Factorizable enriched categories and applications

F A CTORIZABLE ENRICHED CA TEGOR IE S AND APPLICA TIONS AURA B ˆ ARDES ¸ AND DRAGOS ¸ S ¸ TEF AN Abstract. W e deﬁne the t wisted te nsor pro duct o f t wo enrich ed categories, which generalizes v arious sorts of ‘pro duct s’ of algebraic structures, including the bicrossed pro duc t of groups, the t wisted tensor pro duct of (co)algebras and the double cross product of bialgebras. The k ey ingredien t i n the deﬁnition is the notion of simple t wisting systems betw een t w o enric hed categories. T o give examples of sim ple twisted tensor products w e int ro duce matc hed pairs of enrich ed categories. Sev eral other examples related to ordinary catego ries, p oset s and group oids are also discussed. Contents Int ro duction 1 1. Preliminarie s a nd notation. 3 2. F actor izable M -categor ies and twisting systems. 7 3. Matched pa irs of enric hed categ ories. 15 4. Examples. 18 References 25 Introduction The most conv enient wa y to explain what we mean by the facto rization problem of an algebraic structure is to consider a concrete ex ample. Chronolog ically sp eaking, the ﬁrs t pr oblem of this t yp e was studied for groups, see for instance [Mai, O re, Za, Sz, T ak]. Let G b e a g r oup. Let H and K de no te tw o subgro ups of G. One says that G factor izes throug h H and K if G = H K and H ∩ K = 1 . Ther efore, the factor ization problem for gr o ups mea ns to ﬁnd nece s sary and suﬃcient conditions whic h ensure that G factorizes thro ug h the giv en s ubg roups H a nd K . Note that, if G factor izes thro ugh H and K then the mult iplication induces a canonical bijective map ϕ : H × K → G, which can b e used to transp ort the group struc tur e of G on the C a rtesian pro duct of H and K . W e sha ll ca ll the re s ulting gro up structure the bicro ssed pro duct of H and K , and we shall de no te it by H ⋊ ⋉ K . The iden tit y element of H ⋊ ⋉ K is (1 , 1), a nd its g roup law is uniquely determined by the ‘twisting’ map R : K × H → H × K, R ( k , h ) := ϕ − 1 ( k h ) . Obviously , R is induced b y a couple of functions ⊲ : K × H → H and ⊳ : K × H → K such that R ( k , h ) = ( k ⊲ h, k ⊳ h ) . Using this notation the m ultiplica tion on H ⋊ ⋉ K ca n be w r itten as ( h, k ) · ( h ′ , k ′ ) = ( h ( k ⊲ h ′ ) , ( k ⊳ h ′ ) k ′ ) . The group axio ms easily imply that ( H , K, ⊲, ⊳ ) is a matc hed pa ir of gr oups, in the sense of [T ak]. Conv ersely , any bicrosse d pro duct H ⋊ ⋉ K factorizes thr ough H and K . In conc lus ion, a group G facto rizes through H and K if and only if it is isomor phic to the bicros sed pr o duct H ⋊ ⋉ K asso ciated to a certain matc hed pair ( H , K, ⊲, ⊳ ) . 2000 Mathematics Subje c t Classiﬁc ation. Primary 18D20; Secondary 18D10; 16Sxx. Key wor ds and phr ases. Enriched cat egory , t wisting system, twisted te nsor pr oduct, matche d pair, bicrossed product. 1 2 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN Similar ‘products’ are kno wn in the literature for man y o ther alg ebraic structures. In [Be], for a distributive la w λ : G ◦ F → F ◦ G b et ween t wo monads, Jon Beck deﬁned a mona d str ucture on F ◦ G, which can b e regar ded as a sort of bicro ssed pro duct of F and G with resp ect to the t wisting na tural tr ansformation λ. The twisted tensor pro duct of tw o K - algebras A and B with r espect to a K -linear twisting map R : B ⊗ K A → A ⊗ K B was inv estigated for instance in [Ma1], [T am], [CSV], [CIMZ], [LP oV] and [JLPvO]. It is the ana logous in the categor y of a ssocia tiv e a nd unital algebr as of the bicross ed pro duct o f gr o ups. The clas sical tensor pro duct of tw o a lgebras, the gr aded tensor pro duct of tw o graded a lgebras, skew algebr as, smash pr oducts, Ore extens io ns, generaliz ed quaternion a lgebras, quantum a ﬃne space s and q uan tum tori a re all e x amples o f twisted tensor pr oducts. Another clas s o f exa mples, including the Drinfeld double and the do uble cr o ssed pro duct of a matched pair o f bialgebras, comes from the theory of Hopf algebra s, see [Ma2]. Some of these constructions hav e b e en generalized for bia lgebras in monoidal categor ies [BD] and bimona ds [BV]. Enriched categor ies have be e n playing an increa singly imp ortant role not only in Algebra , but also in Algebra ic T op ology and Mathema tica l Ph ysics, for instance. They gener alize usual ca te- gories, linear ca tegories, Hopf mo dule ca tegories and Hopf como dule categories. Monoids, algebras , coalgebr as and bialgebr a s may b e r egarded a s enr iched categor ies with one ob ject. Our aim in this pap er is to ‘catego rify’ the factorizatio n pr oblem, i.e. to answer the question when an enriched catego r y factorizes thr ough a couple o f enr ic hed subca tegories. Finding a solution at this level of g eneralit y would allow us to approa c h in an unifying w ay a ll factorization pro blems that we hav e alr eady mentioned. Mor e over, it would also provide a gener al metho d for pro ducing new non- tr ivial examples of enriched ca tegories. In order to deﬁne factorizable enriched categorie s , we need some notation. Let C be a s mall enriched categ o ry ov er a monoidal category ( M , ⊗ , 1 ) . Let S denote the set of ob jects in C . F or the hom- o b jects in C we use the notation x C y . The comp osition of morphisms and the identit y morphisms in C ar e deﬁned b y the maps x c y z : x C y ⊗ y C z → x C z and 1 x : 1 → x C x , respectively . F or details, the r eader is r eferred to the next section. W e assume that A and B are tw o enriched sub c ategories of C . The inclusion functor α : A → C is given b y a family { x α y } x,y ∈ S of morphisms in M , wher e x α y : x A y → x C y . If β is the inclusion of B in C , then for x, y and u in S we deﬁne x ϕ u y : x A u ⊗ u B y → x C y , x ϕ y z = x c u y ◦ ( x α u ⊗ u β y ) . Assuming that all S -indexed families of ob jects in M ha ve a co product it follows that the maps { x ϕ u y } u ∈ S yield a unique morphism x ϕ y : L u ∈ S x A u ⊗ u B y → x C y . W e say that C factorizes through A and B if a nd only if all x ϕ y are invertible. An enriched category C is called factoriz a ble if it factorizes through A and B , for some A and B . In Theor em 2.3, o ur ﬁrst main result, under the additiona l assumption that the tenso r pro duct o n M is distributive ov er the direct sum, we show tha t to every M - c ategory C that factorizes through A and B cor responds a t wisting sys tem betw een B and A , that is a family R := { x R y z } x,y ,z ∈ S of morphisms x R y z : x B y ⊗ y A z → L u ∈ S x A u ⊗ u B z which a re compatible with the co mposition and iden tit y ma ps in A and B in a certain sense. T rying to a ssocia te to a twisting system R := { x R y z } x,y ,z ∈ S an M -catego r y we encountered some diﬃculties due to the fact that, in general, the image of x R y z is to o big. Consequently , in this pap er we fo cus on the pa rticular class o f twisting sys tems for which there is a function |· · · | : S × S × S → S s uch that the imag e of x R y z is included into x A | xy z | ⊗ | xy z | B z , for every x, y , z ∈ S . Thes e twisting sy stems are characterized in Prop osition 2.5. A more precise description of them is g iv en in C o rollary 2.7, provided that M satisﬁes an additional condition ( † ), see § 2.6. A similar result is obta ined in Corollary 2.9 for a linea r mo noidal catego ry . In this wa y we are led in § 2.1 0 to the deﬁnition o f simple twisting systems. F or such a t w is ting system R b et ween B and A , in Theorem 2.14 we construct a n M -ca tegory A ⊗ R B which factorizes F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 3 through A and B . Since it gener alizes the twisted tenso r pro duct o f a lgebras, A ⊗ R B will be ca lled the twisted tensor pro duct of A a nd B . In the third section w e consider the cas e when M is the monoidal catego ry of coalge bras in a braided category M ′ . In this setting, we prove that there is an o ne-to-one corr espondence b et ween simple twisting systems and matched pair of enriched categor ies, see § 3.6 for the deﬁnition of the latter notio n. W e shall r efer to the twisted tensor pr oduct of a matched pair as the bic r ossed pro duct. By construction, the bicro ssed pr oduct is a catego ry enriched ov er M ′ , but we pr o ve that it is enr ic hed ov er M a s well. More examples of twisted tenso r pr oducts of enriched categor ie s are given in the last part of the pap er. By deﬁnition, usual categor ies are enriched ov er S et , the category o f sets. Actually , they are enriched ov er the monoidal catego r y o f coalg ebras in S et . Hence, simple twisting sys tems and matched pairs are eq uiv alent notions for usual ca tegories. Moreover, if A and B a r e thin categorie s (that is their hom- s ets contain at most one morphism), then we show that any twisting system b et ween B and A is simple, so it co r respo nds to a uniquely determined ma tched pair of categorie s. W e use this res ult to in vestigate the t wis ting systems b et ween tw o p osets. Our results may b e applied to alg e bras in a monoidal categ ory M , which are enriched ca tegories with one o b ject. Ther efore, we are als o able to recover all bicros s ed pro duct constructio ns tha t we discussed at the b eginning of this in tro duction. Finally , we pr o ve tha t the bicross ed pr oduct of t wo group oids is also a g roupo id, and we g ive an example o f fa c torizable g r oupoid with t wo ob jects. 