Projective and injective objects in symmetric categorical groups

PR OJECTIVE AND INJECTIV E OBJECTS IN SYMMETRIC CA TEGORICAL GR OUPS TEIMURAZ PIRASHVILI De dic ate d to the memory of Pr of. V. K. Bentkus Categorical rings (called also 2-rings) w ere introduced in [4]. Categorical mo dules (called also 2-mo dules) ov er a categor ical ring w ere in tro duced in [7]. Categorical mo dules a nd symmetric categorical groups [1] are examples of ab elian 2-categories studied in [2] ( for other examples of ab elian 2-categories see [5]). Pro jectiv e ob jects in the framew o rk of symmetric categorical groups w ere in tro duced in [1] hav e an obv ious generalization to the case o f a b elian 2-categories. Hence by duality one can also ta lk on injectiv e ob jects in an y ab elian 2- category . Moreo v er in [1] the a utho rs conjectured that the ab elian 2- category of symmetric categorical groups ha v e enough pro jectiv e ob jects. In the course of our w ork [6 ] w e noted t ha t the 2-category o f symmetric categorical groups hav e enough pro jectiv e and injectiv e ob jects. Moreo ve r t his statemen t is a tr ivial consequences of the cohomological description of abelian 2-cat ego ry of symmetric categorical groups obtained first in [8]. Using base c hange a r g umen t this result also yields that the 2-category o f catego rical mo dules ov er an y categorical ring hav e enough pro jectiv e and injectiv e ob jects. Quite recen tly these results we r e announced in [3 ] but with wrong pro ofs (Lemmata 3 and 1 1 of lo c. cit. are b oth wrongs). Here w e giv e our orig ina l pro of s. A gro up oid enric h categor y T is a 2-category suc h tha t any 2-ar ro w is in ve rtible. If T is a group oid enric h category then we use the w or d ”morphism” for 1-morphisms and w e use the w ord ”tra ck” for 2- morphisms. W e let Ho ( T ) b e the corresp onding homotopy category . If f , g : A → B are morphism in T , then w e sa y that f is homotopic to g if there exists a track from f to g . Let H o ( T ) b e the corresp onding ho mo t op y catego r y . Th us ob jects o f Ho ( T ) are the same as of T , while morphisms in Ho ( T ) are homotopic classes of morphisms in T . Symmetric categorical groups fo r m a group oid enric h category whic h is denoted b y SymCatGr . In particular one can form the homotopy category o f the 2-category of sym- metric catego rical groups. W e will denote the corresp onding homotopy category b y H o . This category has the follow ing nice description which follows from the classical results o f H.X.Sinh [8]. First we fix some notations. If S is a symmetric categorical group then w e let π 0 ( S ) and π 1 ( S ) denote resp ectiv ely the ab elian group of connected comp onen ts of S and t he ab elian group of all a utomorphisms of the neutral ob ject of S . F or symmetric categorical groups S 1 and S 2 w e ha ve a group oid (in fact a symme tric categorical group [1]) Hom ( S 1 , S 2 ). T o Research was pa rtially suppo rted b y the GNSF Grant ST08/ 3 -387. 