The Coherence Theorem for Ann-Categories

THE COHERENCE THEOREM F OR ANN-CA TEGORIES Nguy en Tien Quang No v ember 18, 2018 abstract This pap er 1 presents the proof of th e coherence theorem for An n -categories whose set of axioms and original basic prop erties were giv en in [9]. Let A = ( A , A , c, (0 , g , d ) , a, (1 , l, r ) , L , R ) b e an A nn-category . The coherence theorem states that in the category A , any morphism b uilt from the ab ov e isomorphisms and the identification b y composition and th e tw o op erations ⊗ , ⊕ only dep ends on its source and its target. The first coherence th eorems w ere built for monoidal and symmetric monoidal categorie s by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. F o r Ann-categories, applying Hoang Xuan Sinh’s ideas used fo r Gr-categories in [2], the proof of t he coherence theorem is constructed by faithfully “embedding” each arbitrary Ann-category into a quite strict Ann -category . Here, a qui te strict An n -categogy is an A n n-category whose all constraints are strict, except for the commutativit y and left distribu tivity ones. This p aper is the w ork contin uin g from [9]. If there is no ex planation, t he terminologies and notations in this pap er mean as in [9]. ˇ 0.5cm 1 Canonical isomorphisms In this section, we define some canonical iso morphisms induced b y iso morphisms c, L , b L A and the ident ifica tion, laws ⊗ , ⊕ on the quite strict Ann-catego r y A . Let I b e a fully ordered limited set. If I 6 = ∅ and α is the maximal of I , w e will denote I ′ = I \ { α } ; and the notation | I | refers to the cardinal o f I . Definition 1.1. [1] The c anonic al sum P I A i wher e A i ∈ O b A , i ∈ I is define d inductively as fol lows 1. P I A i = 0 if I = ∅ and P I A i = A α if I = { α } . 2. P I A i = ( P I ′ A i ⊕ A α ) if | I | > 1 . Definition 1 . 2. We define the isomorphi sm ν P A i , P B i : ( X I A i ) ⊕ ( X I B i ) − → X I ( A i ⊕ B i ) , (which is abbr eviate d by ν I ) by induction on | I | as fol lows 1. ν I = id if | I | ≤ 1 . 2. if | I | > 2 , ν I is define d by the fol l lowing c ommutative diagr am 1 This pap er has b een pub l ished (in Vietnamese) in Vietnam Journal of Mathemat i cs V ol. XVI, No 1, 1988. 1 2 The coherence theorem for Ann-categorie s ( P I A i ) ⊕ ( P I B i ) id − − − − → ( P I ′ A i ) ⊕ A α ⊕ ( P I ′ B i ) ⊕ B α ν I   y   y ν P I ( A i ⊕ B i ) ν I ′ ⊕ id A α ⊕ B α − − − − − − − − − → ( P I ′ A i ) ⊕ ( P I ′ B i ) ⊕ A α ⊕ B α wher e ν = id ⊕ c ⊕ id . We c an se e that the isomorphism ν I is built only fr om isomorph isms c, id by law ⊕ . Mor e over, the isomorphism ν I is n atur al. Definition 1 . 3. We define fol lowing isomorphisms u I ,J : X I X J ( A i ⊗ B j ) − → X J X I ( A i ⊗ B j ) by induction on | I | as fol lows 1. u I ,J = id if | I | ≤ 1 or J = ∅ . 2. u I ,J = ν J ( u I ′ ,J ⊕ id ) if | I | > 1 . So, isomorphisms u I ,J ar e also built fr om the isomorphisms c, i d by law ⊕ and these morphisms ar e functorial. Definition 1 . 4. We define fol lowing isomorphisms F I ,J : ( X I A i ) ⊗ ( X J B j ) − → X I X J ( A i ⊗ B j ) = X I × J ( A i ⊗ B j ) wher e I × J is o der e d alphab etic al ly as fol lows 1. F I ,J = id : 0 ⊗ ( P J B i ) → 0 if I = ∅ . (sinc e R = i d we have b R A = id for al l A ∈ Ob A ) 2. F I ,J = b L X : X ⊗ 0 → 0 wher e X = P I A i if J = ∅ . 