1. Preliminaries and not a tion. Mainly for ﬁxing the notatio n and the terminolo gy , in this section we recall the deﬁnition of enriched categ ories, a nd then w e give so me exa mple that are useful for our work. 1.1. Monoidal categories . Thr oughout this paper ( M , ⊗ , 1 , a, l , r ) will denote a monoidal cat- egory with as sociativity constra in ts a X,Y ,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) and unit c o nstrain ts l X : 1 ⊗ X → X and r X : X ⊗ 1 → X . The clas s of ob jects o f M will b e denoted b y M 0 . Mac Lane’s Cohere nce Theorem states that given tw o parenthesized tensor pro ducts o f some o b jects X 1 , . . . , X n in M (with pos sible arbitrary insertions of the unit ob ject 1 ) there is a unique mor - phism b et ween them tha t can be written as a co mposition of asso ciativity a nd unit constraints, and their in verses. Consequently , a ll these parenthesized tenso r pro ducts can b e identiﬁed coher en tly , and the parenthesis, asso ciativity constr ain ts and unit co nstrain ts may be omitted in computa- tions. Henceforth, we sha ll always ignore them. The identit y mor phism of a n ob ject X in M will be denoted by the same sym bo l X . By deﬁnition, the tensor pro duct is a functor. In particular, for an y morphisms f ′ : X ′ → Y ′′ and f ′′ : X ′′ → Y ′′ in M the follo wing equatio ns hold ( f ′ ⊗ Y ′′ ) ◦ ( X ′ ⊗ f ′′ ) = f ′ ⊗ f ′′ = ( Y ′ ⊗ f ′′ ) ◦ ( f ′ ⊗ X ′′ ) . (1) If the co product of a family { X i } i ∈ I of ob jects in M exists , then it will b e denoted a s a pair  L i ∈ I X i , { σ i } i ∈ I  , where the maps σ i : X i → L i ∈ I X i are the canonical inclusions. 1.2. The opp osite monoidal category . If ( M , ⊗ , 1 ,a, l , r ) is a monoidal categ o ry , then o ne constructs the mono idal ca tegory ( M o , ⊗ o , 1 o , a o , l o , r o ) as follows. By deﬁnition, M o and M share the same ob jects and identit y morphisms. On the other hand, for t wo ob jects X , Y in M , one takes Hom M o ( X, Y ) := Hom M ( Y , X ). The co mposition o f mo rphisms in M o • : Hom M o ( Y , Z ) × Hom M o ( X, Y ) → Hom M o ( X, Z ) is deﬁned b y the formula f • g := g ◦ f , for any f : Z → Y and g : Y → X in M . The monoidal structure is deﬁned b y X ⊗ o Y = X ⊗ Y and 1 o = 1 . The a ssocia tivit y a nd unit constraints in M o are giv en by a o X,Y ,Z = a − 1 X,Y ,Z , l o = l − 1 X and r o = r − 1 X . If, in addition M is braided monoidal, with braiding χ X,Y : X ⊗ Y → Y ⊗ X then M o is also braided, with resp ect to the braiding χ o deﬁned by χ o X,Y := ( χ X,Y ) − 1 . 4 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN Deﬁnition 1.3. Let S b e a set. W e say that a monoidal catego ry M is S -distributive if every S -indexe d family of o b jects in M ha s a copro duct, and the tensor pro duct is distributive to the left and to the right over any such copro duct. More precise ly , M is S -distributive if for a n y family { X i } i ∈ S the copr oduct ( L i ∈ S X i , { σ i } i ∈ S ) exists and, for an a rbitrary ob ject X , ( X ⊗ ( L i ∈ S X i ) , { X ⊗ σ i } i ∈ S ) and (( L i ∈ S X i ) ⊗ X , { σ i ⊗ X } i ∈ S ) are the copr oducts o f { X ⊗ X i } i ∈ S and { X i ⊗ X } i ∈ S , r espectively . Note that all mono idal categories are S -distributive, provided that S is a singleton (i.e. the car dinal o f S is 1 ). 1.4. Enriched categories. An enriche d c ate gory C over ( M , ⊗ , 1 ) , o r an M -categor y for short, consists of: (1) A class o f o b jects, that we denote by C 0 . If C 0 is a set w e say that C is smal l . (2) A hom-ob ject x C y in M , for each x a nd y in C 0 . It plays the sa me role as Hom C ( y , x ), the set o f mo rphisms fr o m y to x in an ordinary categor y C . (3) A morphism x c y z : x C y ⊗ y C z → x C z , for all x, y and z in C 0 . (4) A morphism 1 x : 1 → x C x , for all x in C 0 . By deﬁnition one assumes that the diagra ms in Figure 1 are commutativ e, for all x, y , z a nd t in C 0 . The commutativit y of the square means that the comp osition o f morphisms in C , deﬁned b y { x c y z } z ,y,z ∈ C 0 , is asso ciative . W e shall say tha t 1 x is the identity morphism of x ∈ C 0 . x C y ⊗ y C z ⊗ z C t x c y z ⊗ z C t / / x C y ⊗ y c z t   x C z ⊗ z C t x c z t   x C y ⊗ y C t x c y t / / x C t x C y ⊗ y C y x c y y % % J J J J J J J J J x C y x C y ⊗ 1 y o o 1 x ⊗ x C y / / x C x ⊗ x C y x c x y y y t t t t t t t t t x C y Figure 1. The deﬁnition of enriched categor ies. An M -functor α : C → C ′ is a couple ( α 0 , { x α y } x,y ∈ C 0 ), where α 0 : C 0 → C ′ 0 is a function and x α y : x C y → x ′ C ′ y ′ is a mo r phism in M for any x, y ∈ C 0 , where for simplicity w e deno ted α 0 ( u ) by u ′ , for an y u ∈ C 0 . By deﬁnition, α 0 and x α y m ust satisfy the following conditions x α x ◦ 1 C x = 1 D x ′ and x ′ d y ′ z ′ ◦ ( x α y ⊗ y α z ) = x α z ◦ x c y z . 1.5. T o work e asier with tensor pro ducts of hom-ob jects in M - categories we int ro duce so me new notation. Let S b e a set and for ev ery i = 1 , . . . , n + 1 w e pick up a family  x X i y  x,y ∈ S of ob jects in M . If x 1 , . . . , x n +1 ∈ S then the tenso r pro duct x 0 X 1 x 1 ⊗ x 1 X 2 x 2 ⊗ · · · ⊗ x n − 1 X n x n ⊗ x n X n +1 x n +1 will be denoted by x 0 X 1 x 1 X 2 x 2 · · · x n X n +1 x n +1 . Assuming that M is S -distributive a nd ﬁxing x 0 and x n +1 , one can co nstruct inductively the iterated copro duct x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 := L x 1 ∈ S · · · L x n ∈ S x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 . (2) It is not diﬃcult to see that this ob ject is a copr o duct o f { x 0 X 1 x 1 · · · x n X n +1 x n +1 } ( x 1 ,...,x n ) ∈ S n . Moreov er, as a c o nsequence of the fact that the tens o r pr oduct is distributiv e over the dir e ct sum, w e hav e x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 ∼ = L x 1 ∈ S · · · L x n ∈ S x 0 X 1 x π (1) X 2 x π (2) · · · x π ( n ) X n +1 x n +1 (3) for any p ermutation π of the s et { 1 , 2 , . . . , n } . The inclusion of x 0 X 1 x 1 · · · x n X n +1 x n +1 int o the copro duct deﬁned in (2) is also inductiv ely constructed as the compo sition o f the following tw o arrows x 0 X 1 x 1 ⊗ x 1 X 1 x 2 · · · x n X n +1 x n +1 − → x 0 X 1 x 1 ⊗ x 1 X 1 x 2 · · · x n X n +1 x n +1 ֒ → L x 1 ∈ S x 0 X 1 x 1 ⊗ x 1 X 1 x 2 · · · x n X n +1 x n +1 , where the ﬁrst mor phis m is the tensor pro duct b etw een the identit y of x 0 X 1 x 1 and the inclusion of x 1 X 1 x 2 · · · x n X n +1 x n +1 int o x 1 X 1 x 2 · · · x n X n +1 x n +1 . Clear ly , for ev ery x n +1 ∈ S , x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 := L x 0 ∈ S x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 5 is the copro duct of { x 0 X 1 x 1 · · · x n X n +1 x n +1 } ( x 0 ,x 1 ,...,x n ) ∈ S n . The ob jects x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 and x 0 X 1 x 1 · · · x n − 1 X n x n X n +1 x n +1 are analog ously deﬁned. A similar notation will b e used for mo rphisms. Let us supp ose that x α i y is a morphis m in M with sour ce x X i y and targe t x Y i y , where x, y ∈ S and i ∈ { 1 , . . . , n + 1 } . W e set x 1 α 1 x 2 α 2 x 3 · · · x n α n +1 x n +1 := x 0 α 1 x 1 ⊗ · · · ⊗ x n α n +1 x n +1 . By the univ ersal prop erty of copro ducts, { x 0 α 1 x 1 · · · x n α n +1 x n +1 } ( x 1 ··· x n ) ∈ S n induces a unique map x 0 α 1 x 1 · · · x n − 1 α n − 1 x n α n x n +1 that commutes with the inclusions. In a similar wa y o ne co nstructs x 0 α 1 x 1 · · · x n − 1 α n x n α n +1 x n +1 , x 0 α 1 x 1 · · · x n − 1 α n x n α n +1 x n +1 and x 0 α 1 x 1 · · · x n − 1 α n x n α n +1 x n +1 . T o make the above no tation clearer, let us hav e a lo ok at some examples. Let A a nd B b e tw o M -categ ories suc h that A 0 = B 0 = S . Reca ll that the ho m- ob jects in A a nd B ar e denoted by x A y and x B y . Hence, x A y = L x ∈ S x A y . W e also have x A y B z A t = x A y ⊗ y B z ⊗ z A t and x A y B z A t = L y ∈ S L z ∈ S x A y B z A t ∼ = L z ∈ S L y ∈ S x A y B z A t ∼ = L y ,z ∈ S x A y B z A t . Since we hav e ag reed to use the s a me notation fo r an o b ject and its identit y ma p, we can write x B y α z A t β u instead of I d x B y ⊗ y α z ⊗ I d z A t ⊗ t β u , for any morphisms y α z and t β u in M . The maps x a y z : x A y A z − → x A z and x a y z : x A y A z − → x A z are induced by the comp osition in A , that is by the set { x a y z } z ∈ S . F or example, the fo rmer map is uniquely deﬁned such that its restr iction to x A y A z and σ x,z ◦ x a y z coincide for all x ∈ S, where σ x,z is the inclusion of x A z int o x A z . Similar ly , x a y z : x A y A z − → x A z is the unique map whose re striction to x A y A z is x a y z , for all y ∈ S. F or more details o n enriched categor ie s the reader is re ferred to [Ke]. W e end this section g iving some examples of enr ic hed categor ies. 1.6. The category S et . The categor y of sets is mo noidal with resp ect to the Cartesia n pro duct. The unit ob ject is a ﬁxed s ingleton set, s a y {∅} . The copro duct in S et is the disjoint union. Since the disjoint union and the Car tesian pro duct c o mm ute, S et is S -distributive for a n y se t S . Clear ly , a S et -ca teg ory is an ordinary category . If C is suc h a catego ry , then an ele men t f ∈ x C y will be thought o f as a morphism from y to x, and it will be denoted b y f : y → x, as usual. In this case we shall s a y that y (r espectively x ) is the domain or the sour ce (resp ectively the codomain or the target) of f . The same notation a nd ter minology will b e used for a rbitrary M -catego ries, whose ob jects a re sets. 1.7. The category K - M od . Let K be a comm utative r ing. The categ o ry of K -mo dules is monoidal with r e s pect to the tens or pro duct of K -mo dules. The unit ob ject is K , regarded as a K -mo dule. This monoida l category is S - dis tributiv e for any S . By deﬁnition, a K - line ar c ate gory is an enric hed catego ry ov er K - M o d . 1.8. The category Λ - M o d - Λ . Let Λ be a K -algebra and let Λ- M od -Λ denote the categor y of left (or rig ht) mo dules over Λ ⊗ K Λ o , where Λ o is the o pposite algebra of Λ . Th us, M is a n ob ject in Λ- M od -Λ if, and only if, it is a left a nd a right Λ-mo dule and these s tructures are compatible in the sense that a · m = m · a and ( x · m ) · y = x · ( m · y ) for a ll a ∈ K , x, y ∈ Λ and m ∈ M . A morphis m in Λ- M o d -Λ is a map of left and rig h t Λ-mo dules. The categ ory o f Λ-bimo dules is monoidal with r espect to ( − ) ⊗ Λ ( − ) . The unit o b ject in Λ- M od -Λ is Λ , regarded as a Λ-bimodule. This mono idal ca tegory also is S -distr ibutive for any S . 