COALGEBRAS, BRAIDINGS, AND DISTRIBUTIVE LAWS

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1 COALGEBRAS, BRAIDINGS, AND DISRIBUIVE LAWS SEFANO KASANGIAN, SEPHEN LACK, AND ENRICO M. VIALE o Aurelio Carboni on his 60th birthday ABSRAC. We show, for a monad, that oalgebra strutures on a -algebra an be desribed in terms of braidings, provided that the monad is equipped with an invertible distributive law satisfying the Yang-Baxter equation. 1. Introdution he aim of this note is to provide an equivalent desription of -oalgebra strutures on a -algebra, for a monad equipped with a speial kind of distributive law on a ategory C, and the omonad indued by the adjuntion C L R Alg() L R Our interest for suh oalgebras is motivated mainly by lassial desent theory: let f : R S be a morphism of ommutative unital rings, and onsider the indued funtor f!: R-mod S-mod defined by f!(n) = N R S (where S is seen as an R-module by restrition of salars). he desent problem for f onsists in reognizing when an S-module is of the form f!(n) for some R-module N. A lassial theorem [6, 2, 7], whih establishes a deep link between desent theory and the theory of (o)monads, asserts that, if f is faithfully flat, then an S-module is of the form f!(n) if and only if it is equipped with a -oalgebra struture, where is the omonad on S-mod indued by the adjuntion R-mod f! S-mod f f! f and f is the restrition of salars funtor. (For this reason, a -oalgebra struture on an S-module M is sometimes alled a desent datum for M.) It is also well-known that, for any morphism f : R S of ommutative unital rings, there are several equivalent ways Seond author supported by the Australian Researh Counil; third author by FNRS grant Mathematis Subjet Classifiation: 18C15, 18C20, 18D10, 16B50. Key words and phrases: Desent data, monads, distributive laws, Yang-Baxter equation. 1

2 of desribing what a -oalgebra struture for an S-module is, and a natural problem is to lift to a ategorial level these other desriptions of -oalgebra strutures. In a reent paper [11], Menini and Stefan, extending results by Nuss [12] on nonommutative rings, replae the situation 2 R-mod f! S-mod f f! f by C L R Alg() L R where is a monad on an arbitrary ategory C (indeed, even in the non-ommutative situation, f : S-mod R-mod is a monadi funtor, so that S-mod is equivalent to the ategory of algebras for the monad on R-mod indued by the adjuntion f! f ). In this ontext, they prove that, if the monad is equipped with a ompatible flip K : 2 2, then to give a -oalgebra struture on a -algebra X is equivalent to giving a symmetry on X, that is an involution X X satisfying some suitable onditions. Unfortunately, the following natural example, whih is a diret generalization of the lassial ase of ommutative rings, does not fit into their general ontext: let C be a braided monoidal ategory and let S be a monoid in C, then the braiding S,S : S S S S indues a natural isomorphism K : 2 2 on the monad = S : C C, but this natural isomorphism is not a flip unless the braiding is a symmetry and the monoid is ommutative. In this note we adapt the notions of ompatible flip and symmetry to enompass the previous example, as well as another example oming from the theory of bialgebras. In Setion 2 we introdue the notion of BD-law on a monad as a speial ase of distributive law in the sense of Bek [1]. Using a BD-law K, we an define K-braidings on a -algebra, and we want to show that K-braidings orrespond bijetively to -oalgebra strutures. here are two different methods: in Setion 3 we use K-braidings to define a ategory Brd(, K) equipped with a forgetful funtor V : Brd(, K) Alg(). We show, using the Bek riterion [10], that V is omonadi; and that the orresponding omonad is, so that Brd(, K) is isomorphi to Coalg( ). In Setion 4 we give a different proof based as far as possible on general fats about monads and distributive laws. his seond proof is quite long, but it seems to us of some interest, sine it shows that the bijetion between K-braidings and -oalgebra strutures is the natural bijetion indued by a pair of adjoint funtors. he desription of K-braidings we obtain in this way is slightly different, but in fat equivalent, to that used in Setion BD-laws on a monad o begin, we fix notation. A monad on a ategory C is a triple = ( : C C, m: 2, e: Id C )

