Limits of Coalgebras, Bialgebras and Hopf Algebras

LIMITS OF COALGEBRAS, B IALGEBRAS AND HOPF ALGEBRAS A.L. AGORE De d ic at e d to the memory of Pr ofessor S. Ianu¸ s Abstra ct. W e giv e the explicit construction of the prod uct of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the prod uct of an arbitrary family of coalgebras (resp. b ialgebras, Hopf algebras) is the sum of a family of coalge- bras (resp. bialgebras, H opf algebras). The equalizers of tw o morph isms of coalgebras (resp. bialgebras, Hopf algebras) are also describ ed ex plicitly . As a consequ ence the categories of coalgebras, bialgebras and Hopf algebras are shown to b e complete and a complete d escription for limits in the ab ov e categories is given. Introduction It i s w ell kno wn that the catego ry k -Alg of k -algebras is complete and cocomplete: that is an y functor F : I → k -Alg has a limit and a colimit, for all small catego ries I . This is immediately implied b y the existence of pro du cts, copro ducts, equalizers and co equalizers in the catego ry k -Alg. The categories of coalgebras, bialgebras or Hopf algebras ha v e arbitrary copro ducts and co equalizers (see [5, Prop ositon 1.4.19], [4, Prop osition 2.10], [8, Corollary 2.6.6] and [2, R emark 2.1, Theorem 2.2]), hence these categories are co complete. Related to th e question of whether these categories are complete (i.e. if th ey hav e arbitrary pro du cts and equalizers) we could not fin d similar results in th e classical Hopf algebra textb o oks ([1], [10]), not even in the more recen t ones ([4], [5]). F o r example, [5, Prop ositon 1.4.21] pr o v es only the existence of finite pr o ducts (namely the tensor pro duct of coalgebras) and only in the categ ory of c o c omm utative coalge bras, as a dual r esu lt to the one concernin g comm utativ e algebras. Only r ecen tly in [9, Theorem 9] it is pro v ed that the cat egory of coa lgebras or, more generally , the catego ry of corings is locally presen table, thus they are co mplete by the d efinition of lo cally presenta ble categories. How eve r the pro of of [9, Theorem 9] do es not constru ct explicitly the limits (in particular the pro d ucts) of an arbitrary family of coalgebras. In this note we shall fill this gap: using the fact that the f orgetful functor f r om the catego ry of coalge bras to the category of ve ctor spaces has a righ t adjoint, namely the so called cofree coalg ebra, we sh all constru ct exp licitly the pro duct of an arbitrary family of coalg ebras. As a consequence, the pro duct of an arb itrary family of bialgebras and Hopf algebras is constructed. The equalizers of t w o morphisms of coalgebras (bialgebras, 2000 Mathematics Subje ct Classific ation. 16W30, 18A30, 18A40. Key wor ds and phr ases. p ro duct of coalgebras, bialgebras, Hopf algebras. The author ac knowl edges partial supp ort from CNCSI S grant 24/28.09.07 of PN I I ”Groups, quantum groups, corings and representatio n theory”. 