1.9. The category H - M o d . Let H b e a bialgebra ov er a c o mm uta tiv e r ing K . The categ ory of left H -mo dules is monoidal with res p ect to ( − ) ⊗ K ( − ). If M and N are H -mo dules, then the H -actio n on M ⊗ N is given by h · m ⊗ n = P h (1) · m ⊗ h (2) · n. 6 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN In the a bov e equation we us ed the Σ-notation ∆ ( h ) = P h (1) ⊗ h (2) . The unit ob ject is K , which is an H -mo dule with the tr ivial action, induced by the counit of H . This catego r y is S - distributiv e, for any S . An e nr ic hed categor y ov e r H - M od is called H - mo dule c ate gory . 1.10. The category C omod - H . Dually , the catego r y of right H -como dules is monoidal with resp ect to ( − ) ⊗ K ( − ). The coaction on and M ⊗ K N is deﬁned by ρ ( m ⊗ n ) = P m h 0 i ⊗ n h 0 i ⊗ m h 1 i n h 1 i , where ρ ( m ) = P m h 0 i ⊗ n h 0 i , and a similar Σ-notation was used fo r ρ ( n ) . This categor y is S - distributive, for any set S . By deﬁnition, an H - c omo dule c ate gory is an enriched category over C omod - H . 1.11. The category [ A , A ] . Let A b e a small categor y , and let [ A , A ] denote the catego ry of all endofunctor s o f A . Therefore, the ob jects in [ A , A ] are functors F : A → A , while the set F [ A , A ] G contains a ll natural transfor mations µ : G → F . The comp osition in this ca tegory is the compo sition of natural transformations. The category [ A , A ] is monoidal with r espect to the comp osition of functors. If µ : F → G and µ ′ : F ′ → G ′ are natural trans formations, then the natural tra nsformations µF ′ and Gµ ′ are given by µF ′ : F ◦ F ′ → G ◦ G ′ , ( µF ′ ) x := µ F ′ ( x ) , Gµ ′ : G ◦ F ′ → G ◦ G ′ , ( Gµ ′ ) x := G ( µ ′ x ) . W e can now deﬁne the tensor pro duct of µ and µ ′ by µ ⊗ µ ′ := Gµ ′ ◦ µF ′ = µG ′ ◦ F µ ′ . Even if A is S -distr ibutive, [ A , A ] may not hav e this pr operty . In spite of the fact that, by assumption, any S -indexed family in [ A , A ] has a copr o duct, in gener al this do es no t c omm ute with the compo sition of functors. Nevertheless, as we have already noticed, [ A , A ] is S -distributive if | S | = 1. This r emark will allow us to apply our ma in results to an [ A , A ]-ca tegory C with one o b ject x . Hence F := x C x is a n endofunctor o f A , and the co mposition and the identit y morphisms in C are uniquely deﬁned b y natural tra nsformations µ : F ◦ F → F and ι : Id A → F . The commut ativity of the dia g rams in Figure 1 is equiv alent in this case with the fact that ( F , µ, ι ) is a monad , see [Be] for the deﬁnitio n o f mona ds. In conclusio n, monads are in one-to-one corre- sp ondence to [ A , A ]-categories with one ob ject. 1.12. The category O pmon ( M ) . Let ( M , ⊗ , 1) b e a monoidal catego r y . An opmono idal functor is a tr iple ( F, δ, ε ) that consists of (1) A functor F : M → M . (2) A natural tr a nsformation δ := { δ x,y } ( x,y ) ∈ M 0 × M 0 , with δ x,y : F ( x ⊗ y ) → F ( x ) ⊗ F ( y ) . (3) A map ε : F ( 1 ) → 1 in M . In additio n, the tra nsformations δ and ε are assumed to render commut ative the diagrams in Figure 2. An o pmonoidal transformation α : ( F, δ, ε ) → ( F ′ , δ ′ , ε ′ ) is a na tural map α : F → F ′ such that, for a rbitrary ob jects x a nd y in M , ( α x ⊗ α y ) ◦ δ x,y = δ ′ x,y ◦ α x ⊗ y and ε ′ ◦ α 1 = ε . Obviously the compo sition of t wo opmonoidal transformations is opmono idal, and the identit y of an opmonoidal functor is an opmonoida l transfo rmation. The resulting catego r y will be denoted by O pmon ( M ) . F or tw o opmono idal functors ( F , δ, ε ) and ( F ′ , δ ′ , ε ′ ) one deﬁnes ( F, δ, ε ) ⊗ ( F ′ , δ ′ , ε ′ ) := ( F ◦ F ′ , δ F ′ ,F ′ ◦ F ( δ ′ ) , ε ◦ F ( ε ′ )) , where δ F ′ ,F ′ =  δ F ′ ( x ) ,F ′ ( y )  x,y ∈ M 0 . O n the other hand, if µ : F → G a nd µ ′′ → G ′ are opmonoidal transformatio ns, then µ ⊗ µ ′ := µG ′ ◦ F µ ′ is o pmo noidal to o. One ca n s ee ea sily that ⊗ deﬁnes a monoidal str uc tur e on Opmon ( M ) with unit o b ject (Id M , { Id x ⊗ y } x,y ∈ M 0 , Id 1 ) . F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 7 F ( x ⊗ y ⊗ z ) δ x ⊗ y,z / / δ x,y ⊗ z   F ( x ⊗ y ) ⊗ F ( z ) δ x,y ⊗ F ( z )   F ( x ) ⊗ F ( y ⊗ z ) F ( x ) ⊗ δ y,z / / F ( x ) ⊗ F ( y ) ⊗ F ( z ) F ( x ) ⊗ F ( 1 ) F ( x ) ⊗ ε 1 / / F ( x ) F ( 1 ) ⊗ F ( x ) ε 1 ⊗ F ( x ) o o F ( x ) δ x, 1 e e L L L L L L L L L L δ 1 ,x 9 9 r r r r r r r r r r Figure 2. The deﬁnition of opmono idal functor s . 1.13. The categories Alg ( M ) and C oalg ( M ) . Let ( M , ⊗ , 1 , χ ) b e a braided monoidal cate- gory with braiding χ := { χ x,y } ( x,y ) ∈ M 0 × M 0 , where χ x,y : x ⊗ y → y ⊗ x. The categ o ry Al g ( M ) of all algebra s in M is monoida l to o. Recall that an alge bra in M is a n M - category with one ob ject. As in § 1.11, s uc h a c a tegory is uniquely deter mined b y a n o b ject X in M and tw o mor phisms m : X ⊗ X → X (the multiplication) and u : 1 → X (the unit). The commutativit y o f the diagrams in Figure 1 means that the algebra is asso ciative and unital. If ( X, m, u ) and ( X ′ , m ′ , u ′ ) are alg ebras in M , then X ⊗ X ′ is a n algebra in M with m ultiplica tion ( m ⊗ m ′ ) ◦ ( X ⊗ χ X ′ ,X ⊗ X ′ ) : ( X ⊗ X ′ ) ⊗ ( X ⊗ X ′ ) → X ⊗ X ′ and unit u ⊗ u ′ : 1 → X ⊗ X ′ . The monoidal catego ry C o a lg ( M ) of coalgebras in M ca n be deﬁned in a similar wa y . Alter- natively , one may take C oal g ( M ) := Al g ( M o ) o . Note that the monoidal categor y of coalg ebras in M and the monoidal catego ry of alg ebras in M o are opp osite each other. It is no t ha rd to see tha t C o alg ( M ) is S -distributive, provided that M is s o. 2. F actorizable M -ca tegories a nd twisting systems. In this section we deﬁne factorizable M -categor ie s and twisting systems. W e shall prov e that to every facto rizable system corresp onds a certain twisting system. Under a mild extra a s sumption on the monoidal catego ry M , we shall also pro duce enr ic hed catego ries using a sp ecial class o f t wisting s y stems that w e ca ll simple. Throughout this section S deno tes a ﬁxed set. W e assume that all M -categor ies that w e work with ar e small, and that their set of o b jects is S . 2.1. F actorizable M -categories. Let C b e a small enr iched ca tegory ov er ( M , ⊗ , 1 ). W e assume that M is S -distr ibutive. Suppos e that A and B ar e M - subcatego ries o f C . Note tha t, by assumption, A 0 = B 0 = C 0 = S. F or x, y and u in S we deﬁne x ϕ u y : x A u B y → x C y , x ϕ u y := x c u y ◦ x α u β y , (4) where α : A → C and β : B → C denote the cor r esponding inclusion M -functors. By the universal pro p erty o f coproducts, for every x and y in S, there is x ϕ y : x A u B y → x C y such tha t x ϕ y ◦ x σ u y = x ϕ u y , (5) where x σ u y is the canonical inclusio n of x A u B y int o x A u B y . Note that by the universal prop erty of copro ducts x ϕ y = x c u y ◦ x α u β y , as we have x α u β y ◦ x σ u y = x τ u y ◦ x α u β y and x c u y ◦ x τ u y = x c u y , where x τ u y denotes the inclusion o f x C u C y int o x C u C y . W e sha ll say that C factorizes through A and B if x ϕ y is an is omorphism, for a ll x and y in S . By deﬁnition, an M -ca tegory C is factorizable if it facto r izes throug h A and B , w he r e A and B are cer tain M -sub c a tegories of C . 2.2. The t wisti ng system asso ciated to a factorizable M -category . Let C be an enriched category over a monoidal category ( M , ⊗ , 1 ). W e assume that M is S - distributiv e. T he family R := { x R y z } x,y ,z ∈ S of morphis ms x R y z : x B y A z → x A u B y is called a twisting system if the four diagrams in Figure 3 ar e commutativ e for all x, y , z a nd t in S . Let us brieﬂy expla in the notation that w e used in these dia grams. As a general rule, w e omit all subscripts and supe r scripts denoting elemen ts in S , and which a re attached to a mo rphism. The symbol ⊗ is a lso omitted. F or example, a and 1 A (resp ectiv ely b and 1 B ) stand for the suitable 8 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN x B y B z A t RI ◦ I R   bI / / x B z A t R   x A v B u B t I b / / x A v B t x B y A z A t I R ◦ RI   I a / / x B y A t R   x A v A u B t aI / / x A u B t x A y 1 B I / / I 1 B   x B x A y R   x A y B y σ / / x A u B y x B y I 1 A / / 1 A I   x B y A y R   x A x B y σ / / x A u B y Figure 3. The deﬁnition of twisting systems. comp osition maps and identit y mo r phisms in A (resp ectiv ely B ). The iden tity mor phism of an ob ject in M is deno ted by I . Thus, by I a : x B y A z A t → x B y A t we mean x B y ⊗ y a z t . On the o ther hand, a I : x A v A u B t → x A u B t is a shorthand notation for x a v u B t , which in tur n is the unique map induced by { x σ u t ◦ x a v u B t } u,v ∈ S . W e shall keep the for egoing nota tion in all diagr a ms that we shall work with. W e claim that to every fa c torizable M - category C co rresp o nds a certain twisting s y stem. By deﬁnition, the map x ϕ y constructed in (5) is inv ertible for all x a nd y in S. Let x ψ y denote the inv er se of x ϕ y . F or x , y and z in S , w e can now deﬁne x R y z : x B y A z → x A u B z , x R y z := x ψ z ◦ x c y z ◦ x β y α z . (6) Theorem 2.3. If C is a factor izable enriched categor y over a n S - dis tributiv e mono idal category M , then the maps in (6) deﬁne a twisting system. Pr o of. Let us ﬁrs t pr ove that the ﬁr st diagra m in Figure 3 is commutativ e. W e ﬁx x, y , z and t in S, and w e consider the following diag ram. x B y A u B t /.-, ()*+ A β αβ / / β αI   x C y C u C t /.-, ()*+ F I c / / x C y C t c   x C y C u B t I I β / / /.-, ()*+ B cI   x C y C u C t cI   x C u B t /.-, ()*+ C I β / / ψ I   x C u C t x A v B u B t /.-, ()*+ D αβ β / / I b   x C v C u C t /.