1 2 T. PIRASHV ILI describe π i ( Hom ( S 1 , S 2 )) , i = 0 , 1 we need to in tro duce the categor y Types . Ob jects of the category Typ es are triples A = ( A 0 , A 1 , α ), where A i is an ab elian group, i = 0 , 1 and α ∈ hom( A 0 / 2 A 0 , A 1 ) = hom ( A 1 , 2 A 1 ) Here for an ab elian group A we set 2 A = { a ∈ A | 2 a = 0 } A morphism f fr o m A = ( A 0 , A 1 , α ) t o B = ( B 0 , B 1 , β ) in Types is given b y a pair f = ( f 0 , f 1 ), where f 0 : A 0 → B 0 and f 1 : A 1 → B 1 are ho mo mo r phisms of ab elian groups suc h that β f 0 = f 1 α . Let S b e a symmetric categorical group. W e put ty pe ( S ) := ( π 0 ( S ) , π 1 ( S ) , k S ) where k S is the homomorphism induced b y the comm utativit y constrants in S . Both categories Ho and Types are a dditiv e and the functor ty pe : Ho → T ypes is additive . According to [8] the f unctor ty pe is full, essen tially surjectiv e on o b jects and the k ernel of ty pe (morphisms whic h go es to zero) is a square zero ideal of Ho . It f o llo ws then that the functor ty pe reflects isomorphisms and induces a bijection o n the isomorphism classes of ob jects. More precisely , for any symmetric categorical groups S 1 and S 2 one has a short exact sequence of ab elian groups (1) 0 → Ext ( π 0 ( S 1 ) , π 1 ( S 2 )) → π 0 ( Hom ( S 1 , S 2 )) → Types ( ty pe ( S 1 ) , ty pe ( S 2 )) → 0 F urthermore one ha s also an isomorphism o f ab elian groups (2) π 1 ( Hom ( S 1 , S 2 )) ∼ = hom( π 0 ( S 1 ) , π 1 ( S 1 )) These facts greatly simplifies to work with symmetric categorical gr oups. F or a giv en ob ject A of the category Typ es we c ho ose a symmetric categorical group H ( A ) suc h that ty pe ( H ( A )) = A . Suc h ob ject exist and is unique up to equiv alence. Moreo ve r , for a n y morphism f : A → B w e c ho ose a morphism of symmetric categor ical groups H ( f ) : H ( A ) → H ( B ), suc h that ty pe ( H ( f ) = f . The reader mus t b e aw are that the assignmen ts A → H ( A ), f 7→ H ( f ) do es NOT define a functor Types → Ho . Recall that [1] a morphism F : S 1 → S 2 of symmetric categorical groups is called essential ly surje ctive (resp. faithful ) if it is epimorphism on π 0 (resp. monomorphism on π 1 ). A symm etric categorical gro up S is called pr oje ctive prov ided for an y essen tially surjectiv e functor F : S 1 → S 2 and a morphism G : S → S 2 there exist a morphism L : S → S 1 and a tra ck f rom F L → G . Dually a symmetric categor ical group S is called inje ctive pro vided for an y faithful functor F : S 1 → S 2 and a morphism G : S 1 → S there exist a morphism L : S 2 → S and a tra c k from LF to G . W e can dev elop same sort of languag e in the category Types . A morphism f = ( f 0 , f 1 ) in Types is essential ly surje ctive if f 0 is epimorphism of ab elian groups. Moreo v er an o b j ect PROJE CTIVE AND INJECTIVE OBJECTS IN SYMMETRIC CA TEGORICAL GROUPS 3 P in T ypes is pr oje ctive of for a n y essen tially surjectiv e morphism f : A → B in Types the induced map Types ( P , A ) → Types ( P , B ) is surjectiv e. Dually , a morphism f in Type s is f aithful provided f 1 is injectiv e and an ob ject I = ( I 0 , I 1 , ι ) of Types is inje ctive if for an y faithful morphism f : A → B in Types the induced map Types ( B , I ) → Types ( A , I ) is surjectiv e. It is clear that a morphism F : S 1 → S 2 of symmetric categorical g roups is essen tial ly surje ctive (resp. faithful ) iff ty pe ( F ) : ty pe ( S 1 ) → ty pe ( S 2 ) is so in Ty pes . F or an ab elian group M we in tro duce t wo ob jects in T ypes : l ( M ) := ( M , M / 2 M , id M / 2 M ) r ( M ) = ( 2 M , M , id 2 M ) Lemma 1. i) If M is an ab elian gr oup and A = ( A 0 , A 1 , α ) is an obje ct in