1 2 A.L. AGORE Hopf algebras) are also describ ed exp licitly . Thus we sh all obtain that the categories of coalge bras, b ialgebras and Hopf algebras are complete. Throughout th is pap er, k will b e a field. Unless sp ecified otherwise, all v ect or spaces, algebras, coalge bras, bialge bras, tensor pro ducts an d homomorphism s are o v er k . Our notation for the standard categories is as follo ws: k M ( k -vect or s p aces), k -Alg (as- so ciativ e unital k -algebras), k -CoAlg (coalgebras o ve r k ), k -BiAlg (bialg ebras o ve r k ), k -HopfAlg (Hopf algebras ov er k ), M C (righ t C -como dules). F or a coalgebra C , we will use Swee dler’s Σ-notation, that is, ∆ ( c ) = c (1) ⊗ c (2) , ( I ⊗ ∆)∆( c ) = c (1) ⊗ c (2) ⊗ c (3) , etc. Giv en a ve ctor space V , ( K ( V ) , p ) stands for the cofree coalgebra on V , where K ( V ) is a coalge bra and p : K ( V ) → V is a k -linear map. W e r efer to [1], [5], [10] for further details concerning Hopf alge bras. A category C is called (c o)c omplete if an y fu nctor F : I → C has (co)limits, w here I is a small category . A category C is (c o)c omplete if and only if C has (co)equalizers of all pairs of arrows an d all (co)pro ducts [7 , Th eorem 6.10]. Giv en a morphism f ∈ C w e denote by dom ( f ) and cod ( f ) the domain, r esp ectiv ely th e co domain of f . If C is a small category w e denote b y H om ( C ) the set of all morp h isms of C . 1. Limits for coalgebras, bialgebras a nd Hopf al gebras First, w e explicitly construct the pro du ct of an arbitrary family of coalge bras. Theorem 1.1. The c at e gory k - CoA lg of c o algebr as has arbitr ary pr o ducts and e qualizers. In p articular, the c ate gory k -CoAlg of c o algebr as is c om plete. Pr o of. Let f , g : C → D b e t w o coalgebra maps and S := { c ∈ C | f ( c ) = g ( c ) } , w hic h is a k -subs pace of C . Let E b e the sum of all sub coalgebras of C included in S . Note that the family of sub coalgebras of C includ ed in S is n ot empty since it con tains the n ull coalgebra. It is immediate that E is a sub coalgebra of C . W e shall p r o v e that ( E , i ) is the equalizer of the p air ( f , g ) in k -CoAlg, where i : E → C is the canonical inclus ion. Let E ′ b e a coalgebra and h : E ′ → C a coalgebra map suc h that f ◦ h = g ◦ h . Then f  h ( x )  = g  h ( x )  , for all x ∈ E ′ , hence h ( E ′ ) ∈ S and since h ( E ′ ) is a sub coalgebra in C we obtain h ( E ′ ) ⊆ E . Thus there exists a unique coalge bra map h : E ′ → E su c h that i ◦ h = h . Hence ( E , i ) is the equalizer of the p air ( f , g ) in the category k -CoAlg of coalge bras. No w let ( C i ) i ∈ I b e a family of coalg ebras and  Q i ∈ I C i , ( π i ) i ∈ I  b e the pr o duct of the k -mo dules ( C i ) i ∈ I . Let  K ( Q i ∈ I C i ) , p  b e the cofree coalg ebra o v er th e vec tor space Q i ∈ I C i . Let D b e th e s u m of all su b coalgebras E of K  Q i ∈ I C i  suc h that π i ◦ p ◦ j