-, ()*+ E cI / / I c   x C u C t c   x A v B t αβ / / x C v C t c / / x C t x C t Since the tensor pr oduct in a monoidal category is a functor, that is in view of (1), we hav e x C y C u β t ◦ x β y α u B t = x β y α u β t , (7) for any u in S. Hence by the universal prop ert y of the copro duct and the co nstruction o f the maps x C y C u β t , x β y α u B t and x β y α u β t we deduce tha t the r elation which is obtained by replacing u with u in (7) holds true. This means that the square (A) is commutativ e. P rocee ding similarly one s ho ws tha t (B) is commut ative as well. F urthermor e , x c v u C t , x α v β u β t and x ψ u B t are induced by  x c v u ⊗ u C t  u ∈ S , { x α v β u ⊗ u β t } u ∈ S and { x ψ u ⊗ u B t } u ∈ S , r espectively . Hence their comp osite λ := x c v u C t ◦ x α v β u β t ◦ x ψ u B t is induced by { λ u } u ∈ S , where λ u =  x c v u ⊗ u C t  ◦ ( x α v β u ⊗ u β t ) ◦ ( x ψ u ⊗ u B t ) =  x c v u ◦ x α v β u ◦ x ψ u  ⊗ u β t = ( x ϕ u ◦ x ψ u ) ⊗ u β t . Since x ψ u is the inverse of x ϕ u it follows that λ u = x C u β t , for ev ery u ∈ S. In conclusion x c v u C t ◦ x α v β u β t ◦ x ψ u B t = x C u β t , so (C) is a comm utative square. Since β is an M - functor it follows that { x C v c u t ◦ x α v β u β t } u,v ∈ S and { x α v β t ◦ x A v b u t } u,v ∈ S are equa l. Ther efore these families induce the same morphism, that is x C v c u t ◦ x α v β u β t = x α v β t ◦ x A v b u t . F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 9 Hence (D) is commutativ e to o. Since the compositio n of morphisms in C is as s ociative, we hav e x c v t ◦ x C v c u t = x c u t ◦ x c v u C t and x c y t ◦ x C y c u t = x c u t ◦ x c y u C t . These equatio ns imply that (E) and (F) are commutativ e. Summarizing, we have just pr o ved that all diagrams (A)-(F) are commutativ e. By diag r am chasing it res ults that the outer s quare is commutativ e as well, that is x ϕ t ◦ x A v b u t ◦ x R y u B t = x c y t ◦ x β y ϕ t . Left compo sing a nd r ig h t c o mposing b oth sides of this equatio n by x ψ t and x B y R z t , resp ectiv ely , yield x A v b u t ◦ x R y u b t ◦ x B y R z t = x ψ t ◦ x c y t ◦ x β y ϕ t ◦ x B y R z t = x ψ t ◦ x c y t ◦ x β y ϕ t ◦ x B y ψ t ◦ x B y c z t ◦ x B y β z α t = x ψ t ◦ x c y t ◦ x C y c z t ◦ x β y β z α t , where for the seco nd a nd third relations we us ed the deﬁnition of y R z t and that y ϕ t and y ψ t are inv er ses each other. O n the other hand, the deﬁnitio n o f x R z t , the fact that β is a functor and asso ciativity of the compo s ition in C imply the following sequence o f identities x R z t ◦ x b y z A t = x ψ t ◦ x c z t ◦ x β z α t ◦ x b y z A t = x ψ t ◦ x c z t ◦ x c y z C t ◦ x β y β z α t = x ψ t ◦ x c y t ◦ x C y c z t ◦ x β y β z a t . In conclus ion, the ﬁrst dia gram in Figure 3 is commutativ e. T a king into account the deﬁnitio n of x R x y , the identit y x β x ◦ 1 B x = 1 x and the compatibility relation betw een the comp osition and the ident ity morphisms in a n enriched ca teg ory , we get the following sequence of equa tions x ϕ y ◦ x R x y ◦ 1 B x A y = x ϕ y ◦ x ψ y ◦ x c x y ◦ x β x α y ◦ 1 B x A y = x c x y ◦ 1 x α y = x α y . Analogously , using the deﬁnition of x ϕ y and the prop e r ties o f identit y morphisms, we get x ϕ y ◦ x σ y y ◦ x A y 1 B y = x ϕ y y ◦ x A y 1 B y = x c y y ◦ x α y β y ◦ x A y 1 B y = x c y y ◦ x α y 1 y = x α y . Since x ϕ y is an iso morphisms, in view of the a bov e co mputations, it follows that the third diagra m is commutativ e a s well. One ca n prove in a similar wa y that the remaining tw o diagra ms in Figure 3 are comm utative.  2.4. W e hav e noticed in the introduction that to every twisting s ystem of groups (or , e quiv a len tly , every matched pair of gro ups ) one asso ciates a factoriz able gr oup. T r ying to prove a simila r result for a twisting s y stem R b et ween the M -categories B and A we have encountered so me diﬃculties due to the fact that, in general, the image of the map x R y z : x B y A z → L u ∈ S x A u B z is not included into a summand x A u B z , for some u ∈ S that dep ends on x, y and z . F o r this reason, in this pap er we sha ll inv estigate only those t wisting systems for which there are a function | · · · | : S 3 → S and the maps x e R y z : x B y A z → x A | xy z | B z such that x R y z = x σ | xy z | z ◦ x e R y z , (8) for all x, y , z , ∈ S. F or them w e shall use the notation ( e R, | · · · | ) . Prop osition 2.5. Let M b e a monoidal catego ry whic h is S -distributive. Let | · · · | : S 3 → S and { x e R y z } x,y ,z ∈ S be a function and a set of maps as a bov e. The family { x R y z } x,y ,z ∈ S deﬁned by 10 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN (8) is a t wisting s ystem if and only if, for any x, y , z , t ∈ S, the follo wing rela tions hold: x σ | xy | yz t || t ◦ x A | xy | yz t || b | y zt | t ◦ x e R y | y zt | B t ◦ x B y e R z t = x σ | xz t | t ◦ x e R z t ◦ x b y z A t , (9) x σ | xy z | z t || t ◦ x a | xy z | || xy z | zt | B t ◦ x A | xy z | e R z t ◦ x e R y z A t = x σ | xy t | t ◦ x e R y t ◦ x B y a z t , (10) x σ | xxy | y ◦ x e R x y ◦ (1 B x ⊗ x A y ) = x σ y y ◦ ( x A y ⊗ 1 B y ) , (11) x σ | xy y | y ◦ x e R y y ◦ ( x B y ⊗ 1 A y ) = x σ x y ◦ (1 A x ⊗ x B y ) . (12) Pr o of. W e cla im that { x e R y z } x,y ,z ∈ S satisfy (9) if and only if { x R y z } x,y ,z ∈ S render commutativ e the ﬁrst diagr am in Figur e 3. Indeed, le t us consider the following diag ram. x B y B z A t I e R / / bI   x B y A | y zt | B t I σ / / e RI   x B y A u B t RI   /.-, ()*+ B x B z A t e R   x A | xy | yz t || B | y zt | B t σI / / I b   x A v B u B t I b   /.-, ()*+ C /.-, ()*+ A x A | xy | yz t || B t σ / / x A v B t x A | xz t | B t σ / / x A v B t The squar e s (B) a nd (C) are commutativ e by the deﬁnition of x R y u : x B y A u → x A v B u and v b u t : v B u B t → v B t . Hence the hexago n (A) is co mm utative if and only if the outer square is commutativ e. This pr o ves our claim as (A) and the outer square in Figure 3 are commutativ e if and o nly if (9) holds and the ﬁrst diagra m in Figure 3 is commutativ e, resp ectiv ely . Similarly one shows that the comm utativity o f the second diagram fr om Figure 3 is equiv alent to (10). On the other hand, obviously , the third a nd fourth dia grams in Figure 3 are co mmutative if and only if (11) and (12) hold, so the propo sition is pro ved.  The inclus ion maps make diﬃcult to handle the equa tio ns (9)-(12). In some cases we can remov e these mor phisms by impo sing more conditions on the map | · · · | or on the monoidal catego ry M . 2.6. The ass umption ( † ). Let M be a mono ida l category which is S - distributiv e. W e shall say that M satisﬁes the hypo thesis ( † ) if for any copro duct ( L i ∈ S X i , { σ i } i ∈ S ) in M and a ny morphisms f ′ : X → X i ′ and f ′′ : X → X i ′′ such that σ i ′ ◦ f ′ = σ i ′′ ◦ f ′′ , then either X is a n initial ob ject ∅ in M , o r f ′ = f ′′ and i ′ = i ′′ . The prototype for the clas s of mo noidal ca tegories that satisfy the c o ndition ( † ) is S et . Indeed, let { X i } i ∈ S be a family of sets, a nd let σ i denote the inclus ion of X i int o the disjoint union ` i ∈ S X i . W e assume that f ′ : X → X i ′ and f ′′ : X → X i ′′ are functions suc h that X in not the e mpty set, the initial ob ject of S et , and σ i ′ ◦ f ′ = σ i ′′ ◦ f ′′ . Then in view of the co mputatio n ( i ′ , f ′ ( x )) = ( σ i ′ ◦ f ′ )( x ) = ( σ i ′′ ◦ f ′′ )( x ) = ( i ′′ , f ′′ ( x )) it follows that f ′ = f ′′ and i ′ = i ′′ . Corollary 2.7. Let M b e an S -distributive mo noidal catego ry . Let A and B be tw o M - categorie s s uc h that A 0 = B 0 = S . Given a function |· · · | : S 3 → S and the maps { x e R y z } x,y ,z ∈ S as in § 2.4, let us consider the follo wing four co nditio ns : (i) If x B y B z A t is not an initial ob ject, then | xy | y z t || = | xz t | a nd x A | xz t | b | y zt | t ◦ x e R y | y zt | B t ◦ x B y e R z t = x e R z t ◦ x b y z A t ; (13) (ii) If x B y A z A t is not an initial ob ject, then || xy z | z t | = | xy t | a nd x a | xy z | | xy t | B t ◦ x A | xy z | e R z t ◦ x e R y z A t = x e R y t ◦ x B y a z t ; (14) F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 11 (iii) If x A y is not an initial ob ject, then | xxy | = y and x e R x y ◦ (1 B x ⊗ x A y ) = x A y ⊗ 1 B y ; (15) (iv) If x B y is not an initial ob ject, then | xy y | = x and x e R y y ◦ ( x B y ⊗ 1 A y ) = 1 A x ⊗ x B y . (16) The ab o ve conditions imply the rela tions (9)-(12). Under the a dditional ass umption tha t M satisﬁes the h yp othesis ( † ), the rev ersed implication ho lds as well. Pr o of. Let us prove that the condition (i) implies the re la tion (9). In the case when x B y B z A t = ∅ this is clea r, as both sides of (9) ar e morphisms from an initial o b ject to x A u B t . Let us suppo s e that x B y B z A t 6 = ∅ . By compo sing b oth sides of (13 ) with x σ | xy | yz t || t = x σ | xz t | t we g et the equation (9). Simila rly , the conditions (ii), (iii) a nd (iv) imply the r e la tions (10), (11) and (12), r e s pectively . Let us assume that M s atisﬁes the hypothesis ( † ). W e claim that (9) implies (i). If x B y B z A t is not a n initial ob ject we take f ′ and f ′′ to b e the left hand side and the right hand side o f (13), resp ectiv ely . W e a ls o set i ′ := | xy | y z t || and i ′′ := | xz t | . In view of ( † ), it follows that f ′ = f ′′ and i ′ = i ′′ , so our claim has b een pr o ved. W e conclude the pro of in the same wa y .  