )(1 ⊕ 1) ❅ ❅ ❘ L ⊕ L ( X ( A (1 ⊕ 1)) ⊕ ( X ( B (1 ⊕ 1)) ❍ ❍ ❍ ❍ ❥ a ⊕ a ❄ ( id ⊗ L ) ⊕ ( id ⊗ L ) ( X A ⊕ X B )(1 ⊕ 1) ❆ ❆ ❆ ❆ ❆ ❯ R ✻ L X ( A (1 ⊕ 1) ⊕ B (1 ⊕ 1)) ❄ id ⊗ ( L ⊕ L ) P P P P P P q L ( X ( A ⊕ B ))(1 ⊕ 1)) ❍ ❍ ❍ ❍ ❥ L ⊗ id ✻ L (( X A )1 ⊕ ( X B )1) ⊕ (( X A )1 ⊕ ( X B )1) ❄ v X (( A ⊕ B )(1 ⊕ 1)) ❍ ❍ ❍ ❍ ❥ a ❄ id ⊗ R ✻ id ⊗ L ( X A ⊕ X B )1 ⊕ ( X A ⊕ X B )1 ◗ ◗ ◗ ◗ s R ⊕ R ( X ( A 1) ⊕ X ( B 1)) ⊕ ( X ( A 1) ⊕ X ( B 1))      ✠ ( a ⊕ a ) ⊕ ( a ⊕ a ) ❄ v ( X ( A ⊕ B ))1 ⊕ ( X ( A ) ⊕ B ))1 ❅ ❅ ❘ ( L ⊗ id ) ⊕ ( L ⊗ id ) ( X ⊕ X )( A 1) ⊕ ( X ⊕ X )( B 1) ❄ R ⊕ R ( X ⊕ X )( A 1 ⊕ B 1) ❄ L ✻ R X (( A ⊕ B )1) ⊕ X (( A ⊕ B )1) ✟ ✟ ✟ ✟ ✯ ( id ⊗ R ) ⊕ ( id ⊗ R ) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ☛ a ⊕ a X (( A ⊕ B )1 ⊕ ( A ⊕ B )1) ❍ ❍ ❍ ❍ ❍ ❥ L ✟ ✟ ✟ ✟ ✟ ✯ id ⊗ ( R ⊕ R ) X ( A 1 ⊕ B 1) ⊕ X ( A 1 ⊕ B 1) ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ L ⊕ L X (( A 1 ⊕ B 1) ⊕ ( A 1 ⊕ B 1)) ❍ ❍ ❍ ❍ ❍ ❥ L (I) (I I ) (I I I) (IV) (V) (VI) (VI I) (VI II) (IX) (X) (2.2) N. T. Quang, D. D. Hanh a nd N. T . Th uy 7 X (( A ⊕ B )(1 ⊕ 1)) ( X ⊕ X )( A 1 ⊕ B 1) X (( A ⊕ B )1 ⊕ ( A ⊕ B )1) X ( A 1 ⊕ B 1) ⊕ X ( A 1 ⊕ B 1) X (( A 1 ⊕ B 1) ⊕ ( A 1 ⊕ B 1)) ( X ( A 1) ⊕ X ( B 1)) ⊕ ( X ( A 1) ⊕ X ( B 1)) X ((( A 1 ⊕ B 1) ⊕ A 1) ⊕ B 1) (( X ( A 1) ⊕ X ( B 1) ⊕ X ( A 1)) ⊕ X ( B 1) X (( A 1 ⊕ ( B 1 ⊕ A 1)) ⊕ B 1) X ( A 1 ⊕ ( B 1 ⊕ A 1)) ⊕ X ( B 1) ( X ( A 1) ⊕ X ( B 1 ⊕ A 1)) ⊕ X ( B 1) ( X ( A 1) ⊕ ( X ( B 1) ⊕ X ( A 1))) ⊕ X ( B 1) X ( A (1 ⊕ 1) ⊕ B (1 ⊕ 1)) ( X ⊕ X )( A 1) ⊕ ( X ⊕ X )( B 1) ( X ( A 1) ⊕ X ( A 1)) ⊕ ( X ( B 1) ⊕ X ( B 1)) X (( A 1 ⊕ A 1) ⊕ ( B 1 ⊕ B 1)) X ( A 1 ⊕ A 1) ⊕ X ( B 1 ⊕ B 1) X ((( A 1 ⊕ A 1) ⊕ B 1) ⊕ B 1) (( X ( A 1) ⊕ X ( A 1)) ⊕ X ( B 1)) ⊕ X ( B 1) X (( A 1 ⊕ ( A 1 ⊕ B 1)) ⊕ B 1) X ( A 1 ⊕ ( A 1 ⊕ B 1)) ⊕ X ( B 1) ( X ( A 1) ⊕ X ( A 1 ⊕ B 1)) ⊕ X ( B 1) ( X ( A 1) ⊕ ( X ( A 1) ⊕ X ( B 1))) ⊕ X ( B 1) ❅ ❅ ❅ ❅ ❘ ✲     ✒ ✻ ✻ ❄ ❄ ❄ ❄ ❄ ❄ ✻ ✻ ✻ ✻ ✻ ✻ ❄ ❄     ✒ ✲ ❅ ❅ ❅ ❅ ❘   ✒ ❅ ❅ ❘ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ✲ ✛ L L ⊕ id ( id ⊕ L ) ⊕ id id ⊗ ( a ⊕ id ) a ⊕ id id ⊗ a a id ⊗ ( R ⊕ R ) L ⊕ L id ⊗ L R id ⊗ R L id ⊗ ( L ⊕ L ) R ⊕ R id ⊗ a a id ⊗ ( a ⊕ id ) a ⊕ id L L ⊕ L L L ⊕ id ( id ⊕ L ) ⊕ id id ⊗ (( id ⊕ c ) ⊕ id ) ( id ⊕ c ) ⊕ id L (I) (I I ) (I I I) (IV) (V) (2.3) 8 On the axiomatics o f Ann-categ ories X (( A 1 ⊕ ( B 1 ⊕ A 1)) ⊕ B 1) X (( A 1 ⊕ ( A 1 ⊕ B 1)) ⊕ B 1) X ( A 1 ⊕ ( B 1 ⊕ A 1)) ⊕ X ( B 1) X ( A 1 ⊕ ( A 1 ⊕ B 1 )) ⊕ X ( B 1) ( X ( A 1) ⊕ X ( B 1 ⊕ A 1)) ⊕ X ( B 1) ( X ( A 1) ⊕ X ( A 1 ⊕ B 1)) ⊕ X ( B 1 ) ( X ( A 1) ⊕ ( X ( B 1) ⊕ X ( A 1))) ⊕ X ( B 1) ( X ( A 1) ⊕ ( X ( A 1) ⊕ X ( B 1))) ⊕ X ( B 1) ❄ L (I) ✲ id ⊗ (( id ⊕ c ) ⊕ id ) ❄ L ❄ id (I I ) ✲ ( id ⊗ ( id ⊕ c )) ⊕ id ❄ L⊕ id ❄ ( id ⊕L ) ⊕ id (I I I) ✲ ( id ⊕ ( id ⊗ c )) ⊕ id ❄ ( id ⊕L ) ⊕ id ✲ ( id ⊕ c ) ⊕ id (2.4) 4 Categorical rings and the r elat ionship with Ann-categories In [1], the a uthors presented the definition of categor ical ring s by mo difying some axioms of the definition o f Ann-ca tegories. Let us reca ll this definition. Definition 4 .1. A c ate goric al ring is a symmetric c ate goric al gr oup R to gether with a bifunctor R × R − → R (denote d by mu ltiplic ation), an obje ct 1 ∈ R , and natur al isomorphisms: a r,s,t : ( rs ) t − → r ( st ) (asso ciative law), λ r : 1 r − → r ; ρ r : r 1 − → r (left and right unit), λ r,x,y : r ( x + y ) − → rx + ry (left di stribut ive law), ρ x,y ,s : ( x + y ) s − → xs + y s (right distributive law) such that ( R , a, (1 , λ, ρ )) is a monoidal c ate gory making the diagr ams (1.1), N. T. Quang, D. D. Hanh a nd N. T . Th uy 9 (1.1’), (1.2),(1.3), (1.4), (1.4’) and: r ( x + y ) + r ( z + t ) r (( x + y ) + ( z + t )) ( rx + r y ) + ( r z + rt ) r (( x + z ) + ( y + t )) ( rx + r z ) + ( ry + rt ) r ( x + z ) + r ( y + t ) ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ λ r,x,y + λ r,z,t ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ λ r,x + y,z + t ❄ id ⊗ v x,y,z ,t ❄ v rx,r y,rz ,rt ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ λ r,x + z ,y + t ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ λ r,x,z + λ r,y,t (3.1) ( x + y ) s + ( z + t ) s (( x + y ) + ( z + t )) s ( xs + y s ) + ( z s + ts ) (( x + z ) + ( y + t )) s ( xs + z s ) + ( y s + ts ) ( x + z ) s + ( y + t ) s ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ρ x,y,s + ρ z ,t,s ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ ρ x + y,z + t,s ❄ v x,y,z ,t ⊗ id ❄ v xs,ys,z s,ts ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ρ x + z ,y + t,s ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ ρ x,z ,s + ρ y,t,s (3.1’) c ommut e. The main re s ult o f this pap e r is the relationship of Ann-catego ries and cat- egorica l rings. First, we hav e the following theorem. Theorem 3. Each A nn- c ate gory is a c ate goric al ring. Pr o of. Assume that ( A , ⊕ , ⊗ ) is an Ann-categ ory . W e only need prov e the commutation of the diag r ams (3.1 ) and (3.1’). It can b e deduced from the coherence theor em in an Ann-c a tegory [4]. How ever, we may present a direct pro of as follows. Consider the diagra m: 10 On the axiomatics o f Ann-categ ories ( rx + r y ) + ( r z + rt ) r ( x + y ) + ( rz + r t ) r [( x + y ) + ( z + t )] r [(( x + y ) + z ) + t ] r (( x + y ) + z ) + r t ( r ( x + y ) + rz ) + rt (( rx + r y ) + rz ) + rt r ( x + y ) + r ( z + t ) r [( x + ( y + z )) + t ] r ( x + ( y + z )) + r t ( rx + r ( y + z )) + r t ( rx + ( r y + rz )) + rt r ( x + z ) + r ( y + t ) r [( x + ( z + y )) + t ] r ( x + ( z + y )) + r t ( rx + r ( z + y )) + r t ( rx + ( r z + ry )) + rt ( rx + r z ) + ( ry + rt ) r ( x + z ) + ( r y + rt ) r [( x + z ) + ( y + t )] r [(( x + z ) + y ) + t ] r (( x + z ) + y ) + rt ( r ( x + z ) + r y ) + rt (( rx + r z ) + ry ) + rt ✲ v ◗ ◗ ◗ ◗ ◗ ❦ λ + i d ✑ ✑ ✑ ✑ ✑ ✸ λ + λ ◗ ◗ ◗ ◗ ◗ ❦ λ + λ ✑ ✑ ✑ ✑ ✑ ✸ λ + i d ✛ id + λ ✲ id + λ ✑ ✑ ✑ ✑ ✑ ✸ λ ◗ ◗ ◗ ◗ ◗ ❦ λ ✲ r ⊗ v ✑ ✑ ✑ ✑ ✑ ✸ r ⊗ a + ◗ ◗ ◗ ◗ ◗ ❦ r ⊗ a + ✲ r . ( a + + id ) ✲ r . (( id + c ) + i d ) ✛ r . ( a + + id ) ❄ λ ❄ λ ❄ λ ❄ λ ✲ r .a + + id ✲ r . ( id + c ) + id ✛ r .a + + id ❄ λ + id ❄ λ + i d ❄ λ + i d ❄ λ + i d ✲ ( id + rc ) + id ❄ ( λ + i d ) + id ❄ ( id + λ ) + id ❄ ( id + λ ) + id ❄ ( λ + i d ) + id ✲ a + + id ✲ ( id + c ) + id ✛ a + + id ✲ a + ✲ a + ✛ a + ✛ a + (I) (II) (I II) (IV) (V) (VI) (VII) (VII I) (IX) (X) (XI) (XII) (XII I) (XIV) (XV) N. T. Quang, D. D. Hanh a nd N. T . Th uy 11 In the ab ov e diagram, the regio ns (I) a nd (V) commut e thanks to the nat- urality of a + , the regions (II) and (IV) commute thanks to the comp osition of functors, the regions (VI), (VI I), (XI I) and (XIV) co mm ute thanks to the compatibility of the functor ( L a , ˘ L a ) with the aso ciativity constr ain t a + , the regions (IX), (X), (XI), (XII I) commute thanks to the naturality of the func- tor λ , the regio n (XV) commutes tha nk s to the compatibility of the functors ( L a , ˘ L a ) with the commut ativity constraint c , the region (XI I I ) and the outside region comm ute thanks to the determination of the functor v in the s ymmetric monoidal catego r y ( A , ⊕ ). Therefore, the r egion (I I I) co mm utes, that means the diagra m (3.1) commutes. With a similar pro of, the diagra m (3.1’) comm utes. Now, to pr ove the conv erse, we need add the following axiom into the defi- nition of categor ical rings (U) F or e ach obje ct a ∈ R , t he p airs ( L a , ˘ L a ) ,and ( R a , ˘ R a ) define d by 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 the unitivity c onstra int (0 , g , d ) of the op er ation ⊕ . That me ans ther e exist isomorph isms c L A : A ⊗ O − → O, c R A : O ⊗ A − → O, such that the diagr ams (1.5), (1.5’), (1.6), (1.6’) c ommute. With this addition, we have the fol lowi ng the or em Theorem 4. Each c ate goric al ring s atisfying the c ondition (U) is an Ann- c ate gory. Pr o of. Assume that A is a categor ical ring sa tisfying the condition (U). W e m ust show that A satisfies the axio m (Ann-1) o f an Ann-categor y . Accor ding to Prop osition 2, it remains to show that the functor ( L a , ˘ L a ) is compatible with the a sso c ia tivit y constra in t a + , i.e, the commutation of the following diagr ams: x ( a + ( b + c )) xa + x ( b + c ) x a + ( xb + xc ) x ( a + b ) + c ) x ( a + b ) + xc ( x a + xb ) + xc ✲ λ ❄ id ⊗ a ✲ id ⊕ λ ❄ a ✲ λ ✲ λ ⊕ id (3.2) and a similar diagra m for the pa ir ( R A , ˘ R A ) , for each A. First, since ( L, ˘ L A ) is compatible with the constra in t (0 , g , d ), there ex ists a functor b L A , suc h that these dia g rams: 12 On the axiomatics o f Ann-categ ories A (0 + X ) A 0 + AX AX 0 + AX ✲ ❄ ❄ ✲ A ( X + 0) AX + A 0 AX AX + 0 ✲ ❄ ❄ ✲ commute. In order to prov e the diagram (3.2), let’s consider the diagram ( X A + X C ) + ( X 0 + X D ) ( X A + X 0) + ( X C + X D ) ( X A + X C ) + (0 + X D ) ( X A + 0) + ( X C + X D ) ( X A + X C ) + X D X A + ( X C + X D ) X ( A + C ) + X (0 + D ) X ( A + 0) + X ( C + D ) X ( A + C ) + X D X A + X ( C + D ) X (( A + C ) + D ) X ( A + ( C + D )) X (( A + C ) + (0 + D )) X (( A + 0) + ( C + D )) ✲    ✒ ❅ ❅ ❅ ■ ✲ ❄ ❄ ❄ ❄ ❅ ❅ ❅ ❘    ✠ ❄ ❄ ❄ ❄ ❄ ❄    ✒ ❅ ❅ ❅ ■ ❅ ❅ ❅ ❘    ✠ ✲ ✲ ✲ v id + ( b L + id ) ( id + b L ) + id v ρ + ρ id + g d + id ρ + ρ a + ρ + id ⊗ g id ⊗ d + ρ ρ + id id + ρ id + id ⊗ g id ⊗ d + id ρ ρ ρ ρ id ⊗ a + id ⊗ ( id + g ) id ⊗ ( d + id ) id ⊗ v (I) (I I ) (I I I) (IV) (V) (VI) (VI I) (VI II) (IX) (X) In the ab ov e diag r am, the outside reg ion commutes thanks to the h yp othesis (1.5); the regions (I) and (VII I) commute thanks to the functorica l prop erty of ρ , the reg ions (II ) and (IX) commute thanks to the co mp osition, the regions (II I) and (X) commute thanks to the definition of b L ; the regions (IV) and (VI) commute thanks to the coherence theore m in a symmetric monoida l ca tegory , the region (VI I) comm utes thanks to the functorical proper ty o f v . Ther efore, the region (V) commutes. In other words, the dia gram (3.2) commutes. The compatibility of the functor ( R A , ˘ R A ) with the ass o ciativity co nstraint a + can be prov ed similarly . N. T. Quang, D. D. Hanh a nd N. T . Th uy 13 References [1] M. Jibladze and T. Pir ashvili, Third Mac Lane co homology via categorica l rings, arxiv. math. KT/0608 519 v1, 21 Aug 2006. [2] S. Kasangian and F. R ossi, Some remar ks on symmetry for a mono dial category , Bul l. A ustr al Math. So c. 23, No2 (1981), 209-214. [3] S. Mac L ane, Homolo gie des anneaux et des mo dules, Col lque de T op olo gie algebrique. L ouvain (195 6), 55-80. [4] N. T. Q uang, Intro ductio n to Ann-categories , J . Math. Hanoi, No.15, 4 (1987), 14-24 . [5] N. T. Quang, Coherence for Ann-categories , J. Math. Hanoi, No.16, 1(1988), 17 -26. [6] N. T. Q uang, Ann-categ ories, D o ctor al dissertation, Hanoi, Vietnamese, 1988. [7] N. T. Quang, Structure of Ann-catego ries and Mac Lane-Shukla co homol- ogy of rings, (Russian) Ab elian gr oups and mo dues, No. 11,1 2, T omsk. Gos. Univ., T omsk (1994), 166 -183. [8] U.Shukla, Cohomologie de s algebras asso ciatives. A nn.Sci.Ec ole Norm.,Sup.,7 (1961 ), 163-209.