3. F I ,J = P I f A i if I 6 = ∅ and J 6 = ∅ , wher e f A : A ⊗ ( X J B j ) → X J A ⊗ B j is define d as fol lows: I f | J | = 1 , f A = i d ; wher e as f A is define d by induction on | J | by the fol lowing c ommu tative diagr am A ⊗ ( P J B j ) f A − − − − → ( P J ( A ⊗ B j ) ˘ L A   y    A ⊗ ( P J B j ) ⊕ ( A ⊗ B β ) f ′ A ⊗ id − − − − → ( P J ′ A ⊗ B j ) ⊕ ( A ⊗ B β ) wher e β is the maximal element of J and J ′ = J \ { β } . Definition 1 . 5. We define fol lowing isomorphisms K I ,J : ( X I A i ) ⊗ ( X J B j ) − → X J X I ( A i ⊗ B j ) as fol lows 1. K I ,J = id if I = ∅ . 2. K I ,J = b L X , wher e X = P I A i if J = ∅ . 3. K I ,J = f X , wher e X = P I A i in other c ases. Nguyen Tien Quang 3 Then, we hav e the following pr op osition immediately Prop ositio n 1.6. With c anonic al su ms P I A i , P J A j , we have the r elation K I ,J = u I ,J · F I ,J . Applying this pr op osition we can pr ov e Prop ositio n 1.7. Assume J 1 , J 2 b e non-empty subsets of J such that J = J 1 ` J 2 and j 1 < j 2 if j 1 ∈ J 1 , j 2 ∈ J 2 . Then for sums A = P I A i , P J B j , P J 1 B j , P J 2 B j we have fol lowing r elations F I ,J = ν I · ( F I ,J 1 ⊕ F I ,J 2 ) · ˘ L A F J,I = F J 1 ,I ⊕ F J 2 ,I . W e will g ive the pro of of this propos itio n in detail to illustrate the pro of using comm utative diagrams. Herea fter, for co nvenience, w e write AB instead of A ⊗ B for all A, B ∈ O bs ( A ) . Prop ositio n 1.8. In the A n n-c ate gory A , the fol lowing diagr ams ( P I A i ) ⊗ ( P J B j ) ⊗ ( P T C t ) F I ,J ⊗ id − − − − − → ( P I × J A i ⊗ B j ) ⊗ ( P T C t ) id ⊗ F J,T   y   y F I × J,T ( P I A i ) ⊗ ( P J × T B j ⊗ C t ) − − − − − → F I ,J × T P I × J × T A i ⊗ B j ⊗ C t c ommu te. Pro of. 1. In case I = ∅ , we ha ve the prop osition prov ed since the diagram (1.1) b ecomes the following one 0( P J B j )( P T C t ) id ⊗ id − − − − → 0 ( P T C t ) id ⊗ F J,T   y   y id 0( P J × T B j ⊗ C t ) − − − − → id 0 whose commutativit y follows from b R X = i d and the prop erty of the zer o ob ject (see Prop.3 .2 [9]). In case J = ∅ or T = ∅ the pro po s ition is pro ved similar ly . Hence, we now can s uppos e that I , J , T are all not empty . 2. In cas e | I | = 1. Firstly , co nsider the case in which | J | = 1 . W e prove the pr op osition by induction on | T | . Then If | T | = 1 , the pro of is obvious. If | T | = 2 , the diagra m commutes thanks to the axiom (1.1) of Ann-catego ries (see [9]). If | T | > 2 , consider the diagra m (1.2). 4 The coherence theorem for Ann-categorie s A ( P T ′ B C t ⊕ B C γ ) A ( P T ′ B C t ) ⊕ AB C γ A ( P T ′ B C t ⊕ B C γ ) P T ′ AB C t ⊕ AB C γ AB ( P T C t ) P T ′ AB C t ⊕ AB C γ AB ( P T C t ) AB ( P T ′ C t ) ⊕ AB C γ A ( B P T ′ C t ⊕ B C γ ) AB ( P T C t ) ⊕ AB C γ id ⊗ ( f ′ B ⊕ id ) id id ⊗ f B id id ⊗ ˘ L B ˘ L A f A f AB ˘ L AB ˘ L A f ′ A ⊕ id id f AB ⊕ id id ( id ⊗ f ′ B ) ⊕ id ✻ ❄ ❄ ❄ ✻ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ (2) In this diag ram, the reg ion (I) commutes thanks to the axio m (1.1) in [9]; regio ns (II), (IV), (V) commute thanks to definitions of iso morphisms f AB , f A , f B ; (VI) c o mm ute thanks to the inductive suppo sition; the parameter co mm utes since ˘ L A is a functoria l iso morphism. Therefore, the region (II I) commu tes . This co mpletes the pr o o f. After that, still with the condition | I | = 1 , we can prov e the propo sition with | J | > 1 by inductio n on | J | . 