2.8. K -linear monoidal categorie s . Recall that M is K -linear if its hom-sets a re K -modules , and bo th the comp osition and the tensor pro duct o f morphisms are K -bilinear maps. F or instance, K - M o d , H - M od , C omo d - H and Λ- M o d -Λ are S -distributive linear monoidal ca tegories, for any set S . Note that the ( † ) condition fail in a K -linear monoidal category M . Indeed let us pick up an ob ject X , whic h is not an initial ob ject, and a copr oduct ( L i ∈ S X i , { σ i } i ∈ S ) in M . If f ′ : X → X i ′ and f ′′ : X → X i ′′ are the z ero morphisms , then of course σ i ′ ◦ f ′ = σ i ′′ ◦ f ′′ , but neither i ′ = i ′′ nor f ′ = f ′′ , in general. Nevertheless, the r elations (9)-(12) can also be simpliﬁed if M is a linear monoida l categor y . F or any co pr oduct ( L i ∈ S X i , { σ i } i ∈ S ) in M and every i ∈ S, there is a map π i : L i ∈ S X i → X i such that π i ◦ σ i = X i and π i ◦ σ j = 0, provided that j 6 = i . Hence, supp osing that f ′ : X → X i ′ and f ′′ : X → X i ′′ are morphisms such that σ i ′ ◦ f ′ = σ i ′′ ◦ f ′′ , w e must ha ve either i ′ = i ′′ and f ′ = f ′′ , or i ′ 6 = i ′′ and f ′ = 0 = f ′′ . Using the a b ov e prop ert y of linear monoidal categor ies, and pro ceeding as in the pro of of the previous coro llary , we get the following res ult. Corollary 2.9. Let M b e a n S - dis tributiv e K -linear monoidal category . If A and B are M - categorie s, then the relations (9)-(12) ar e equiv alent to the following conditions : (i) If | xy | y z t || = | xz t | then the rela tion (1 3) holds; other wise, each side of this identit y has to be the zero map; (ii) If || xy z | z t | = | xyt | , then the relation (14) holds; other w is e, each side of this iden tity has to b e the zer o ma p; (iii) If | xxy | = y , then the relation (15) holds; otherwis e, each side of this identit y has to be the zero ma p; (iv) If | xy y | = x, then the relation (16) holds; otherwise, ea c h side of this identit y has to be the zero ma p. 2.10. Simple t wisti ng s ystems. The pro per context for constr ucting an enriched ca tegory A ⊗ R B o ut of a s pecial type of twisting s y stem R is provided by Coro llary 2.7. By deﬁnition, the couple ( e R, | · · · | ) is a simple twisting system b etwe en B and A if the function |· · · | : S 3 → S and the maps { x e R y z } x,y ,z ∈ S as in § 2.4 satisfy the conditions (i)-(iv) in Corollar y 2 .7. As a part of the deﬁnition, we also assume that x A | xy z | B z is not a n initial o b ject whenever x B y A z is not so. The latter technical assumption will be used to prov e the asso ciativity of the comp osition in A ⊗ R B , our categor ical version of the t wisted tenso r pro duct of tw o algebras , which w e ar e going 12 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN to deﬁne in the next subsection. Note that for S et this condition is super ﬂuous (if the source of x e R y z is not empt y , then its targ et cannot be the empt y set). F or a simple twisting system ( e R, | · · · | ) we deﬁne the ma ps x R y z using the relation (8). By Corollar y 2.7 and P ropos ition 2.5 it follows that R := { x R y z } x,y ,z ∈ S is a twisting system. 2.11. The category A ⊗ R B . F o r a simple twisting sy stem ( e R, | · · · | ) we set ( A ⊗ R B ) 0 := S a nd x ( A ⊗ R B ) y := L u ∈ S x A u ⊗ u B y = x A u B y . Let us ﬁx three elements x, y and z in S. B y deﬁnition x A u B y A ¯ v B z := L u,v ∈ S x A u B y A v B z , and x A u B y A v B z ∼ = x A u B y ⊗ y A v B z as M is S -distributive. Via this identiﬁcation, the canonical inclusion of x A u B y A v B z int o the copro duct x A u B y A v B z corres p onds to x σ u y σ v z = x σ u y ⊗ y σ v z . Thus, there is a unique morphism x c y z : x A u B y A v B z → x A u B z such that x c y z ◦ x σ u y σ v z = x σ | uy v | z ◦ x a | uy v | b z ◦ x A u e R y v B z , for all u , v ∈ S . Finally , w e set 1 x := x σ x x ◦ (1 A x ⊗ 1 B x ), and w e deﬁne x α y := x σ y y ◦ ( x A y ⊗ 1 B y ) and x β y := x σ x y ◦ (1 A x ⊗ x B y ) . 2.12. Domains. T o sho w that the ab o ve data deﬁne an enr ic hed monoidal ca tegory A ⊗ R B we need a n extra hyp othesis on M . By deﬁnition, a monoidal category M is a domai n in the case when the tenso r pro duct o f t wo ob jects is an initial ob ject if and only if at least one of them is an initial ob ject. By conv e n tio n, a monoida l category that has no initial ob jects is a domain as w ell. Obviously S et is a do main. If K is a ﬁeld, then K - M od is a doma in. Keeping the assumption on K , the categor ies H - M o d and C o mod - H ar e domains, as their tensor pro duct is induced b y that one of K - M o d . On the other hand, if K is not a ﬁeld, then K - M o d and Λ- M od -Λ are not necessarily doma ins. F or instanc e , Z - M od ∼ = Z - M od - Z is no t a domain. Lemma 2.13 . Let M b e an S -distributive mono idal domain. Let ( e R, | . . . | ) denote a simple t wisting s y stem betw een B and A . (1) If x A u B y A v B z A w B t 6 = ∅ then | uy q | = | pv q | = | pz w | , where p = | uy v | and q = | vz w | . (2) In the following diag r am all sq uares ar e well deﬁned and comm utative. x A u B y A v B z A w B t /.-, ()*+ F I I I e RI / / I e RI I I   x A u B y A v A q B w B t I I I I b / / /.-, ()*+ F I I e RI I ◦ I e RI I I   x A u B y A v A q B t /.-, ()*+ R I I aI / / I I e RI ◦ I e RI I   x A u B y A q B t I e RI   x A u A p B v B z A w B t I I e RI I ◦ I I I e RI , , I I aI I   /.-, ()*+ F x A u A p A | pvq | B q B w B t /.-, ()*+ F I I I I b / / aI I I I   x A u A p A | pvq | B q B t I aI I / / aI I I   /.-, ()*+ A x A u A | pvq | B q B t aI I   x A p B v B z A w B t /.-, ()*+ L I e RI I ◦ I I e RI 2 2 I bI I   x A p A | pvq | B q B w B t I I I b / / I I bI   /.-, ()*+ A x A p A | pvq | B q B t aI I / / I I b   /.-, ()*+ F x A | pvq | B q B t I b   x A p B z A w B t I e RI / / x A p A | pvq | B w B t I I b / / x A p A | pvq | B t aI / / x A | pvq | B t Pr o of. Since M is a domain it follows that any subfactor of x A u B y A v B z A w B t is not an initial ob ject. In par ticular v B z A w 6 = ∅ . Thus, by the deﬁnition of simple twisting systems, v A q B w is not an initial ob ject. In conclusion, v A q and q B w are not initial ob jects in M . Since u B y A v 6 = ∅ it follows that u B y A v A q 6 = ∅ . In view of the deﬁnition of simple twisting sys tems (the second condition) we deduce that | pv q | = | uy q | . The other r elation can b e prov ed in a similar wa y . Let f and g denote the following tw o morphis ms f := x A u a p | pvq | B q B t ◦ x A u A p e R v q B t ◦ x A u e R y v A q B t and g := x A u e R y q B t ◦ x A u B y a v q B t . F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 13 The target of f is x A u A | pvq | B q B t , while the co domain o f g is x A u A | uy q | B q B t . These t wo ob jects may b e diﬀerent for some elements x, u, t, p and q in S . Thus, in general, it do es no t make s e ns e to sp eak a bout the square (R). On the other hand, we hav e s e en that | uy q | = | pv q | , if p = | u y v | and q = | v z w | . Hence (R) is well deﬁned for these v alue s o f p a nd q . F urthermore, since u B y A v A q 6 = ∅ , by deﬁnition of simple twisting systems we hav e u a p | pvq | B q ◦ u A p e R v q ◦ u e R y v A q = u e R y q ◦ u B y a v q . (17) By tensoring b o th sides of the ab o ve relation with x A u on the left and with q B t on the rig h t we ge t that f = g , i.e. (R) is c o mm utativ e. Analogously , one shows tha t (L) is w ell deﬁned a nd commutativ e. All other s quares are well deﬁned by construction, their ar ro ws targeting to the right ob jects. The squar es (F) ar e co mm utativ e s inc e the tenso r pro duct is a functor . The remaining squares (A) are commut ative b y a s sociativity .  Theorem 2.14. Let M be an S -distr ibutive mono ida l doma in. I f ( e R, | . . . | ) is a simple twisting system, then the da ta in § 2.11 deﬁne an M -catego ry A ⊗ R B that factorizes through A and B . Pr o of. Let us assume that x A u B y A v B z A w B t 6 = ∅ . In view of the prev ious le mma, the o uter s quare in the diagram from Lemma 2.13 (2 ) is comm utative. It follows that x c y t ◦ x A u B y c z t ◦ x σ u y σ v z σ w t = x c z t ◦ x c y z A w B t ◦ x σ u y σ v z σ w t . If x A u B y A v B z A w B t = ∅ this identit y obviously holds. Since x A u B y A v B z A w B t is the copro duct of { x A u B y A v B z A w B t } u,v, w ∈ S , with the ca no nical inclusions { x σ u y σ v z σ w t } u,v, w ∈ S , we deduce that the comp osition in A ⊗ R B is asso ciativ e. W e apply the sa me s trategy to show that 1 x := x σ x x ◦ (1 A x ⊗ 1 B x ) is a left identit y map of x, tha t is we hav e x c x y ◦ (1 x ⊗ x A u B y ) = x A u B y for any y . By the universal pr operty of copro ducts and the deﬁnition of the comp osition in A ⊗ R B , it is enough to prov e that x σ | xxu | y ◦ x a x | xxu | b u y ◦ x A x e R x u B y ◦ (1 A x ⊗ 1 B x ⊗ x A u B y ) = x σ u y , (18) for all u ∈ S. If x A u is an initial ob ject we ha ve nothing to prove, as the domains of the sides of the above equation are also initial ob jects (recall that x A u B y = ∅ if x A u = ∅ ). Let us suppo se that x A u is not a n initial ob ject. Then by the deﬁnition of simple twisting sy stems (the third conditio n) we get | xxu | = u and x σ | xxu | y ◦ x a x | xxu | b u y ◦ x A x e R x u B y ◦ (1 A x ⊗ 1 B x ⊗ x A u B y ) = x σ u y ◦ x a x u b u y ◦ (1 A x ⊗ x A u ⊗ 1 B u ⊗ u B y ) . Thu s the equation (18) immediately follo ws b y the fact 1 A x and 1 B u are the identit y morphisms of x and u . The fact that 1 x is a righ t identit y map o f x can b e prov ed a nalogously . W e now claim that { x α y } x,y ∈ S is an M -functor. T aking in to a c coun t the deﬁnition o f α and x c y z we must prov e that x σ | y yz | z ◦ x a x | y yz | a z z ◦ x A y e R y z A z ◦ ( x A y ⊗ 1 A y ⊗ y A z ⊗ 1 A z ) = x σ z z ◦ ( x a y z ⊗ 1 A z ) , (19) for all x, y and z in S. Once again, if y A z = ∅ we have no thing to prov e. In the other case , one can pro ceed as in the pro of of (18) to get this e q uation. Similarly , β is an M -functor. It remains to prove the fact that A ⊗ R B factorize s through A a nd B . As a matter o f fact, for this enriched categor y , we shall show that x ϕ y is the identit y map of x ( A ⊗ R B ) y , for all x and y in S. Recall that x ϕ y is the unique map s uc h that x ϕ y ◦ x σ u y = x c u y ◦ x α u β y , for all u ∈ S. Hence to conclude the pro of of the theor em it is enough to obtain the following relation x σ | uuu | y ◦ x a u | uuu | b u y ◦ x A u e R u u B y ◦ ( x A u ⊗ 1 B u ⊗ 1 A u ⊗ u B z ) = x σ u y , (20) for all u ∈ S. W e may supp ose tha t x A u is no t initial ob ject. Th us | uu u | = u and w e can take x = u and y = u in (16). Hence, using the same reas oning a s in the proof of (18), w e deduce the required identit y .  Corollary 2. 