E is a coalgebra map for all i ∈ I , where j E : E → K ( Q i ∈ I C i ) is the canonical inclusion. The family of sub coalgebras of E satisfying this p rop erty is nonemp ty since it con tains the n ull coalge bra. The k -linear map π i ◦ p ◦ j : D → C i is a coalg ebra map for all i ∈ I where j : D → K ( Q i ∈ I C i ) is the canonical inclusion. W e shall prov e that  D , ( π i ◦ p ◦ j ) i ∈ I  is the pro du ct of the family of coalgebras ( C i ) i ∈ I in k -CoAlg. Indeed, let D ′ b e a coalge bra and g i : D ′ → C i , i ∈ I , a family of coalgebra maps. Using the unive rsal p r op ert y of the p ro duct in k M we obtain that there exists a un ique LIMITS OF COALGEBRAS, BIALG EBRAS AND HOPF ALGEBRAS 3 k − linear map g : D ′ → Q i ∈ I C i suc h that π i ◦ g = g i , for all i ∈ I . F urth ermore, since  K ( Q i ∈ I C i ) , p  is the cofree coalgebra ov er the k − mo du le Q i ∈ I C i , there exists a unique coalge bra map f : D ′ → K ( Q i ∈ I C i ) suc h that p ◦ f = g . T hus we ha v e the follo wing comm utativ e diagram: D ′ f % % K K K K K K K K K K g * * V V V V V V V V V V V V V V V V V V V V V V V V g i , , X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X D j / / K  Q i ∈ I C i  p / / Q i ∈ I C i π i / / C i It follo ws that ( π i ◦ p )( f ( x )) = g i ( x ), for all x ∈ D ′ and i ∈ I . So, since eac h g i is a coalge bra map, we ha v e f ( D ′ ) ⊆ D . Hence, we p ro v ed that for an y coalgebra D ′ and any family g i : D ′ → C i , i ∈ I , of coalgebra map s there exists a coalgebra map f : D ′ → D suc h that ( π i ◦ p ◦ j ) ◦ f = g i , for all i ∈ I . Let h : D ′ → D b e another coalgebra map suc h that ( π i ◦ p ◦ j ) ◦ h = g i for all i ∈ I . F rom th e uniquen ess of g w e obtain p ◦ j ◦ h = g . Moreo ver, from the uniqueness of f w e obtain j ◦ h = f , hence h = f . Th us  D , ( π i ◦ p ◦ j ) i ∈ I  is the pro du ct of the family ( C i ) i ∈ I in the category k -C oAlg of coalge bras.  Remark 1.2 . 1) In [3, Lemma 1.1 .3] a description for th e equalize rs in the category k -HopfAlg is giv en. W e can use the same metho d in order to obtain another d escription for the equalizer of a pair of coalgebra (or bialgebra) map s . Let f , g : C → D b e tw o coalge bra maps. It can b e easily prov ed that ( E , i ) is the equalizer of the pair ( f , g ) in the category k -CoAlg of coalge bras, where E = { c ∈ C | c (1) ⊗ f ( c (2) ) ⊗ c (3) = c (1) ⊗ g ( c (2) ) ⊗ c (3) } and i : E → C is the canonical in clusion. This equiv ale n t descrip tion of equalizers in th e catego ry k -CoAlg will turn out to b e more efficien t for compu tations. Example 1.3. Let G b e a m ultiplicativ e group and k G the k -v ector s pace w ith basis { g | g ∈ G } endow ed with the classical coalgebra stru cture : ∆( g ) = g ⊗ g and ε ( g ) = 1 for all g ∈ G . Thus any elemen t x ∈ k G has the form x = P g ∈ G k g g where ( k g ) g ∈ G is a family of elemen ts in k w ith only a fin ite num b er of n on-zero element