3. Now if | I | > 1 , consider the diagr am (1.3). In this diag ram, the reg ion (I) c omm ute since R = id is a functoria l isomorphism; the region (I I) commutes thanks to the inductive s uppos ition for the first co mpo nen t of sums, for the second comp o nent since the case | I | = 1 has just b een aprov ed above; the reg ion (II I) co mm utes thanks to the prop erty o f the isomorphism F I ,J (see the Prop.1.7 ); reg ions (IV), (V) commute thanks to definitions o f F I ,J and F I ,J × T . So the pa rameter commutes. This completes the pro of. Nguyen Tien Quang 5 ( P I ′ × J A i B j ⊕ P J A α B j )( P T C t ) P I × J × T A i B j C t ( P I ′ × J A i B j )( P T C t ) ⊕ ( P J A α B j )( P T C t ) P I ′ × J × T A i B j C t ⊕ P J × T A α B j C t ( P I A i )( P J B j )( P T C t ) ⊕ A α ( P J B j )( P T C t ) ( P I ′ A i )( P J × T B j C t ) ⊕ A α ( P J × T B j C t ) ( P I A i )( P J B j )( P T C t ) ( P I A i )( P J × T B j C t ) F I × J,T id ( F I ′ ,J ⊗ id ) ⊕ ( F A α ⊗ id ) id F I ,J ⊗ id F I ′ × J,T ⊕ F J,T ( id ⊗ F J,T ) ⊕ ( id ⊗ F J,T ) id ⊗ F J,T id F I ′ ,J × T ⊕ f A α id F I ,J × T ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲ 6 The coherence theorem for Ann-categorie s 2 The coherence theorem for Ann-categories Let A be a quite s trict Ann-categ ory . Ass ume X s , s ∈ Ω be a non-empty , limited family of ob jects o f A and Y b e an expressio n of the family X s with op era tions ⊗ and ⊕ . With distributivit y constraints L , R = id, induced is omorphisms b L A and is o morphisms A = id , g , d = id , a = id , l , r = id , we ca n write Y as a s um o f monomials of o b jects of X s by using the following isomorphism h : Y → X I A i where A i 6 = 0 for all i if I 6 = ∅ and h is built from the identification, and isomo rphisms L , b L A . A such a pair ( h, P I A i ) is called an exp ansion form o f Y . W e now define a c anonic al exp ansion form of Y by induction on its length, where the length of expansion form Y is the total num b er of times of app ear ances of ob jects A i in Y . It is easily to see that any Y whose length is mor e than 1 can be written in the form of U ⊗ V or U ⊕ V . That implies Definition 2 . 1. The c anonic al exp ansion form h : Y → X I A i of Y is define d as fol lows 1. If Y = P I X i , h = i d : P I X i → P I ′ X i , wher e I is a subset of Ω ; wher e as I ′ is t he set of indexes i su ch that X i 6 = 0 . 2. If Y = U ⊗ V , the isomorphism h is the c omp osition Y = U ⊗ V u ⊗ v − − − − → ( P J B j ) ⊗ ( P T C t ) F J,T − − − − → P J × T B j C t wher e ( u, P J B j ) , ( v , P T C t ) ar e, r esp e ctively, c anonic al exp ansion forms of U , V and define d by the induction su pp osition. 3. If Y = U 1 ⊕ U 2 , h is the c omp osition Y = U 1 ⊕ U 2 u 1 ⊕ u 2 − − − − → ( P I 1 B i ) ⊗ ( P I 2 B i ) id − − − − → P I ′ B i wher e u 1 , u 2 ar e define d by the inductive supp osition; I = I 1 ∐ I 2 ; i 1 < i 2 if i 1 ∈ I 1 , i 2 ∈ I 2 and I ′ is t he set of indexes i ∈ I such that B i 6 = 0 . Prop ositio n 2.2. If Y 1 , Y 2 ar e ex pr essions of obje cts X s , s ∈ Ω and ϕ : Y 1 → Y 2 is the morphism built fr om morphisms c, L , b L A , the identific ation and laws ⊗ , ⊕ to gether with t he c omp osition, then they c an b e emb e dde d into the fol lowing c ommutative diagr am Y 1 h 1 − − − − → P I A i ϕ   y   y u Y 2 h 2 − − − − → P I A σ ( i ) wher