1 5. Let A and B b e enric hed categories over an S -distributive mono ida l categor y M . Let us supp ose that for all x, y , z and t in S the function |· · · | : S 3 → S satisﬁes the e quations | xy | y z t || = | xz t | , || xy z | z t | = | xy t | , | xxy | = y and | xy y | = x. (21) 14 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN If { x e R y z } x,y ,z ∈ S is a family of maps which satisﬁes the identities (13)-(16) fo r all x, y , z and t in S, then the data in § 2.11 deﬁne a n M -catego ry A ⊗ R B that factorizes through A and B . Pr o of. Let x, y , u, v , z , w and t b e a rbitrary elements in S. By using the ﬁr s t tw o identities in (21) we get | uy q | = | pvq | = | pz w | , wher e p = | uy v | and q = | v z w | . Hence the ﬁrst statement in Lemma 2.1 3 is true. In particular , the squar es (R) and (L) in the diagr a m fr om Lemma 2.13(2) are well deﬁned. On the o ther hand, under the assumptions of the coro llary , the relation (17) hold. Therefore we ca n c on tin ue as in the pro of of the second par t o f Lemma 2.13 to show tha t (R) is commutativ e. Similarly , (L) is co mm utative to o. It follows that the outer square o f is co mm uta tiv e to o. B y the universal prop erty of the copro duct we deduce that the comp osition is asso ciative, see the ﬁrst pa ragraph o f the pro of of Theorem 2.14. F urthermor e, the relations in (21) together with the identit ies (13)-(16) imply the equa tions (18), (19) and (20). Pr oceeding as in the proo f of Theorem 2.14 we conclude that A ⊗ R B is an M -categ ory that fa c to rizes through A and B .  R emark 2 .1 6 . Throughout this remark w e assume that M is a T -distributive monoidal category , where T is an ar bitrary s e t. In other words, any family of ob jects in M has a copro duct a nd the tensor product is distributive ov er a ll copr oducts. It was noticed in [R W, § 2.1 and § 2.2] that, for such a mo no idal ca tegory M , one can deﬁne a bicategor y M - m at as follo ws. By construction, its 0 -cells ar e arbitra ry se ts. If I and J a r e tw o sets, then the 1- cells in M - mat from I to J are the J × I -indexed families o f ob jects in M . A 2-cell with source { X j i } ( j,i ) ∈ J × I and targe t { Y j i } ( j,i ) ∈ J × I is a family { f j i } ( j,i ) ∈ J × I of mor phisms f j i : X j i → Y j i . The compos ition of the 1-cells { X kj } ( k,j ) ∈ K × J and { Y j i } ( j,i ) ∈ J × I is the family { Z ki } ( k,i ) ∈ K × I , where Z ki := L j ∈ J X kj ⊗ Y j i . The vertical co mposition in M - mat of { f j i } ( j,i ) ∈ J × I and { g j i } ( j,i ) ∈ J × I makes sense if and only if the sour ce of f j i and the ta r get o f g j i are equa l for all i and j. If it exists, then it is deﬁned b y { f j i } ( j,i ) ∈ J × I • { g j i } ( j,i ) ∈ J × I = { f j i ◦ g j i } ( j,i ) ∈ J × I . Let { f j i } ( j,i ) ∈ J × I and { f ′ kj } ( k,j ) ∈ K × J be 2 -cells such that f j i : X j i → Y j i and f ′ kj : X ′ kj → Y ′ kj . By the univ ersal prope r t y of copr oducts, for each ( k , i ) ∈ K × I , there exists a unique morphism h ki : L j ∈ J X ′ kj ⊗ X j i → L j ∈ J Y ′ kj ⊗ Y j i whose restriction to X ′ kj ⊗ X j i is f ′ kj ⊗ f j i . By deﬁnition, the horizontal comp osition of { f ′ kj } ( k,j ) ∈ K × J and { f j i } ( j,i ) ∈ J × I is the family { h ki } ( k,i ) ∈ K × I . The ident ity 1-cells and 2-cells in M - mat are the ob vious ones . As p oin ted out in [R W], a mo nad on a set S in M - ma t is an M -catego ry with the set of o b jects S , and conv ersely . In particular , given t wo M -ca teg ories with the same set o f ob jects, one may sp eak a bout distributive laws b et ween the c orresp onding monads in M - mat . In our terminology , they are pr e cisely the twisting systems. In view of [R W, § 3.1], factor izable enriched c ategories generalize stric t factorization systems. In conclus ion, the Theo rem 2.3 may b e reg arded as a version of [R W, P ropos ition 3.3 ] for enriched categorie s. F or a simple twisting system ( e R, |· · · | ) b et ween B and A , the enriched categor y A ⊗ R B that w e constr uc ted in Theorem 2.14 can also b e des cribed in terms of monads . Let ρ : B ◦ A → A ◦ B denote the distr ibutiv e law asso ciated to ( e R, |· · · | ) , where ( A, m A , 1 A ) and ( B , m B , 1 B ) ar e the monads in M - mat corre s ponding to A a nd B , respectively . By the general theo ry of monads in a bicategor y , it follows that A ◦ B is a monad in M - mat with resp ect to the multiplication and the unit giv en by the form ulae: m := ( m A ◦ m B ) • (Id A ◦ ρ ◦ Id B ) and 1 := 1 A ◦ 1 B . It is no t diﬃcult to show that A ⊗ R B is the M -categ ory asso ciated to ( A ◦ B , m, 1) . By r eplacing S et - ma t with a suitable bicategory , o ne o btains similar results for other algebraic structures, suc h a s PROs and PR OPs; s ee [La]. W e also w ould like to note that distributive laws betw een pseudomonads are inv estigated in [Mar]. W e are indebted to the r e feree for pointing the pap ers [La, Mar, R W] out to us. F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 15 3. Ma tched p airs of enriched ca tegories. Throughout this se c tion ( M ′ , ⊗ , 1 , χ ) denote a braided categor y a nd we take M to be the monoidal category C oal g ( M ′ ) . Our aim is to characterize s imple twisting systems b et w een tw o categorie s tha t a re enriched ov er M . W e start by investigating some prop erties of the morphisms in M . F o r the momen t, we imp ose no co nditio ns on M ′ . A slig h tly more general version o f the following lemma is stated in [La , P ropo sition 3.2]. F or the sake o f completeness we include a pr oof o f it. Lemma 3.1. Let ( C, ∆ C , ε C ) , ( D 1 , ∆ D 1 , ε D 1 ) and ( D 2 , ∆ D 2 , ε D 2 ) b e coa lgebras in M ′ . L e t f : C → D 1 ⊗ D 2 be a mo rphism of coa lgebras. Then f 1 := ( D 1 ⊗ ε D 2 ) ◦ f and f 2 := ( ε D 1 ⊗ D 2 ) ◦ f are co a lgebra morphisms and the following rela tions hold: ( f 1 ⊗ f 2 ) ◦ ∆ C = f , (22) ( f 2 ⊗ f 1 ) ◦ ∆ C = χ D 1 ,D 2 ◦ ( f 1 ⊗ f 2 ) ◦ ∆ C . (23) Conv ersely , let f 1 : C → D 1 and f 2 : C → D 2 be coalgebra morphis ms such that (23) holds. Then f := ( f 1 ⊗ f 2 ) ◦ ∆ C is a coalgebra map such that ( D 1 ⊗ ε D 2 ) ◦ f = f 1 and ( ε D 1 ⊗ D 2 ) ◦ f = f 2 . (24) Pr o of. Let us a s sume that f : C → D 1 ⊗ D 2 is a coalg ebra mor phism. Let ε i := ε D i , for i = 1 , 2 . Clearly , D 1 ⊗ ε D 2 and ε D 1 ⊗ D 2 are co algebra morphisms. In conclusion f 1 and f 2 are mo r phisms in M . On the other hand, as f is a morphism in M we have ( D 1 ⊗ χ D 1 ,D 2 ⊗ D 2 ) ◦ (∆ D 1 ⊗ ∆ D 2 ) ◦ f = ( f ⊗ f ) ◦ ∆ C . (25) Hence, using the deﬁnition o f f 1 and f 2 , the relation (2 5), the fact that the braiding is a natur a l transformatio n and the compatibility relation b et ween the com ultiplication and the co unit we get ( f 1 ⊗ f 2 ) ◦ ∆ C = ( D 1 ⊗ ε 2 ⊗ ε 1 ⊗ D 2 ) ◦ ( f ⊗ f ) ◦ ∆ C =  D 1 ⊗  ( ε 2 ⊗ ε 1 ) ◦ χ D 1 , D 2 ⊗ D 2  ◦ (∆ D 1 ⊗ ∆ D 2 )  ◦ f = ( D 1 ⊗ ε 1 ⊗ ε 2 ⊗ D 2 ) ◦ (∆ D 1 ⊗ ∆ D 2 ) ◦ f = f . By applying ε 1 ⊗ D 2 ⊗ D 1 ⊗ ε 2 to (2 5) a nd using o nc e ag ain the co mpatibilit y b et w een the c o m ul- tiplication a nd the counit we obtain ( f 2 ⊗ f 1 ) ◦ ∆ C = ( ε 1 ⊗ D 2 ⊗ D 1 ⊗ ε 2 ) ◦ ( f ⊗ f ) ◦ ∆ C = ( ε 1 ⊗ D 2 ⊗ D 1 ⊗ ε 2 ) ◦ ( D 1 ⊗ χ D 1 ,D 2 ⊗ D 2 ) ◦ (∆ D 1 ⊗ ∆ D 2 ) ◦ f = χ D 1 ,D 2 ◦ ( ε 1 ⊗ D 1 ⊗ D 2 ⊗ ε 2 ) ◦ (∆ D 1 ⊗ ∆ D 2 ) ◦ f = χ D 1 ,D 2 ◦ f . Conv ersely , let us assume that f 1 : C → D 1 and f 2 : C → D 2 are morphisms in M such that (23) holds. Let f := ( f 1 ⊗ f 2 ) ◦ ∆ C . By the deﬁnition of the comultiplication on D 1 ⊗ D 2 and the fact that f 1 and f 2 are mor phisms in M , we get ∆ D 1 ⊗ D 2 ◦ f = ( D 1 ⊗ χ D 1 ,D 2 ⊗ D 2 ) ◦ (∆ D 1 ⊗ ∆ D 2 ) ◦ ( f 1 ⊗ f 2 ) ◦ ∆ C = ( D 1 ⊗ χ D 1 ,D 2 ⊗ D 2 ) ◦ ( f 1 ⊗ f 1 ⊗ f 2 ⊗ f 2 ) ◦ (∆ C ⊗ ∆ C ) ◦ ∆ C . T aking in to account (2 3) and the fact that the comultiplication is coasso ciative, it follows that ∆ D 1 ⊗ D 2 ◦ f = [ f 1 ⊗ ( χ D 1 ,D 2 ◦ ( f 1 ⊗ f 2 ) ◦ ∆ C ) ⊗ f 2 ] ◦ ( C ⊗ ∆ C ) ◦ ∆ C = [ f 1 ⊗ (( f 2 ⊗ f 1 ) ◦ ∆ C ) ⊗ f 2 ] ◦ ( C ⊗ ∆ C ) ◦ ∆ C = [(( f 1 ⊗ f 2 ) ◦ ∆ C ) ⊗ (( f 1 ⊗ f 2 ) ◦ ∆ C )] ◦ ∆ C = ( f ⊗ f ) ◦ ∆ C . The for m ula that deﬁnes f together with ε i ◦ f i = ε C yield ( ε 1 ⊗ ε 2 ) ◦ f = ( ε 1 ◦ f 1 ⊗ ε 2 ◦ f 2 ) ◦ ∆ C = ( ε C ⊗ ε C ) ◦ ∆ C = ε C . Thu s f is a morphism of coalgebra s, so the lemma is prov ed. The equations in (24) ar e obvious, as ε i ◦ f i = ε C .  16 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN R emark 3 .2 . Let f ′ , f ′′ : C → D 1 ⊗ D 2 be coalgebra morphisms. By the prece ding lemma, f ′ and f ′′ are equal if and only if ( ε 1 ⊗ D 2 ) ◦ f ′ = ( ε 1 ⊗ D 2 ) ◦ f ′′ and ( D 1 ⊗ ε 2 ) ◦ f ′ = ( D 1 ⊗ ε 2 ) ◦ f ′′ . 3.3. The morphisms x ⊲ y z and x ⊳ y z . Le t A and B denote t wo M -ca tegories whose ob jects ar e the elements of a set S . The ho m- ob jects of A and B are coalgebra s , which will be denoted b y ( x A y , x ∆ A y , x ε A y ) and ( x B y , x ∆ B y , x ε B y ). By deﬁnition, the comp osition and the iden tit y ma ps in A and B are coa lgebra morphisms. Note that the comultiplication of x B y A z is g iven by ∆ x B y A z = ( x B y ⊗ χ x B y , y A z ⊗ y A z ) ◦ x ∆ B y ∆ A z . Let | · · · | : S 3 → S b e a function and let e R denote an S 3 -indexed family of coalgebr a mo r phisms x e R y z : x B y A z → x A | xy z | B z . W e deﬁne x ⊲ y z : x B y A z → x A | xy z | and x ⊳ y z : x B y A z → | xy z | B z by x ⊲ y z := x A | xy z | ε B z ◦ x e R y z and x ⊳ y z := ( x ε A | xyz | B z ) ◦ x e R y z . (26) In view of Lemma 3.1, x ⊲ y z and x ⊳ y z are co a lgebra morphisms and they satisfy the r elations ( x ⊲ y z ⊗ x ⊳ y z ) ◦ ∆ x B y A z = x e R y z , (27) ( x ⊳ y z ⊗ x ⊲ y z ) ◦ ∆ x B y A z = χ x A | xyz | , | xyz | B z ◦ ( x ⊲ y z ⊗ x ⊳ y z ) ◦ ∆ x B y A z . (28) Conv ersely , if one starts with ⊲ := { x ⊲ y z } x,y ,z ∈ S and ⊳ := { x ⊳ y z } x,y ,z ∈ S , tw o families of coalgebra maps that satisfy (28), then by formula (27) we get the s et e R := { x e R y z } x,y ,z ∈ S whose elements are coalgebra maps, cf. Lemma 3.1. Ther efore, there is a n one-to- o ne co rrespo ndence b et ween the couples ( ⊲, ⊳ ) and the sets e R as ab o ve. O ur g oal is to characteriz e those couples ( ⊲, ⊳ ) that corres p onds to a simple t wisting sys tem in M ′ . Lemma 3.4. The statement s b elow ar e true. (1) If | xy | y z t | | = | xz t | then the relation (1 3) is equiv alent to the following equa tions: x ⊳ z t ◦ x b y z A t = | xz t | b | yz t | t ◦ x ⊳ y | y zt | B t ◦ ( x B y ⊗ y ⊲ z t ⊗ y ⊳ z t ) ◦ ( x B y ⊗ ∆ y B z A t ) , (29) x ⊲ z t ◦ x b y z A t = x ⊲ y | y zt | ◦ x B y ⊲ z t . (30) (2) If | xyz | z t || = | xy t | then the relation (14) is equiv alent to the following equatio ns: x ⊲ y t ◦ x B y a z t = x a | xy z | | xy t | ◦ x A | xy z | ⊲ z t ◦ ( x ⊲ y z ⊗ x ⊳ y z ⊗ z A t ) ◦ (∆ x B y A z ⊗ z A t ) , (31) x ⊳ y t ◦ x B y a z t = | xyz | ⊳ z t ◦ x ⊳ y z A t . (32) (3) If | xyy | = x then the re lation (15) is equiv alent to the following eq