s. W e use the follo w ing formal notation x − 1 := P g ∈ G k g g − 1 and 0 − 1 = 0. Consider the coalgebra maps f = I d k G and h : k G → k G giv en by h ( g ) = g − 1 for all g ∈ G . Then, in the ligh t of the ab o v e remark, it follo ws that th e equalizer of the morphisms ( f , g ) is given by the p air ( E , i ) w here E = { x ∈ k G | x ⊗ x ⊗ x = x ⊗ x − 1 ⊗ x } and i is the canonical inclusion. As an easy consequence of [6, Ch apter 5 § 2, Theorem 1] we obtain the follo wing descrip- tion for limits in the category k -CoAlg of coalge bras: Remark 1.4 . Let J b e a small categ ory , F : J → k -CoAlg b e a functor,  Π j ∈ J F ( j ) , ( p j ) j ∈ J  ,  Π u ∈ H om ( J ) F ( cod ( u )) , ( p u ) u ∈ H om ( J )  b e the pro d u ct in k -CoAlg of the families  F ( j )  j ∈ J , resp ectiv ely  F ( cod ( u ))  u ∈ H om ( J ) and f , g : Π j ∈ J F ( j ) → Π u ∈ H om ( J ) F ( cod ( u )) 4 A.L. AGORE b e the u nique coalge bra m ap s suc h that p u ◦ f = p cod ( u ) and p u ◦ g = F ( u ) ◦ p dom ( u ) for all u ∈ H om ( J ). W e defin e D = { x ∈ Π j ∈ J F ( j ) | x (1) ⊗ f ( x (2) ) ⊗ x (3) = x (1) ⊗ g ( x (2) ) ⊗ x (3) } . Then the p air  D , ( ϕ j = p j ◦ e ) j ∈ J  is the limit of th e f unctor F , where e : D → Π j ∈ J F ( j ) is the canonical inclus ion. In what follo ws w e w ill make use of T heorem 1.1 in order to construct the pro du ct in the category of k -BiAlg of bialgebras. Theorem 1.5. The c ate gor y k - BiAlg of bialgebr as has arbitr ary pr o ducts and e qualizers. In p articular, the c ate gory k -BiAlg of bialgebr as is c omplete. Pr o of. Let  B i , m i , η i , ∆ i , ε i  i ∈ I b e a family of bialgebras and  ( Q i ∈ I B i , ∆ , ε ) , ( π i ) i ∈ I  b e the pr o duct of this family in the category k -CoAlg of coalge bras. Since ( B i , m i , η i , ∆ i , ε i ) is a b ialgebra it follo ws th at m i : B i ⊗ B i → B i and η i : k → B i are coalgebra m aps for all i ∈ I . Then there exists a uniqu e coalgebra map η : k → Q i ∈ I B i suc h that the follo wing diagram : (1) k η   η i # # H H H H H H H H H H Q i ∈ I B i π i / / B i is commutat iv e for all i ∈ I . Also there exists a un ique coalgebra map m : Q i ∈ I B i ⊗ Q i ∈ I B i → Q i ∈ I B i for w hic h the d iagram : (2) Q i ∈ I B i ⊗ Q i ∈ I B i m   m i ◦ ( π i ⊗ π i ) ' ' N N N N N N N N N N N N N Q i ∈ I B i π i / / B i is commutati v e for all i ∈ I . First, we w ill pro v e that ( Q i ∈ I B i , m, η ) is a k -algebra. Since π i ◦ m ◦ ( m ⊗ I d ) is a coalge bra map b y the u niv ersal prop erty of the pro duct we obtain that there exists a unique coalgebra map ψ :  Q i ∈ I B i  ⊗ 3 → Q i ∈ I B i suc h th at the follo wing diagram:  Q i ∈ I B i  ⊗ 3 ψ   π i ◦ m ◦ ( m ⊗ I d ) $ $ J J J J J J J J J J Q i ∈ I B i π i / / B i is comm utativ e for all i ∈ I . It is easy to s ee that the coalgebra map m ◦ ( m ⊗ I d ) makes the ab o v e d iagram commute. Thus, u sing the un iqueness of ψ , in order to prov e that LIMITS OF