e ( h 1 , P I A i ) , ( h 2 , P I A σ ( i ) ) ar e, r esp e ctively, c anonic al exp ansion forms of Y 1 , Y 2 ; σ is a p ermu tation of the set I and u is an isomorph ism bu ilt fr om t he morph ism c, the identific ation, the law ⊕ to gether with the c omp osition. Pro of. W e can prove the pr o po sition in case ϕ is one of isomorphisms c, L , b L A , A = id , g = d = id , a = id , l = r = id . Next, we can prove it easily in case ϕ is the sum ⊗ or the pro duct ⊕ of tw o morphisms of a bove-men tioned o nes. Nguyen Tien Quang 7 W e now sta te the coher ence tho erem. Theorem 2 .3. L et Y 1 , Y 2 , ..., Y n b e expr essions of the family X s , s ∈ Ω of obje cts in t he quite strict Ann-c ate gory A . L et ϕ i,i +1 : Y i → Y i +1 (i=1, 2, ..., n), ϕ n, 1 : Y n → Y 1 b e isomorphisms built fr om morphisms c, L , b L A , identific ation by laws ⊗ , ⊕ and the c omp osition. Then, the fol lowing diagr am ϕ n, 1 ϕ 1 , 2 ϕ 2 , 3 Y 1 Y 2 Y 3 ... Y n (3) c ommu tes. Pro of. Let ( h i , P J A σ i ( j ) ) deno te the c a nonical expansio n form of Y i . Consider the following diagram P J A σ 1 ( j ) P J A σ 2 ( j ) ... P J A σ n − 1 ( j ) P J A σ n ( j ) Y 1 Y 2 ... Y n − 1 Y n h 1 u 1 , 2 ϕ 1 , 2 h 2 u n, 1 ϕ n, 1 h n − 1 h n u n − 1 ,n ϕ n − 1 ,n ❄ ❄ ✻ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ (4) Then we hav e the dia gram (2.2), where morphis ms u i,i +1 ( i = 1 , 2 , ..., n − 1) , u n, 1 make regio ns (1), ..., (n) and the par ameter commute a ccording to Prop.2 .2. They a re built from e, id and the laws ⊗ , s o a ccording to the coher ence theor em for a s y mmetric monoidal catego r y , the reg ion (b) commutes. Therefo r e, the regio n (a) commutes. This completes the pr o of. 3 The general case In this last section, we assume that A , A ′ are Ann-categor ies with, r esp ectively , co nstraints ( A , c , (0 , g , d ) , a, (1 , l , r ) , L , R ) ( A ′ , c ′ , (0 ′ , g ′ , d ′ ) , a ′ , (1 ′ , l ′ , r ′ ) , L ′ , R ′ ) and ( F , ˘ F , e F ) : A → A ′ is a faithful Ann-functor such that the pair ( F, e F ) is co mpa tible with the unitivity constraints (1 , l, r ), (1 ′ , l ′ , r ′ ). In addition, let ´ F : F 1 → 1 ′ denote the isomo rphism induced by the ab ov e compatibility . Let ( X i ) , i ∈ I b e a non-empty , limited family of ob jects o f A , a nd Y = H ( X i ) be a cer tain expression of the family ( X i ) , i ∈ I . Then, the expr ession Y ′ = H ( X ′ i ) is called the c anonic al 8 The coherence theorem for Ann-categorie s image of Y = H ( X i ) under F if X ′ i = 0 ′ when X i = 0 X ′ i = 1 ′ when X i = 1 X ′ i = F X i otherwise F ro m this notio n we g ive the following definition Definition 3 . 1. We define a c anonic al isomorphism f : F Y = F ( H ( X i )) → F ( H ( X ′ i )) by induction on Y ’s length as fol lows 1. If Y ’s length is e qual to 1, Y = X α then f = b F : F 0 → 0 ′ in c ase X α = 0 f = ´ F : F 1 → 1 ′ in c ase X α = 1 f = id : F X α → F X α in other c ases 2. If Y ’s length is mor e than 1, Y = U 1 ⊗ U 2 or Y = U 1 ⊕ U 2 . Then, the isomorphism f is, r esp e ct ively, the fol lowing c omp ositions F Y = F ( U 1 ⊗ U 2 ) e F − − − − → F U 1 ⊗ F U 2 f 1 ⊗ f 2 − − − − → H 1 ( X ′ i ) ⊗ H 2 ( X ′ i ) F Y = F ( U 1 ⊕ U 2 ) ˘ F − − − − → F U 1 ⊕ F U 2 f 1 ⊕ f 2 − − − − → H 1 ( X ′ i ) ⊕ H 2 ( X ′ i ) wher e f 1 , f 2 ar e c anonic al isomorphisms determine d by inductive supp