uations: x ⊳ x y ◦  1 B x ⊗ x A y  = x ε A y ⊗ 1 B y , (33) x ⊲ x y ◦  1 B x ⊗ x A y  = x A y . (34) (4) If | xxy | = y then the r e lation (16) is equiv alent to the following eq uations: x ⊳ y y ◦  x B y ⊗ 1 A y  = x B y , (35) x ⊲ y y ◦  x B y ⊗ 1 A y  = 1 A x ⊗ x ε B y . (36) Pr o of. In o r der to prov e the ﬁrst statement we apply the Remark 3.2 to f ′ := x e R z t ◦ x b y z A t and f ′′ := x A | xz t | b | y zt | t ◦ x e R y | y zt | B t ◦ x B y e R z t . Note that f ′′ is w ell deﬁned and its target is x A | xz t | B t , since the c odomain of x e R y | y zt | B t ◦ x B y e R z t is x A | xy | yz t || B t and | xy | y z t || = | xz t | . Cle arly , f ′ and f ′′ are coalgebra mo rphisms, since the F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 17 comp osite and the tensor pr oduct of tw o morphisms in M remain in M . An easy co mputation, based on the equation (27) and the formulae of x ⊲ y z and x ⊳ y z , yields us x ε A | xz t | B t ◦ f ′ = x ⊳ z t ◦ x b y z A t , x A | xz t | ε B t ◦ f ′ = x ⊲ z t ◦ x b y z A t , x ε A | xz t | B t ◦ f ′′ = | xz t | b | yz t | t ◦ x ⊳ y | yz t | B t ◦ ( x B y ⊗ y ⊲ z t ⊗ y ⊳ z t ) ◦ ( x B y ⊗ ∆ y B z A t ) . T aking in to account tha t x b y z is a co a lgebra mo rphism a nd using the deﬁnition of x ⊲ y z we get x A | xz t | ε B t ◦ f ′′ = x A | xz t | ε B | yz t | ◦ x e R y | y zt | ◦  x B y ⊗ ( y A | y zt | ε B t ◦ y e R z t )  = x ⊲ y | y zt | ◦ x B y ⊲ z t . In view of the Remark 3 .2 , we hav e f ′ = f ′′ if and only if x A | xz t | ε B t ◦ f ′ = x A | xz t | ε B t ◦ f ′′ and x ε A | xz t | B t ◦ f ′ = x ε A | xz t | B t ◦ f ′′ . Thu s, if | xy | y z t || = | xz t | , then (13) is equiv alent to (29) together with (3 0) . W e omit the pr o of o f the seco nd statement, b eing s imilar. T o prov e the third part of the lemma we reiterate the ab o ve reasoning. W e now take f ′ and f ′′ to b e the co a lgebra mo rphisms f ′ := x e R x y ◦  1 B x ⊗ x A y  and f ′′ := x A y ⊗ 1 B y . Since | xxy | = y b oth f ′ and f ′′ target in x A y B y . It is easy to see that (33) tog ether with (34) ar e equiv alent to (15). Similarly , o ne sho ws that the fourth statement is true.  Theorem 3.5. W e keep the no ta tion and the as s umptions from § 3.3. The set e R is a simple t wisting s y stem in M ′ if and only if the families ⊲ and ⊳ satisfy the follo wing conditions: (i) If x B y A z is not an initial ob ject then x A | xy z | B z is not an initial o b ject as well. (ii) If x B y B z A t is no t an initial ob ject in M ′ , then | xy | y z t || = | xz t | a nd the equa tions (29) and (30) hold. (iii) If x B y A z A t is not an initial ob ject in M ′ , then || xy z | z t | = | xy z | and the equations (3 1) and (32) hold. (iv) If x A y is not a n initial ob ject in M ′ , then | xxy | = y and the equations (33) a nd (3 4) ho ld. (v) If x B y is not an initial ob ject in M ′ , then | xy y | = x and the equations (35) and (36) ho ld. Pr o of. The co ndition (i) is a part of the de ﬁnitio n of simple twisting systems. If x B y B z A t is not a n initial ob ject in M ′ then we may ass ume that | xy | y z t || = | xz t | . Thus, by Lemma 3.4, the r elation (13) and the eq ua tions (29) a nd (30) are equiv alent. T o conclude the pro of we pro ceed in a similar wa y .  3.6. Matc hed pairs and the bicrossed pro duct. Let ⊲ := { x ⊲ y z } x,y ,z ∈ S and ⊳ := { x ⊳ y z } x,y ,z ∈ S be t wo families of maps as in § 3.3. W e sha ll say that the quintuple ( A , B , ⊲, ⊳, |· · · | ) is a matche d p air o f M -categ o ries if and only if ⊲ and ⊳ satisfy the conditions (i)-(v) from the above theorem. F or a matc hed pair ( A , B , ⊲, ⊳, |· · · | ) w e hav e just se en that ( e R, |· · · | ) is a simple t wisting system in M ′ , where e R := { x e R y z } x,y ,z ∈ S is the set of co algebra morphisms which a re deﬁned by the form ula (27). Hence, supp osing that M ′ is a n S -dis tributiv e domain, we may constr uct the twisted tensor pro duct A ⊗ R B , which is a n enriched catego ry ov er M ′ . W e shall call it the bicr osse d pr o duct of ( A , B , ⊲, ⊳, |· · · | ) and we s hall denote it b y A ⋊ ⋉ B . Prop osition 3.7 . The bicros sed pro duct of a matched pair ( A , B , ⊲, ⊳, |· · · | ) is enriched ov er the monoidal ca teg ory M := C o alg ( M ′ ) . Pr o of. Let { C i } i ∈ i be a family of coalgebras in M ′ . Le t us assume that the under lying family of ob jects ha s a co product C := L x ∈ S C i in M ′ . Let { σ i } i ∈ I by the s et of ca nonical inclusions in to C. T he r e a r e unique maps ∆ : C → C ⊗ C and ε : C → 1 s uc h that ∆ ◦ σ i = ( σ i ⊗ σ i ) ◦ ∆ i and ε ◦ σ i = ε i , 18 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN for a ll i ∈ I , where ∆ i and ε i are the comultiplication and the counit of C i . It is easy to see that ( C, ∆ , ε ) is a coalge br a in M ′ . Note that, by the construction of the coalgebr a str uc tur e on C, the inclusio n σ i is a coa lgebra map, for a n y i ∈ I . F urthermore, let f i : C i → D b e a coa lgebra morphism for every i ∈ I . By the univ ersal prop e rt y o f the copro duct there is a unique map f : C → D in M ′ such that f ◦ σ i = f i , for all i ∈ I . It is not diﬃcult to s ee that f is a mor phism of co a lgebras, so ( C, { σ i } i ∈ I ) is the copro duct o f { C i } i ∈ I in M . In par ticular, x A u B y = L u ∈ S x A u B y has a unique coalgebra structure such that the inclusio n x σ u z : x A u B y → x A u B y is a co algebra map, for all x, y and u in S . Recall that for the construction of the co mposition map x c y z : x A u B y A v B z → x A w B z one applies the universal prop erty of the copro duct to { f u,v } u,v ∈ S 2 , where f u,v = x σ | uy v | z ◦ x a | uy v | b z ◦ x A u e R y v B z . Since A and B ar e M - categories and u e R y v is a coalgebr a map, in v iew of the foregoing remarks, it follows that x c y z is a morphism in M , for a ll x, y , z ∈ S. The identit y ma p of x in A ⋊ ⋉ B is the coalgebr a map x σ x x ◦ (1 A x ⊗ 1 B x ) . In conclusion, A ⋊ ⋉ B is enr ic hed ov er M .  4. Examples. In this sectio n we give so me examples o f (simple) twisting systems. W e sta rt by co ns idering the case of S et -ca tegories, that is usua l categories . 4.1. Sim p l e t wisting systems of enric hed categories o v er S et. The categor y S et is a braided monoidal ca tegory with resp ect to the ca r tesian pro duct, its unit ob ject b eing {∅} . Clear ly , the empt y set is the initial ob ject in S et , and this categor y is an S -distributive domain, fo r any set S. W e ha ve already noticed that the ( † ) h yp othesis holds in S et . Let C be an enriched categor y over S et . Thus, by deﬁnition, C is a ca tegory in the usual sense, that is x C y is a set for all x, y ∈ S. An element in x C y is r egarded a s a function from y to x . It is easy to see that a given set X can b e s een in a unique wa y a s a coalg ebra in S et . As a matter of fact the comultiplication and the counit of this c o algebra are given by the diagonal map ∆ : X → X × X and the c onstan t map ε : X → {∅} , ∆( x ) = x ⊗ x a nd ε ( x ) = ∅ . Obviously , any function f : X → Y is mor phism of coalge bras in S et . Co ns equen tly , a ny category C may b e seen a s a n e nric hed categor y ov er C oal g ( S e t ) , Our aim is to desc r ibe the simple twisting systems b et ween t w o categor ies B and A . In view of the foregoing disc ussion a nd of our results in the previous section, for any simple twisting system R := { x e R y z } x,y ,z ∈ S there is a unique matc hed pair ( A , B , ⊲ , ⊳, |· · · | ) , a nd conv ersely . These structures a re related eac h other by the for m ula e (26) and (27). Since A is a n usual categ ory , the co mposition of morphisms will be denoted in the traditional wa y g ◦ g ′ , for an y g ∈ x A y and g ′ ∈ y A z (recall that the domain a nd the co domain o f g are y and x, resp ectiv ely). The same nota tion will be used for B . O n the other hand, for any f ∈ x B y and g ∈ y A z we shall write f ⊲ g := x ⊲ y z ( f , g ) and f ⊳ g = x ⊳ y z ( f , g ) . Since the comultiplication in this cas e is always the diagonal map, and the counit is the constant map to {∅} , the conditions of T he o rem 3 .5 and the following ones a re equiv alent. (i) If x B y A z is not empt y then x A | xy z | B z is not empt y as well. (ii) F o r a n y ( f , f ′ , g ) ∈ x B y B z A t we hav e | xy | y z t || = | xz t | , and ( f ◦ f ′ ) ⊲ g = f ⊲ ( f ′ ⊲ g ) and ( f ◦ f ′ ) ⊳ g = [ f ⊳ ( f ′ ⊲ g )] ◦ ( f ′ ⊳ g ) . (iii) F or a n y ( f , g , g ′ ) ∈ x B y A z A t we hav e || xy z | z t | = | xy t | , and f ⊳ ( g ◦ g ′ ) = ( f ⊳ g ) ⊳ g ′ and f ⊲ ( g ◦ g ′ ) = ( f ⊲ g ) ◦ [( f ⊳ g ) ⊲ g ′ ] . (iv) F or any g ∈ x A y we hav e | xxy | = y , and 1 B x ⊲ g = g and 1 B x ⊳ g = 1 B y . F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 19 (v) F or any f ∈ x B y we hav e | xy y | = x , and f ⊲ 1 A y = 1 B x and f ⊳ 1 A y = f . In this ca se the bicrossed pro duct A ⋊ ⋉ B is the category whose hom-s ets ar e given by x ( A ⋊ ⋉ B ) y = ` u ∈ S x A u B y . The identit y of x in A ⋊ ⋉ B is (1 A x , 1 B x ) . F o r ( g , f ) ∈ x A u B y and ( g ′ , f ′ ) ∈ y A v B z we hav e ( g , f ) ◦ ( g ′ , f ′ ) = ( g ◦ ( f ⊲ g ′ ) , ( f ⊳ g ′ ) ◦ f ′ ) . R emark 4 .2 . R. Rese br ugh a nd R.J. W o od sho wed that every twisting sys tems b et ween tw o S et - categorie s B a nd A is c o mpletely determined by a left a ction ⊲ of B o n A and a r igh t a ction ⊳ of A on B . Mor e pr ecisely , given a t wisting system R = { x R y z } x,y ,z ∈ S and the morphisms f ∈ x B y and g ∈ y A z , then x R y z ( f , g ) is an elemen t in x A u B z , wher e u is a certain element of S. Hence, there are unique morphisms f ⊲ g ∈ x A u and f ⊳ g ∈ u B z such that x R y z ( g , f ) = ( f ⊲ g , f ⊳ g ) . The ac tio ns ⊲ and ⊳ m us t satisfy se veral compatibility conditions, w hich ar e simila r to those that app ear in the above characterization of simple t wisting sys tems. F or details the r eader is r eferred to the s e c ond sectio n of [R W]. 