COALGEBRAS, BIALG EBRAS AND HOPF ALGEBRAS 5 m ◦ ( m ⊗ I d ) = m ◦ ( I d ⊗ m ) it is enough to sho w that π i ◦ m ◦ ( m ⊗ I d ) = π i ◦ m ◦ ( I d ⊗ m ) for all i ∈ I . W e hav e : π i ◦ m ◦ ( m ⊗ I d ) ( 2 ) = m i ◦ ( π i ⊗ π i ) ◦ ( m ⊗ I d ) = m i ◦  ( π i ◦ m ) ⊗ π i  ( 2 ) = m i ◦  m i ◦ ( π i ⊗ π i )  ⊗ π i  = m i ◦ ( m i ⊗ I d ) ◦ ( π i ⊗ π i ⊗ π i ) = m i ◦ ( I d ⊗ m i ) ◦ ( π i ⊗ π i ⊗ π i ) = m i ◦  π i ⊗  m i ◦ ( π i ⊗ π i )  ( 2 ) = m i ◦  π i ⊗ ( π i ◦ m )  = m i ◦ ( π i ⊗ π i ) ◦ ( I d ⊗ m ) ( 2 ) = π i ◦ m ◦ ( I d ⊗ m ) Hence m ◦ ( m ⊗ I d ) = m ◦ ( I d ⊗ m ), i.e. m is asso ciativ e. Consider no w th e coalgebra map π i ◦ m ◦ ( η ⊗ I d ). F rom the universal pr op ert y of the pro du ct, we obtain that there exists a unique coalgebra map ϕ : k ⊗ Q i ∈ I B i → B i suc h that the follo wing d iagram : k ⊗ Q i ∈ I B i ϕ   π i ◦ m ◦ ( η ⊗ I d ) % % J J J J J J J J J J Q i ∈ I B i π i / / B i is commutat iv e for all i ∈ I . By the argument ab o v e, in order to pro v e that m ◦ ( η ⊗ I d ) = s it will b e enough to sho w that π i ◦ m ◦ ( η ⊗ I d ) = π i ◦ s , where w e d en ote by s the scalar multiplicatio n. W e ha v e: π i ◦ m ◦ ( η ⊗ I d ) ( 2 ) = m i ◦ ( π i ⊗ π i ) ◦ ( η ⊗ I d ) = m i ◦  ( π i ◦ η ) ⊗ π i  ( 1 ) = m i ◦ ( η i ⊗ π i ) = m i ◦ ( η i ⊗ I d ) ◦ ( I d ⊗ π i ) = s ◦ ( I d ⊗ π i ) F urth ermore, let k 1 ⊗ b ∈ k ⊗ Q i ∈ I B i . Ha ving in min d that π i is a k -linear map we obtain : s ◦ ( I d ⊗ π i )( k 1 ⊗ b ) = k 1 π i ( b ) = π i ( k 1 b ) = π i ◦ s ( k 1 ⊗ b ) Th us w e pr ov ed that π i ◦ m ◦ ( η ⊗ I d ) = π i ◦ s . In the same wa y it follo ws that m ◦ ( I d ⊗ η ) = s . Hence ( Q i ∈ I B i , m, η ) is an algebra and since m and η are coalgebra m ap s we obtain that ( Q i ∈ I B i , m, η , ∆ , ε ) is actually a bialgebra. 6 A.L. AGORE T o end the pro of we still n eed to sho w that ( Q i ∈ I B i , m, η , ∆ , ε ) is the pr o duct of the family  B i , m i , η i , ∆ i , ε i  i ∈ I in the category k -BiAlg. Let ( B , m B , η B , ∆ B , ε B ) b e a bial- gebra and ( g i ) i ∈ I b e a family of b ialgebra maps, g i : B → B i for all i ∈ I . F rom the unive rsal prop erty of the pro du ct, we obtain that there exists an un ique coalgebra map θ : B → Q i ∈ I B i suc h th at the follo wing diagram comm utes : (3) B θ   g i # # H H H H H H H H H H Q i ∈ I B i π i / / B i W e only n eed to prov e that θ is also an algebra map. By the argument used ab o v e, it is enough to sh o w that: (4) π i ◦ θ ◦ m B = π i ◦ m ◦ ( θ ⊗ θ ) and π i ◦ θ ◦ η B = π i ◦ η Ha ving in mind th at g i is an algebra m ap, we hav e: π i ◦ m ◦ ( θ ⊗ θ ) ( 2 ) = m i ◦ ( π i ⊗ π i ) ◦ ( θ ⊗ θ ) = m i ◦  ( π i ◦ θ ) ⊗ ( π i ◦ θ )  ( 3 ) = m i ◦ ( g i ⊗ g i ) = g i ◦ m B ( 3 ) = π i ◦ θ ◦ m B Moreo ver, π i ◦ θ ◦ η B ( 3 ) = g i ◦ η B = η i ( 1 ) = π i ◦ η hen ce (4 ) holds. In wh at follo ws w e construct equalizers. Let ( A, m A , η A , ∆ A , ε A ), ( B , m B , η B , ∆ B , ε B ) b e tw o bialgebras and f , g : B → A b e