osition. Prop ositio n 3.2. Supp ose that ϕ : Y 1 → Y 2 is a morphism built fr om isomorphi s m s A , c, g , d, a, l , r , L , R , b L A , b R A in the Ann-c ate gory A . Then, ϕ c an b e emb e dde d int o t he fol lowing c ommu tative diagr am F Y 1 f 1 − − − − → Y ′ 1 F ( ϕ )   y   y ϕ ′ F Y 2 f 2 − − − − → Y ′ 2 wher e f i ar e c anonic al isomorphisms c orr esp onding to c anonic al images Y ′ i of Y i (i=1, 2), wher e as ϕ ′ is a morphism built fr om isomorphisms A ′ , c ′ , g ′ , d ′ , a ′ , l ′ , r ′ , L ′ , R ′ , b L A ′ , b R A ′ in the Ann-c ate gory A ′ . Pro of. The pro of is completely similar to the one of Pro po sition 2.2. F ollowing is the ma in result of this pap er . Theorem 3. 3. L et Y 1 , Y 2 , ..., Y n b e ex pr essions of t he limite d family of obje cts ( X i ) i ∈ I of an Ann- c at e gory A . L et ϕ i,i +1 : Y i → Y i +1 (i=1, 2, ..., n-1), ϕ n, 1 : Y n → Y 1 b e isomorphisms built fr om the isomorph isms A , c, g , d, a, l, r , L , R , b L A , b R A , the identific ation and laws ⊗ , ⊕ . Th en, t he dia gr am (2.1) c ommut es. Pro of. F r om the theorem 2.4 [9 ], the Ann-ca tegory A can be faithfully embedded int o a quite strict Ann-catego ry A ′ by the faithful Ann-functor ( F , ˘ F , e F ). Mo reov er , ( F, e F ) is co mpa tible with the unitivit y constraints. In order to prove the diag ram (2.1) c o mm utative, we consider its image under F Nguyen Tien Quang 9 Y ′ 1 Y ′ 2 ... Y ′ n − 1 Y n F Y 1 F Y 2 ... F Y n − 1 F Y n f 1 f 2 f n − 1 f n ϕ ′ 1 , 2 F ( ϕ 1 , 2 ) ϕ ′ n − 1 ,n F ( ϕ n − 1 ,n ) ϕ ′ n, 1 F ( ϕ n, 1 ) ❄ ❄ ✻ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ where f i are canonical iso morphisms, whereas ϕ i,i +1 ( i = 1 , 2 , ..., n − 1), ϕ n, 1 are morphisms making regions from (1) to (n) and the par ameter commut e accor ding to the Pro p.3 .2. These morphisms are built fr o m iso morphisms c ′ , L ′ , b L A ′ , i d and by laws ⊗ , ⊕ . Applying Theorem 2 .3, the r e g ion (b) commutes. This implies that the r egion (a) co mm utes. This co mpletes the pro o f. Remark. The coherence theor em can b e stated in another w ay as follows: Bet ween tw o ob- jects of the Ann-categor y A , there exis ts no mor e tha n one mor phism built from morphisms A , c, g , d, a, l, r , L , R , b L A , b R A and laws ⊗ , ⊕ . ( P I ′ × J A i B j ⊕ P J A α B j )( P T C t ) P I × J × T A i B j C t ( P I ′ × J A i B j )( P T C t ) ⊕ ( P J A α B j )( P T C t ) P I ′ × J × T A i B j C t ⊕ P J × T A α B j C t ( P I A i )( P J B j )( P T C t ) ⊕ A α ( P J B j )( P T C t ) ( P I ′ A i )( P J × T B j C t ) ⊕ A α ( P J × T B j C t ) ( P I A i )( P J B j )( P T C t ) ( P I A i )( P J × T B j C t ) F I ,J ⊗ id id ( F I ′ ,J ⊗ id ) ⊕ ( F A α ⊗ id ) id F I × J,T F I ′ × J,T ⊕ F J,T ( id ⊗ F J,T ) ⊕ ( id ⊗ F J,T ) id ⊗ F J,T id F I ′ ,J × T ⊕ f A α id F I ,J × T ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲ 10 The coherence theorem for Ann-categorie s References [1] H. X. Sinh. Gr-c ate gories. These , Paris (19 75). [2] H. X. Sinh. Gr-c ate gories stricte. Acta Math. Vietnamica, T om I I I, no 2 (1978), 4 7- 59. [3] G. M. Kelly a nd M. L. Laplaza. Coher enc e for c omp act clo set c ate gories. J. P ure and Appl. Algebra, 1 9 (1 9 80), 19 3 - 21 3. [4] G. M. Kelly a nd S. Mac Lane. Coher enc e in close t c ate gories. J. Pure and Appl. 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