3 3 onsisting of a funtor, and natural transformations m and e making the diagrams e 2 e (1) (2) m 3 m 2 m (3) 2 m m ommute. A -algebra is a pair (X, x: X X) in C suh that the diagrams ex X X (4) x 1 X x X (5) X x X x ommute. Given two monads and S on the same ategory C, a distributive law of over S is a natural transformation K : S S suh that the diagrams e (6) e K S S S es (7) Se K S S S 2 KS S S SK S 2 2 S K S K S 2 m S (8) K S m ms S (9) K S Sm ommute. We refer to [1, 3] for more details on monads and distributive laws. When the natural transformation K is an isomorphism, the definition of distributive law an be simplified, as in the following lemma: Lemma 1 Consider two monads and S on a ategory C, and a natural isomorphism K : S S. hen (8) implies (6) and (9) implies (7). Moreover, K satisfies (8) and (9) iff K 1 does. Proof. We prove that (9) implies (7); the proof of the other impliation is similar, and the rest of the statement is obvious. he proof is ontained in the following diagram, in

4 4 whih unlabelled regions ommute by naturality: es S K S S es Sm (2) Se S K S S e S 2 Se S es K 1 S K S K 1 2 S K S (9) (1) ms S. e S S K Definition 2 Let be a monad on a ategory C. A BD-law on is a natural transformation K : 2 2 suh that (B) K satisfies the Yang-Baxter equation: 3 K 3 K 3 K 3 K (10) 3 K 3 K (D) K is a distributive law (that is, it satisfies equations (6 9).) A BCD-law on is a BD-law suh that (C) the monad is K-ommutative: 2 K 2 (11) m m A BD-law or BCD-law is said to be invertible if the natural transformation K is so. Remark 3 One again, as stated in Lemma 1 for the distributivity onditions, a natural isomorphism K : 2 2 satisfies onditions (B) or (C) iff K 1 does. Remark 4 If C is an arbitrary monoidal ategory, and is a monoid in C, one an define a BD-law or BCD-law on as above. A monoid equipped with an invertible BCD-law has been alled a quasi-ommutative monoid by Davydov [5], sine the BCD-law provides a kind of loal braiding with respet to whih the monoid is ommutative.

5 Remark 5 If K is involutive that is, K 2 = 1 eah of the onditions (8) and (9) implies the other. his fat, together with Lemma 1, means that ompatible flips in the sense of Menini and Stefan [11] are preisely the involutive BCD-laws. Example 6 Let C = (C,, I,...) be a monoidal ategory and S = (S, m S, e S ) a monoid in C. he monoid S indues a monad on C in the following way: = S : C C = 1 m S : X S S X S ex = 1 e S : X X I X S 2.1. If C is braided, with braiding = { X,Y : X Y Y X}, then there is an invertible BD-law K on defined by = 1 S,S : X S S X S S. In this ase, ondition (B) is preisely the Yang-Baxter equation; by naturality of the braiding, onditions (8) and (9) redue to the following equations, whih hold in any braided monoidal ategory: = = his BD-law is a BCD-law preisely when the monoid S is ommutative; it is involutive if and only if S,S is so, in partiular if the braiding is a symmetry Let K be a field and take as C the ategory of K-vetor spaes. Let H be a obraided bialgebra with universal form r : H H I; there is a natural isomorphism of H- omodules r V,W : V W W V defined by V W τ W V H W H V 1 τ 1 H H W V r 1 1 W V where is the oation and τ is the standard twist: see [9]. If S is any H-omodule algebra (in partiular, one an take S = H), then we have an invertible BD-law K on defined by = 1 r S,S : X S S X S S. Indeed, onditions (8-10) follow from [9, Proposition VIII.5.2], using the fat that the multipliation m S : S S S is a homomorphism of H-omodules. Example 7 If f : R S is a morphism of unital rings, with R ommutative, we an speialize Example 2.1 by taking C = R-mod, so that K is defined by the standard twist S S S S. his is possible also if R is not ommutative, provided its image lies in the entre of S : taking now C = R-R-bimod, the standard twist on S an be defined and gives one again an invertible BD-law on. If we drop the entrality ondition the standard twist an no longer be defined, but one an use the additivity of the ategory of bimodules to define a different BD-law. In fat, if C is an additive ategory and is any monad on C, then there is an involutive BCD-law K on defined by K = (e + e) m 2 ; see [11]. his ase generalizes results on non-ommutative rings established in [4, 12]. 5