t w o b ialgebra maps. W e den ote b y S := { b ∈ B | f ( b ) = g ( b ) } . Let D b e the sum of all su b coalgebras of B con tained in S . W e already noticed b efore that the family of sub coalgebras of B with this pr op ert y is nonempty and that D is a sub coalgebra of B . The pair ( D , i ) is the equalizer of the morp hisms ( f , g ) in k -BiAlg, where i : D → B is the canonical inclusion. W e only n eed to pr ov e th at D is actually a s ubbialgebra of B . Consider q = P n k =1 d i k ⊗ d j k ∈ D ⊗ D . W e th en h a v e: m A ◦ ( f ⊗ f )( q ) = m A  n X k =1 f ( d i k ) ⊗ f ( d j k )  = m A  n X k =1 g ( d i k ) ⊗ g ( d j k )  = m A ◦ ( g ⊗ g )( q ) No w ha ving in mind that f and g are algebra maps we obtain f  m B ( D ⊗ D )  = g  m B ( D ⊗ D )  , hence m B ( D ⊗ D ) ⊆ S and since m B ( D ⊗ D ) is a sub coalgebra it follo ws that m B ( D ⊗ D ) ⊆ D . Thus D is a su bbialgebra of B and it can b e sho wn as in Theorem 1.1 that the pair ( D , i ) is the equalizer of the morphism s ( f , g ) in k -BiAlg.  LIMITS OF COALGEBRAS, BIALG EBRAS AND HOPF ALGEBRAS 7 As remarke d b efore, we can obtain a description for the equalizers in k -BiAlg similar to the one in Remark 1.2. Thus, w e h a v e the follo wing descrip tion for limits in k -BiAlg: Remark 1.6. Let J b e a small cate gory , F : J → k -BiAlg b e a fun ctor,  Π j ∈ J F ( j ) , ( p j ) j ∈ J  ,  Π u ∈ H om ( J ) F ( cod ( u )) , ( p u ) u ∈ H om ( J )  b e the pro duct in k -BiAlg of the families  F ( j )  j ∈ J , resp ectiv ely  F ( cod ( u ))  u ∈ H om ( J ) and f , g : Π j ∈ J F ( j ) → Π u ∈ H om ( J ) F ( cod ( u )) b e the u nique bialgebra m ap s suc h that p u ◦ f = p cod ( u ) and p u ◦ g = F ( u ) ◦ p dom ( u ) for all u ∈ H om ( J ). W e defin e D = { x ∈ Π j ∈ J F ( j ) | x (1) ⊗ f ( x (2) ) ⊗ x (3) = x (1) ⊗ g ( x (2) ) ⊗ x (3) } . Then the p air  D , ( ϕ j = p j ◦ e ) j ∈ J  is the limit of th e f unctor F , where e : D → Π j ∈ J F ( j ) is the canonical inclus ion. Theorem 1.7. The c ate g ory k -HopfAlg of Hop f algebr as has arbitr ary pr o ducts and e qualizers. In p articular, the c ate gory k - HopfAlg of Hopf algebr as is c o mplete. Pr o of. Let  H i , m i , η i , ∆ i , ε i , S i  i ∈ I b e a family of Hopf algebras and  ( B := Q i ∈ I H i , ∆ , ε, m, η ) , ( π i ) i ∈ I  b e the pro d uct of this family in the category k -BiAlg of bialgebras. The unive rsal pr op ert y of the pro du ct yields an un ique bialgebra map S : B op,cop → B suc h that the follo wing d iagram commutes for all i ∈ I : (5) B π i / / B i B op,cop π i / / S O O B op,cop i S i O O Let H b e the sum of all sub coalgebras C of the bialgebra B suc h that : S ( c (1) ) c (2) = c (1) S ( c (2) ) = η ◦ ε ( c ) for all c ∈ C . The family of sub coalgebras C satisfying th e ab ov e prop erty is nonempt y by the same argumen t used in the pro of of Theorem 1.1. Moreo ver, it is easy to see that (6) S ( h (1) ) h (2) = h (1) S ( h (2) ) = η ◦ ε ( h ) for all h ∈ H . W e will pro v e that H is a b ialgebra and it will follo w b y (6) that H is actually a Hopf algebra w ith the antip o de S | H . First note that η ( k ) = k 1 B ⊆ H . Let h, g ∈ H . W e then ha v e: S  ( hg ) (1)  ( hg ) (2) = S ( h (1) g (1) ) h (2) g (2) = S ( g (1) ) S ( h (1) ) h (2) g (2) = S ( g (1) )( η ◦ ε )( h ) g (2) =  η ◦ ε ( h )  η ◦ ε ( g )  = η ◦ ε ( hg ) In the same w a y it can b e pr o v ed that ( hg ) (1) S  ( hg ) (2)  = η ◦ ε ( hg ). Thus hg ∈ H and H is in deed a b ialgebra. In order to conclude th at S | H is an an tipo d e for H we need to 8 A.L. AGORE pro v e that S ( H ) ⊆ H . Let h ∈ H ; we obtain: S  S ( h ) (1)  S ( h ) (2) = S  S ( h (2) )  S ( h (1) ) = S  h (1) S ( h (2) )  = S  η ◦ ε ( h )  = η ◦ ε ( h ) = η ◦ ε  S ( h )  A similar computation shows that we also ha v e S ( h ) (1) S  S ( h ) (2)  = η ◦ ε  S ( h )  for all h ∈ H . Hence H is a Hopf algebra w ith S | H as an tip o de. T o end the pro of we still need to show th at  ( H , m, η , ∆ , ε, S | H )  , ( q i ) i ∈ I  is the pr o duct of the family  H i , m i , η i , ∆ i , ε i , S i  i ∈ I in the category k -HopfAlg, where q i := π i ◦ j for all i ∈ I and j : H → B is the canonical inclusion. Let K b e a Hopf algebra with antipo d e S K and f i : K → H i b e a family of Hopf algebra maps for all i ∈ I . Since B is the pro d uct in k -BiAlg of the ab ov e family of Hopf algebras, there exist a un ique morph ism of b ialgebras f : K → B suc h that th e follo wing diagram comm utes: (7) K f   f i A A A A A A A A B π i / / H i Using the f act that f i is a Hopf algebra m ap we hav e : π i ◦ S ◦ f ( 5 ) = S i ◦ π i ◦ f ( 7 ) = S i ◦ f i = f i ◦ S K ( 7 ) = π i ◦ f ◦ S K for all i ∈ I . By the same argumen t used in the p ro of of theorem Theorem 1.5 it follo ws that: (8) S ◦ f = f ◦ S K Th us, for all k ∈ K we hav e: S ( f ( k ) (1) ) f ( k ) (2) = S ( f ( k (1) )) f ( k (2) ) ( 8 ) = f  S K ( k (1) )  f ( k (2) ) = f  S K ( k (1) ) k (2)  = f ( k ) Hence f ( K ) ⊆ H . Th us, w e obtained an unique Hopf alg ebra map f : K → H su c h that q i ◦ f = f i for all i ∈ I . No w, since the forgetful functor U : k -HopfAlg → k - BiAlg h as a left adjoint (see [11 ]) it f ollo ws that, in p articular, U p reserv es pro d u cts. That is, H = B and the map S obtained in (5) is actually an ant ip o d for B . Thus,  ( B , m, η , ∆ , ε, S )  , ( π i ) i ∈ I  is the pro d u ct of the family  H i , m i , η i , ∆ i , ε i , S i  i ∈ I in th e LIMITS OF COALGEBRAS, BIALG EBRAS AND HOPF ALGEBRAS 9 catego ry k -HopfAlg. No w let f , g : H → K b e tw o Hopf algebra morphisms and S := { h ∈ H | f ( h ) = g ( h ) } , whic h is just a k -su b space of H . Let D b e the su m of all sub coalgebras of H con tained in S . Again, the f amily of sub coalgebras of H included in S is not empty by the same argumen t used in Theorem 1.1. A simple computation shows that D is in fact a Hopf subalgebra of H . Moreo ver, ( D , i ) is the equalizer in the catego ry k -HopfAlg of the pair ( f , g ) where i : D → H is the canonical in clusion.  