Typ es , then one has fol lowing functorial isomorphism s of ab elian gr oups Types ( l ( M ) , A ) = hom( M , A 0 ) , Types ( A , r ( M )) = hom( A 1 , M ) . ii) F or any fr e e a b elian gr oup P the obje ct l ( P ) ∈ Types is pr oje ctive in Types , dual ly for a ny divisible ab elian gr oup Q the obje ct r ( Q ) ∈ Types is in je ctive. iii) F o r any fr e e ab elian gr oup P the symmetric c ate goric al gr oup H ( l ( P )) is pr oje ctive symmetric c ate goric al gr oup and dual ly for any divisibl e ab elian gr oup Q the triple r ( Q ) is inje ctive. Pr o of. i) and ii) are o bvious. Let F : S 1 → S 2 b e an essen tially surjectiv e mor phism of symmetric categorical groups and G : H ( l ( P )) → S 2 b e a morphism o f symmet- ric categorical groups. Apply the functor ty pe t o g et a essen tially surjectiv e mo r phism ty pe ( F ) : ty pe ( S 1 ) → ty pe ( S 2 ) in Types and a mo r phism ty pe ( G ) : l ( P ) → ty pe ( S 2 ) in Types . Since π 0 ( F ) : π 0 ( S 1 ) → π 0 ( S 2 ) is an epimorphism of ab elian groups, F is a f r ee ab elian group the homomorphism π 0 ( G ) : P → π 0 ( S 2 ) ha s a lifting to the homo mo r phism P → π 0 ( S 1 ) Since P is free ab elian it follow s from the exact sequence (1) that for i = 0 , 1 one has an isomorphism (3) π 0 ( Hom ( H ( l ( P )) , S i )) ∼ = Types ( l ( P ) , ty pe ( S i )) ∼ = hom( P , π 0 ( S i )) T ak e a morphism L : H ( l ( P ) → S 1 of symmetric categorical groups whic h corresp onds to the homomorphism P → π 0 ( S 1 ). By our construction one has an equalit y ty pe ( F L ) = ty pe ( G ) , which imply that the class of F L and of G in π 0 ( Hom ( H ( l ( P )) , S 1 )) are the same. Th us there exist a trac k from F L to G . This sho ws that H ( l ( P )) is a pro j ective symmetric categorical group. a dual argumen t works for injectiv e o b jects.  4 T. PIRASHV ILI Prop osition 2. The 2- c ate gory of symm etric c ate goric al gr oups have enough pr oje ctive and inje ctive obje cts. Pr o of. Let S b e a symmetric categor ical g roup. C ho ose a f r ee ab elian group P and epi- morphism o f a b elian gro ups f 0 : P → π 0 ( S ). By Lemma 1 it has a unique extension to a morphism f = ( f 0 , f 1 ) : l ( P ) → ty pe ( S ) whic h is essen tially surjectiv e. Since P is is a f r ee ab elian group, w e ha ve the isomorphism (3), whic h show that there exist a morphism of symmetric categorical groups H ( l ( P )) → S whic h realizes f 0 on the lev el of π 0 . Clearly this morphism do es the job. Dually , choose a monomorphism g 1 : π 1 ( S ) 1 → Q with divisible ab elian gro up Q . By Lemma 1 it has the unique extension as a morphism g : ty pe ( S ) → r ( Q ) whic h is fa it hful b y the construction. Sinc e π 1 ( r ( Q )) = Q is injectiv e ob ject in the category of ab elian groups by the short exact sequence (1) w e hav e π 0 (Hom( S , H ( r ( Q )))) ∼ = Types ( ty pe ( S ) , r ( Q )) ∼ = hom( π 1 ( S ) , Q ) whic h sho ws that g can b e realized as a morphism of symmetric categorical groups and w e get t he result.  