6 3. Coalgebras and braidings For the reader s onveniene, let us reall how the definition of oalgebra for a omonad speializes when the omonad is of the form. Definition 8 Let = (, m, e) be a monad on a ategory C and onsider the omonad on Alg() indued by the adjuntion 6 C L R Alg() L R A -oalgebra struture on a -algebra (X, x: X X) is a morphism r : X X suh that X r 2 r X X X X x (12) 1 X r X (13) x r (14) r r X X X ex We denote by -oalg(x, x) the set of -oalgebra strutures on a -algebra (X, x). Remark 9 For all X C, the morphism ex : X is a -oalgebra struture on the free -algebra L X = ( X, ). his is the (objet part of the) anonial omparison funtor C Coalg( ). Definition 10 Let = (, m, e) be a monad on a ategory C and let K : 2 2 be a BD-law. A K-braiding on a -algebra (X, x: X X) is a morphism : X X suh that X X ex (15) X 1 X x (16) x (17) X X We denote by K-Brd(X, x) the set of K-braidings on a -algebra (X, x). Remark 11 If K is invertible, ondition (17) means that is a morphism : (X, x) (X, x) in Alg(), where is the lifting of on Alg() indued by the distributive law K 1. Remark 12 We shall see in Proposition 16 that if K is invertible or involutive then the same is true of any K-braiding, whene by Corollary 18 it will follow that if K is an involutive BCD-law, then K-braidings are preisely symmetries in the sense of Menini and Stefan [11].

7 Remark 13 If K is a BD-law on, then is a K-braiding on L X, for all X C. Indeed, onditions (16) and (17) orrespond respetively to onditions (10) and (9), while ondition (15) is the pasting of (1) and (6). In the bijetion stated in Corollary 15, orresponds to the -oalgebra struture ex of Remark 9. If is a monad on a ategory C and K : 2 2 is a BD-law, we write Brd(, K) for the ategory having pairs (X, x) Alg(), K-Brd(X, x) as objets. An arrow f : (X, x), (X, x ), in Brd(, K) is an arrow between the underlying -algebras suh that the diagram ommutes. We are ready to state our main result. X f X X f X heorem 14 Let = (, m, e) be a monad on a ategory C and let K : 2 2 be a BD-law on. he forgetful funtor V : Brd(, K) Alg() is omonadi, and the orresponding omonad on Alg() is the omonad indued by the adjuntion L R between C and Alg(). Proof. We show that V has a left adjoint and then apply Bek s theorem. he free - algebra funtor L : C Alg() fatorizes as L = V J, where J : C Brd(, K) is the funtor sending an objet X of C to the algebra ( X, ) equipped with the K-braiding :. he ounit ɛ : L R 1 may be seen as a natural transformation V JR = L R 1. For an objet (X, x, ) of Brd(, K), write β : X X for the omposite and observe that, by ommutativity of X ex X X 7 X e X x ex x (17) X ex X X

8 8 and X e X ex (16) X e X ex this makes β a map in Brd(, K) from (X, x, ) to ( X,, ). his gives the omponent at (X, x, ) of a natural transformation β : 1 JR V. he triangle equation ɛv.v β = 1 is preisely equation (15), while the other triangle equation JR ɛ.βjr = 1 follows easily from the definitions of BD-law and of -algebra. hus there is an adjuntion V JR, whih learly indues the same omonad as L R. It remains to verify the Bek ondition. Let f, g : (X, x, ) (Z, z, ) be morphisms in Brd(, K), and let (W, w) i (X, x) u f g v (Z, z) be a split equalizer diagram in Alg(), with ui = 1, iu = gv, and fv = 1. Sine i is (split) moni, the only possibility for a K-braiding : W W on (W, w) ompatible with i is given by W i X X u W and we need only hek that this does indeed give a braiding; the fat that the resulting diagram is an equalizer in Brd(, K) is then obvious. hus we must hek equations (15,16,17); instead, we allow the reader to ontemplate the following diagrams at his or her leisure: W i X X u W ew ex W i X 1 X u W 1 x w