Remark 1.8. Let J b e a small category , F : J → k -HopfAlg b e a functor,  Π j ∈ J F ( j ) , ( p j ) j ∈ J  ,  Π u ∈ H om ( J ) F ( cod ( u )) , ( p u ) u ∈ H om ( J )  b e the pr o duct in k -HopfAlg of th e fami- lies  F ( j )  j ∈ J , resp ectiv ely  F ( cod ( u ))  u ∈ H om ( J ) and f , g : Π j ∈ J F ( j ) → Π u ∈ H om ( J ) F ( cod ( u )) b e the u nique Hopf algebra maps suc h that p u ◦ f = p cod ( u ) and p u ◦ g = F ( u ) ◦ p dom ( u ) for all u ∈ H om ( J ). W e d efine D = { x ∈ Π j ∈ J F ( j ) | x (1) ⊗ f ( x (2) ) ⊗ x (3) = x (1) ⊗ g ( x (2) ) ⊗ x (3) } . Then the p air  D , ( ϕ j = p j ◦ e ) j ∈ J  is the limit of th e f unctor F , where e : D → Π j ∈ J F ( j ) is the canonical inclus ion. The k ey role in the construction of the pr o duct in the category of coalgebras wa s pla y ed b y the fact that the forgetful functor f rom the category of coalgebras to the category of v ecto r spaces has a r igh t adjoin t. It is therefore natural to ask if the conclusion r emains true for the categ ory of R -corings ([4]). Let R b e a ring, R -Corings b e the category of R -corings and R M R b e the category of R -bimo dules. Problem: Do es ther e e xist a right adjoint for the f or getful functor F : R − C or ing s → R M R ? A cknowledgement The author wishes to thank Professor Gigel Militaru, who suggested the pr oblems studied here, f or his great sup p ort and f or the useful comments from whic h this manuscript has b enefitted. Referen ces [1] E. Ab e, Hopf Algebras, Cambridge Un ivers it y Press, Cambridge, 1977. [2] A.L. Agore, Categorical constuctions for Hopf algebras, to app ear in Commun. in Algebra , arXiv:0905.26 13 v3 [3] N. An druskiewitsc h, J. Devoto, Ex tensions of H opf algebras, Algebra i Analiz , (1995), V olume 7 , Issue 1 , 22-61 [4] T. Brzezi ´ nski, R. Wisbauer, Corings and comodu les, Cam bridge U n ivers it y Press, 2003. [5] S. D ˇ ascˇ alescu, C. Nˇ astˇ asescu, S ¸ . Raianu, Hopf algebras: An introduction, Pure and Applied Math. 235 (2000), Marcel Dekker [6] Mac Lane, S ., Categories for the wo rking mathematician, GTM 5, Springer, New Y ork ( 1998) [7] B. Pareigi s, Ad v anced Algebra. A v ai lable at: http://w ww.mathematik.uni-m uenchen.de/ ∼ pareigis /V orlesungen/01WS/adv alg.p df [8] B. P areigis, Lectures on quantum groups and noncommutative geometry . A v ail able at: http://w ww.mathematik.uni-m uenchen.de/ ∼ pareigis /V orlesungen/02SS/QGandNCG.pd f [9] H.-E. Porst, On corings and como dules, Arch. Math. (Brno) , 42 (2006), 419-425 10 A.L. AGORE [10] M.E. Sweedler, Hopf Algebras, Benjamin New Y ork, 1969. [11] M. T akeuc hi, F ree H opf algebras generated by coalgebras, J. Math. S oc. Japan 23 (1971), 561-582 Dep ar tment of Ma thema tics, A cademy of Economic Studies, Pia t a Ro mana 6, R O-010374 Bucharest 1, Romania E-mail addr ess : ana.a gore@fmi.u nibuc.ro