Prop osition 3. L et R b e a c ate goric al gr oup. Then the c ate gory of c ate goric al right R - mo dules have enough pr oje ctive and inje ctive obje cts. Pr o of. By Y oneda Lemma for symmetric categorical gr o ups the categor ical ring R consid- ered as a r ig h t R -mo dule is pro jectiv e and from this fact one easily deduces the statemen t on pro jectiv e ob jects. F or injectivit y we consider the 2-f unctor Hom ( R , − ) from t he 2- category o f symmetric categorical groups to the 2-category of cat ego rical rig ht R -mo dules. It is a r igh t 2-adjo in t to the forgetful 2-functor. Since the forg etful functor is exact it follo ws that the 2-functor Hom ( R , − ) take s injectiv e ob jects to injectiv e ones. Let M b e a categorical left R -mo dule. Cho ose a faithful morphism M → Q in the 2-category of sym- metric categorical groups with injective symmetric categor ical g r oup Q . Apply no w the 2- functor Hom ( R , − ). It fo llows from the isomorphism (2) that Hom ( R , M ) → H om ( R , Q ) is a fait hf ul morphism o f right R -mo dules. By the same reasons the obvious morphism M → Hom ( R , M ) is also faithful. T aking the comp osite we obta in a faithful morphism M → H om ( R , Q ) and hence the result.  Note that the pro of o f the la st statemen t is essen tially the same as it w as f or classical rings. The same is also true f or the following r esult and b ecause of t his we omit the pro of. Recall that if T is an additive 2-category and M is an ob ject in T then one has the categorical ring Hom ( M ) (compare [2],[7]). Prop osition 4. I f T is a 2-ab elian c ate gory w hich p osses a smal l pr oje ctive gene r ator P , then T is 2- e quivalent to the c ate gory of right c ate goric al m o dules over the c ate goric al ring Hom ( P , P ) . Consider the fo llo wing symmetric categorical group H . Ob jects of the group oid H are in tegers. If n 6 = m then there is no morphism from n t o m , n, m ∈ Z . The group of automorphisms of n is the cyclic group of order tw o with generator ǫ n , n ∈ Z . The monoidal PROJE CTIVE AND INJECTIVE OBJECTS IN SYMMETRIC CA TEGORICAL GROUPS 5 functor is induced by t he addition of in tegers. The a sso ciativit y and unite constran ts are iden tity morphisms, while the comm utativity constran t n + m → m + n equals to ǫ n + m . By our construction H is a small pro jective generator in the 2- category of symmetric categorical groups. Hence w e obtained the follow ing imp ortan t fact. Prop osition 5. The 2 -c ate gory of symm etric c ate goric al gr oups is 2-e quivalent to the c at- e gory of right c ate goric al mo dules over the c ate goric al ring Hom ( H , H ) . Reference s [1] D. Bourn and E. M. V it ale . Ex tensions of symmetric ca t-groups. Homology Homo topy Appl. 4 (2002), no . 1 , 1 03–16 2. [2] M. Dupont. Abelian ca teg ories in dimensio n 2 . arXiv:080 9.1760 [3] F ang Huang , Shao-Han Chen , Wei Chen and Zhu -Jun Zheng . Higher Dimensiona l Homology Algebra II:P ro jectivity . ar Xiv:1 006.4 6 77 [4] M. Jibladze and T. Pirashvil i . Third Ma c L ane cohomolog y via catego rical rings . J. Homotopy Relat. Struct., 2 (2007), pp.187-216 . [5] T. Pirash vili . Ab elian ca teg ories v er sus ab elia n 2 -categor ies. Georgia n Math. J. 1 6 (2009), no. 2, 353–36 8. [6] T. Pirashvil i . The derived category of mo dules over a 2-s tage ring spectr um (in prepr ation). [7] V. Schmitt . Enrichmen ts ov er symmetric Picard ca tegories. arXiv:081 2.0150 . [8] H. X. Sinh . Gr -cat´ egories, Th ` e se de Doctor al d’Eta t. Universit´ e Paris VI I, 1975. Dep ar tment of Ma thema tics, U niversity of Leicester, University R oad, Leicester, LE1 7RH, UK E-mail addr ess : tp59 -at-le .ac.u k