9 9 2 u 2 W KW 2 W 2 g 2 Z 2 i 2 g 2 Z 2 i 2 W KW 2 W 2 i 2 i 2 f 2 v 1 2 f 2 u 2 i 2 u 2 W KW 2 W 2 W 2 i KW 2 W 2u 2 W 2 i w x W i X mw X u W. Corollary 15 Let = (, m, e) be a monad on a ategory C and let K : 2 2 be a BD-law on. For (X, x) a -algebra, onsider the map Ψ (X,x) : K-Brd(X, x) -oalg(x, x) (: X X) (r : X ex X X ) and the funtor Ψ: Brd(, K) Coalg( ) (X, x), f (X, x ), (X, x, Ψ()) f (X, x, Ψ( )) 1. he funtor Ψ is an isomorphism of ategories; 2. he map Ψ (X,x) is bijetive. Proof. Sine the forgetful funtor V : Brd(, K) Alg() is omonadi, the indued omparison funtor Ψ: Brd(, K) Coalg( ) is an isomorphism of ategories and so it indues a bijetion on objets.

10 Proposition 16 For a BD-law K : 2 2 we have the following fats about a K- braiding : X X on an algebra (X, x): 1. he diagrams 10 3 X 3 X 2 3 X X (18) x X 2 x (19) X X x ommute; 2. If K is invertible then so is ; 3. If K is involutive then so is ; 4. If K is a BCD-law then the diagram ommutes. X X (20) x x X Proof. In eah ase the proof goes as follows. Modify the definition of K-braiding so that the extra ondition is assumed part of the struture, then hek that the modified ategory Brd(, K) is still omonadi via the same omonad. his involves (i) proving that the ofree objets ( X,, ) have the required property, and (ii) proving that in the split equalizer diagram in the proof of heorem 14, the indued morphism = u.. i : W W satisfies the ondition if : X X does so. In eah ase (i) is entirely straightforward: for example, the fat that the ofree objets satisfy (18) is preisely (9). We therefore hek only (ii). 1. If satisfies (18), then so does by ommutativity of the diagram 2 W mw 2 i 2u 2 W KW 2 W W i X x 2 u w X u W

11 while (19) is obtained by pasting together (17) and (18). 2 and 3. If is invertible, then a straightforward alulation shows that u. 1. i is inverse to u.. i. 4. If x = x then the diagram ommutes and so w = w. X i x W X i w W 1 x u X u W W w Remark 17 If K is a distributive law, ondition (18) means that is a morphism : (X, x) (X, x) in Alg(), where is the lifting of to Alg() indued by the distributive law K. In fat we an use the Proposition to give two alternative formulations of the definition. One of them will be used in the following setion, the other to make the onnetion with the symmetries of [11]. Corollary 18 In the definition of K-braiding, ondition (17) an be replaed by (19), while if K is a BCD-law then (15) an be replaed by (20). Proof. We have seen that for a K-braiding (19) always holds; onversely (17) follows easily from (19) by omposing with e X : X. Similarly, if K is a BCDlaw then (20) holds for any K-braiding, while (15) follows from (20) by omposing with ex : X X. If K is an involutive BCD-law on a monad, then a symmetry on a -algebra (X, x), was defined in [11] to be an involution : X X satisfying (16, 17, 20); ombining the proposition and the orollary one now sees as promised that this is preisely a K-braiding Another proof We proved our main theorem in the previous setion; here we provide an alternative proof, whih may be of interest to some readers. It exhibits the bijetion whih is the objet part of the isomorphism Φ of Corollary 15 as being part of the natural bijetion (between hom-sets) of an adjuntion. o do this, we use the definition of K-braiding involving (19) rather than (17); see Corollary 18.

12 Proposition 19 Let = (, m, e) be a monad on a ategory C and let K : 2 2 be an invertible BD-law on. For (X, x) a -algebra, we have a bijetion given by Ψ (X,x) : K-Brd(X, x) -oalg(x, x) K-Brd(X, x) = -oalg(x, x) (: X X) (r : X ex X X ) Ψ 1 (X,x) : -oalg(x, x) K-Brd(X, x) (r : X X) (: X r x X ) As explained above, the proposition an be dedued from Corollary 15; our alternative proof oupies the remainder of the paper. Let and S be monads on a ategory C. A morphism of monads is a natural transformation ϕ: S suh that 12 S 2 ϕ 2 2 m S (21) ϕ m e Id C e S (22) ϕ Following [1], suh a morphism ϕ indues a funtor ϕ : Alg() Alg(S) defined by ϕ (X, x: X X) = (X, ϕx x: SX X X). Moreover, ϕ has a left adjoint ϕ!: Alg(S) Alg() sending an S-algebra (Y, y) to the objet ϕ!(y, y) in the oequalizer y SY Y ϕy my 2 Y q ϕ!(y, y) in Alg(), provided that suh a oequalizer exists. Indeed, if f : (Y, y) ϕ (X, x) is a morphism of S-algebras, then x. f : Y X oequalizes y and my. ϕy. Conversely, from g : ϕ!(y, y) (X, x), we get g.q.ey : (Y, y) ϕ (X, x). If K : S S is a distributive law of over S, we an onsider the omposite monad S on C. Expliitly, S = (S : C C, S S SK S 2 2 mm S, Id C ee S ) Moreover, there are two morphisms of monads as in S ɛ 1 =Se ɛ 2 =e S.

13 Now, for any -algebra (X, x: X X) and S-algebra (Y, y : SY Y ), the adjuntion ɛ 2! ɛ 2 gives a natural bijetion 13 Ψ: Alg(S)[ɛ 2!(X, x), ɛ 1!(Y, y)] Alg()[(X, x), ɛ 2ɛ 1!(Y, y)] he bijetion of Proposition 19 will turn out to be a partiular ase of this natural bijetion. o see this, we give an expliit desription of Ψ and of the hom-sets involved. First of all, let us reall from [1] that the oequalizer defining ɛ 2!(X, x) always exists. In fat, it is given by the solid part of the following diagram in Alg(S) Se X SeX S X S x S S X Sx SX with ation on SX given by S SX S S 2 X S 2 x S 2 X SX Indeed, one easily verifies that the dotted arrows satisfy the equations for a split oequalizer [2]. Reall also that if f is a morphism of S-algebras oequalizing S x and S, then the indued S-algebra map out of SX is f.sex. Unfortunately, the existene of the oequalizer defining ɛ 1!(Y, y) is not automati. We need the following lemma, whih makes sense beause of Lemma 1: Lemma 20 Consider two monads S and on a ategory C, and let K : S S be an invertible distributive law of over S. Consider the omposite monad S indued by K and the omposite monad S indued by K 1. hen K : S S is a morphism of monads, and it satisfies the following equations where η 1 = e and η 2 = es. S η 2 η 1 S K ɛ 1 ɛ 2 S Proof. he equations ɛ 1 = Kη 2 and ɛ 2 = Kη 1 are onditions (6) and (7), and they imply ondition (22). As far as ondition (21) is onerned, let us give the idea of the proof, instead of the omplete alulation. (his ould be formalized using the string alulus

14 of [8].) hink of K as a braiding between and S. hen ondition (21) amounts to the following equation = whih an be proved using the distributivity equations three times. Indeed, the distributivity admits the following graphial representation (this is ondition (8)) = 14 As a onsequene of the previous lemma, ɛ 1!: Alg(S) Alg(S) an be obtained as Alg(S) η 2! Alg(S) (K 1 ) Alg(S) and then ɛ 1!(Y, y) an be desribed by the following oequalizer in Alg(S) esy ey (K 1 ) (L S (SY )) KSY L S (SY ) Sy my S y m Y.SKY (K 1 ) (L S (Y )) KY L S (Y ) y Y with ation on Y given by S 2 Y SmY S Y K 1Y SY y Y. We are now ready to desribe the bijetion Ψ. Lemma 21 Consider two monads S and on a ategory C, and let K : S S be an invertible distributive law of over S. For any -algebra (X, x) and S-algebra (Y, y) :

15 15 (i) he hom-set Alg(S)[(Y, y), ɛ 1(ɛ 2!(X, x))] is the set of morphisms r : Y SX suh that SY Sr S 2 X y Y (12) r ommutes. (ii) he hom-set Alg(S)[ɛ 1!(Y, y), ɛ 2!(X, x)] is the set of morphisms : Y SX suh that 2 SY KY S Y S SSX SX 2 y 2 Y (19) SX ommutes. my Y SX Sx S X (iii) he natural bijetion Ψ: Alg(S)[ɛ 1!(Y, y), ɛ 2!(X, x)] Alg(S)[(Y, y), ɛ 1(ɛ 2!(X, x))] indued by the adjuntion ɛ 1! ɛ 1 is given by Ψ(): Y ey Y SX Ψ 1 (r): Y r SX S X Sx SX. Proof. Following the desription of ϕ! ϕ given at the beginning of this setion, we have Ψ() and Ψ 1 (r) given respetively by Y ey Y e Y S Y K 1 SY y Y SX Y e Y S Y S r S SX S SS X n X S X Sx SX whih are easily seen to redue to formulas given in (iii). For r : Y SX, to be a morphism of S-algebras means that SY Sr S 2 X SeSX S SX y Y r SX S 2 X S 2 x S S 2 X ommutes, whih immediately redues to (12) by (6) and (4).

16 16 For : Y SX, to be a morphism of S-algebras means ommutativity of S 2 Y S S SX S SS X SSx SSX SmY S Y K 1 Y SY y Y SX whih, by (8) and (9), redues to (19). his is the best we an do working at this level of generality. From now on, we assume S = and (X, x) = (Y, y). We now ombine the next two lemmas with the previous one to omplete the proof of Proposition 19. Lemma 22 Let be a monad on a ategory C, and let K : 2 2 be an invertible distributive law on. Fix a -algebra (X, x). In the bijetion of Lemma 21, the map r : X X satisfies ondition (13) iff = Ψ 1 (r): X X satisfies ondition (15). Proof. Condition (15) says preisely that Ψ() satisfies (13). Lemma 23 Let be a monad on a ategory C, and let K : 2 2 be an invertible BD-law on. Fix a -algebra (X, x). In the bijetion of Lemma 21, the map r : X X satisfies ondition (14) iff = Ψ 1 (r): X X satisfies ondition (16). Proof. (14) (16) : 2 X 2 r 2 X 2 r 3 X K X 2 r (14) 3 X 3 ex 3 X 3 r 4 X 3 X 3 ex K X 3 X 2 x 3 X 3 r 4 X (1) 1 (12) 2 x 2 2 X 2 r 3 X (10) 3 X 2 X K X K X 2 x 3 X 3 X 2 r ()

17 17 (16) (14) : X ex ex ex X e X 2 e X X X (7) 2 e X X X (6) e X X ex (16) (6) ex he proof of Proposition 19 is now omplete. Referenes [1] J. Bek, Distributive laws, in Seminar on riples and Categorial Homology heory (Leture Notes in Mathematis 80), pp , Springer, Berlin, [2] F. Boreux, Handbook of ategorial algebra I, Cambridge University Press, [3] F. Boreux, Handbook of ategorial algebra II, Cambridge University Press, [4] M. Cipolla, Disesa fedelmente piatta dei moduli, Rend. Cir. Mat. Palermo 25:43 46, [5] A. Davydov, Quasiommutative monoids, Leture at Australian Category Seminar, 21 January [6] A. Grothendiek, ehnique de desente et théorèmes d existene en géométrie algébrique, I : Généralités, desente par morphismes fidèlements plats, Séminaire Bourbaki 190 ( ). [7] G. Janelidze and W. holen, Faets of desent III: monadi desent for rings and algebras, preprint, [8] A. Joyal and R. Street, Geometry of tensor alulus I, Adv. Math. 88:55 112, [9] C. Kassel, Quantum groups (Graduate exts in Mathematis 155) Springer-Verlag, New York, [10] S. Ma Lane, Categories for the working mathematiian (Graduate exts in Mathematis 5), Springer, New York-Berlin, 1971.

18 [11] C. Menini and D. Stefan, Desent theory and Amitsur ohomology of triples, J. Algebra 266: , [12] P. Nuss, Nonommutative desent and non-abelian ohomology, K-heory 12:23 74, Dipartimento di Matematia Università di Milano Via Saldini 50 I Milano, Italia Shool of Quantitative Methods and Mathematial Sienes University of Western Sydney Loked Bag 1797 Penrith South DC NSW 1797 Australia Département de Mathématique Pure et Appliquée Université atholique de Louvain Chemin du Cylotron 2 B 1348 Louvain-la-Neuve, Belgique Stefano.Kasangian@mat.unimi.it, s.lak@uws.edu.au, vitale@math.ul.a.be 18