4.3. The bicross ed pro duct of t w o g roupoi ds. W e now assume that ( A , B , ⊲, ⊳, |· · ·| ) is a matched pair of group oids. Recall that a gro upoid is a categ ory whose morphisms are in v ertible. W e claim that A ⋊ ⋉ B is also a gro upoid. Indeed, a s in the case of monoids , one ca n show that a category is a g roupoid if a nd only if every mor phism is r igh t inv ertible (or left inv er tible). Since x ( A ⋊ ⋉ B ) y = ` u ∈ S x A u B y , it is enoug h to prove that ( g , f ) is r igh t invertible, where g ∈ x A u and f ∈ u B y are arbitrar y morphisms. Ther efore, we are lo oking for a pair ( g ′ , f ′ ) ∈ y A v × v B x such that g ◦ ( f ⊲ g ′ ) = 1 A x and ( f ⊳ g ′ ) ◦ f ′ = 1 B x . Since g is an inv ertible morphism in x A u we get tha t f ⊲ g ′ = g − 1 ∈ u A x . Since f is invertible to o, g ′ = 1 A y ⊲ g ′ = ( f − 1 ◦ f ) ⊲ g ′ = f − 1 ⊲ ( f ⊲ g ′ ) = f − 1 ⊲ g − 1 . As g ′ ∈ y A v and f − 1 ⊲ g − 1 ∈ y A | y ux | we must hav e v = | y ux | . Thus we ca n now take f ′ = [ f ⊳ ( f − 1 ⊲ g − 1 )] − 1 ∈ | y ux | B x . 4.4. The sm ash pro duct category . W e take M to b e the monoidal categor y K - M o d , where K is a commutativ e ring. Hence in this case we work with K -linea r ca tegories. Let H be a K - bialgebra. W e deﬁne a n enriched categ ory H ov er K - M od b y setting x H x = H and x H y = 0 , for x 6 = y . The co mposition of morphisms in H is given b y the multiplication in H and the identit y of x is the unit of H . F or the comultiplication of H we shall use the Σ-notation ∆( h ) = P h (1) ⊗ h (2) . Let A denote an H -mo dule ca tegory , i.e. a catego ry enriched in H - M od . Thus H acts on x A y , for any x, y ∈ S , and the co mposition and the identit y maps in A are H - linear morphisms . O b v iously , A is a K -linear categ o ry . Our aim is to asso ciate to A a simple t wisting system R = { x e R y z } x,y ,z ∈ S . First we deﬁne | · · · | : S 3 → S by | xy z | = z . Then, using the a ctions · : H ⊗ x A z → x A z , we deﬁne x e R x z : H ⊗ x A z → x A z ⊗ H , x e R x z ( h ⊗ f ) = P h (1) · f ⊗ h (2) . F or x 6 = y w e take x e R y z = 0. It is easy to see that R is a simple twisting system of K -linear categorie s. Clearly K - M od is S -distributive, for any set S. If K is a ﬁeld then K - M od is a domain, so in this ca se the twisted tenso r pro duct of A and H with resp ect to R mak es sense , cf. 20 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN Theorem 2.1 4. It is called the smash pro duct of A b y H , and it is denoted b y A # H. By deﬁnition, x ( A # H ) y = x A y ⊗ H and ( f ⊗ h ) ◦ ( f ′ ⊗ h ′ ) = P f ◦ ( h (1) · f ′ ) ⊗ h (2) h ′ , (37) for any f ∈ x A y , f ′ ∈ y A z and h, h ′ ∈ H . 4.5. The se midirect pro duct. Let A b e a catego ry . Let us s uppose that ( B , · , 1) is a monoid that acts to the left on ea c h x A y via ⊲ : B × x A y → x A y . W e de ﬁne the catego ry B so that x B x = B and x B y = ∅ , for x 6 = y . The comp osition of morphisms is given by the multiplication in B . T o this data we asso ciate a matched pair ( A , B , |· · ·| , ⊲, ⊳ ) , setting f ⊳ g = f for an y ( f , g ) ∈ x B x A y , and deﬁning the function |· · · | : S 3 → S by | xy z | = z . Note that if x 6 = y then x B y A z is empty , so x ⊲ y z and x ⊳ y z coincide with the empty function. O ne shows that ( A , B , ⊲, ⊳, |· · ·| ) is a matched pa ir if and only if for any ( g , g ′ ) ∈ x A y A z and f ∈ B f ⊲ ( g ◦ g ′ ) = ( f ⊲ g ) ◦ ( f ⊲ g ′ ) , f ⊲ 1 A x = 1 A x . The corr esponding bicros sed pro duct will b e denoted in this case by A ⋊ B . If | S | = 1 then A ca n be identiﬁed with a mono id and A ⋊ B is the usual s emidirect pro duct of t wo monoids. F o r this reason we shall call A ⋊ B the s emidirect pr oduct of A with B . Note that x ( A ⋊ B ) y = x A y × B . F or an y f , f ′ ∈ B and ( g , g ′ ) ∈ x A y A z , the compositio n of mor phisms in A ⋊ B is given by ( g , f ) ◦ ( g ′ , f ′ ) = ( g ◦ ( f ⊲ g ′ ) , f ◦ f ′ ) . 4.6. Twisting sys te m s b etw een algebras i n M . W e now co ns ider a twisting s ystem R b et ween t wo M -categor ies B a nd A with the prop erty that S = { x 0 } . Obviously , M is S -distributive. W e shall use the no tation A = x 0 A x 0 and B = x 0 B x 0 . The comp osition map a := x 0 a x 0 x 0 and 1 A := 1 A x 0 deﬁne an algebra s tructure on A . A similar notation will b e us ed for the a lgebra corr esponding to the M -category B . L et R be a mor phism from B ⊗ A to A ⊗ B . Since x 0 σ x 0 x 0 is the identit y map of B ⊗ A, by Pro position 2.5, we deduce that x 0 R x 0 x 0 = R deﬁnes a t wisting system betw een B and A if a nd only if R satisﬁes the re la tions (13)-(16) with resp e c t to the unique map |· · · | : S 3 → S. In turn, they are equiv alent to the follo wing identities R ◦ ( b ⊗ A ) = ( A ⊗ b ) ◦ ( R ⊗ B ) ◦ ( B ⊗ R ) , (38) R ◦ ( B ⊗ a ) = ( a ⊗ B ) ◦ ( A ⊗ R ) ◦ ( R ⊗ A ) , (39) R ◦ (1 B ⊗ A ) = A ⊗ 1 B , (40) R ◦ ( B ⊗ 1 A ) = 1 A ⊗ B . ( 41) In conclus io n, in the cas e when | S | = 1 , to give a twisting system b et ween B a nd A is equiv alent to give a twisting map b etw een the a lgebras B and A, that is a morphism R which satisﬁes (3 8)-(41). By applying Cor ollary 2.15 to a twisting map R : B ⊗ A → A ⊗ B (vie w ed as a twisting system betw een tw o M -categories with one ob ject) we get the twiste d tensor algebr a A ⊗ R B . T he unit of this a lgebra is 1 A ⊗ 1 B and its m ultiplica tio n is given by m = ( a ⊗ b ) ◦ ( A ⊗ R ⊗ B ) . Note that, in view o f the fo regoing r emarks, an alge br a C in M factor izes thro ugh A a nd B if a nd only if it is isomo rphic to a twisted tensor a lgebra A ⊗ R B , for a certain twisting map R . An a lgebra in the monoidal c ategory K - M od is by deﬁnition an as sociative and unital K -algebra. Twisted tenso r K -alg ebras were inv estigated for instance in [Ma1], [T am], [CSV], [CIMZ], [LPoV ] and [JLPvO]. Coalgebr a s ov er a ﬁeld K are a lgebras in the monoida l category ( K - M o d ) o . Hence a twisting map b et ween tw o co algebras ( A, ∆ A , ε A ) and ( B , ∆ B , ε B ) is a K -linear map R : A ⊗ K B → B ⊗ K A F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 21 which sa tis ﬁe s the equations that ar e o btained from (38)-(41) by making the substitutions a := ∆ A , b := ∆ B , 1 A := ε A and 1 B := ε B , and reversing the order of the factor s with resp ect to the comp osition in M . F o r ex ample (38 ) s hould b e repla ced with (∆ B ⊗ K A ) ◦ R = ( B ⊗ K R ) ◦ ( R ⊗ K B ) ◦ ( A ⊗ K ∆ B ) . Obviously A ⊗ R B is the K -co algebra ( A ⊗ K B , ∆ , ε ), wher e ε := ε A ⊗ ε B and ∆ = ( A ⊗ R ⊗ B ) ◦ (∆ A ⊗ K ∆ B ) . An algebra in Λ- M od -Λ is called a Λ- ring . Specia lizing M to Λ- M o d -Λ we ﬁnd the de ﬁnitio n o f the twiste d tensor Λ -ring . Dua lly , Λ-corings a re a lgebras in (Λ- M od -Λ) o . Thus in this particular case we are led to the constr uction of the twiste d tensor Λ- c oring . By deﬁnition a monad on a c a tegory A is an a lgebra in [ A , A ]. If ( F , µ F , ι F ) and ( G, µ G , ι G ) are mona ds in M , then a natural transfo r mation λ : G ◦ F → F ◦ G satisﬁes the relations (38)-(41) if and only if λ is a distributive law , cf. [Be]. Let F 2 := F ◦ F . F or every distributive law λ we g e t a monad ( F ◦ G, µ, ι ) , where ι := ι F G ◦ ι G = F ι G ◦ ι F and µ := µ F G ◦ F 2 µ G ◦ F λG = F µ G ◦ µ F G 2 ◦ F λG. Distributive laws be t ween co mo nads can b e deﬁned similarly , o r working in [ A , A ] o . Finally , twisting ma ps in O pmon ( M ) hav e been considered in [BV]. In lo c. cit. the authors deﬁne a bimonad in M as an a lg ebra in O pmon ( M ). Hence a twisting map b et ween tw o bimona ds is an opmonoida l distributive law b et ween the underlying monads. F or any o pmonoidal dis tributiv e law λ b et ween the bimonads G and F , ther e is a ca nonical bimonad structure on the endofunctor F ◦ G . See [B V, Sec tio n 4] for details . 4.7. Matc hed pairs of algebras in C o a lg ( M ′ ) . Let M denote the catego ry of coalg ebras in a braided mono idal categ ory ( M ′ , ⊗ , 1 , χ ) . By deﬁnition, a bialgebra in M ′ is an a lgebra in M . W e ﬁx tw o bialgebras ( A, a, 1 A , ∆ A , ε A ) a nd ( A, b, 1 B , ∆ B , ε B ) in M ′ and take R : B ⊗ A → A ⊗ B to b e a mo r phism in M . By L emma 3.1, there are the coalg ebra maps ⊲ : B ⊗ A → A and ⊳ : B ⊗ A → B such that R = ( ⊲ ⊗ ⊳ ) ◦ ∆ B ⊗ A and χ A,B ◦ ( ⊲ ⊗ ⊳ ) ◦ ∆ B ⊗ A = ( ⊳ ⊗ ⊲ ) ◦ ∆ B ⊗ A . (42) W e hav e seen that R is a t wisting map in M if and only if it satisﬁes the relations (38)-(41). In view of Lemma 3.4, these eq ua tions ar e equiv alent to the fact that ( A, ⊲ ) is a le ft B -module and ( B , ⊳ ) is a r igh t A -mo dule such that the following identities hold: ⊳ ◦ ( b ⊗ A ) = b ◦ ( ⊳ ⊗ B ) ◦ ( B ⊗ ⊲ ⊗ ⊳ ) ◦ ( B ⊗ ∆ B ⊗ A ) , (43) ⊲ ◦ ( B ⊗ a ) = a ◦ ( A ⊗ ⊲ ) ◦ ( ⊲ ⊗ ⊳ ⊗ A ) ◦ (∆ B ⊗ A ⊗ A ) , (44) ⊳ ◦ (1 B ⊗ A ) = ε A ⊗ 1 B , (45) ⊲ ◦ ( B ⊗ 1 A ) = 1 A ⊗ ε B . (46) By deﬁnition, a matche d p air of bialgebr as in M ′ consists of a left B -action ( A, ⊲ ) and a right A -action ( B , ⊳ ) in M s uc h that the s econd equation in (42) and the relations (43)-(46) hold. F o r a matched pair of bialgebras we shall use the notatio n ( A, B , ⊲, ⊳ ) . Summarizing, there is an one to-one-cor respo ndence b et w een t wisting maps o f bialgebra s in M ′ and matched pairs of bialgebras in M ′ . If ( A, B , ⊲, ⊳ ) is a matched pa ir of bialgebras and R is the co r respo nding twisting map, then A ⊗ R B will b e ca lle d the bicr osse d pr o duct of the bialgebr as A and B , and it will b e denoted by A ⋊ ⋉ B . Note that A ⋊ ⋉ B is an a lgebra in M . Thus the bicrossed pro duct of A and B is a bia lgebra in M ′ . The unit of this bia lgebra is 1 A ⊗ 1 B and the m ultiplication is given by m = ( a ⊗ b ) ◦ ( A ⊗ ⊲ ⊗ ⊳ ⊗ B ) ◦ ( A ⊗ ∆ B ⊗ A ⊗ B ) . As a coalgebra A ⋊ ⋉ B is the tensor pr oduct co algebra of A and B . W e a lso conclude that a bialgebra C in M ′ factorizes thr ough the sub-bialgebras A a nd B if and only if C ∼ = A ⋊ ⋉ B . 22 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN As a ﬁrst application, let us take M ′ to be the ca tegory of sets, whic h is bra ided with resp ect to the braiding giv en by ( X , Y ) 7→ ( Y , X ) and ( f , g ) 7→ ( g , f ), for a n y sets X, Y and any functions f , g . W e ha ve a lready noticed that there is a unique coalgebra structure on a given set X ∆( x ) = ( x, x ) , ε ( x ) = ∅ , where ∅ denotes the empty set; recall that the unit o b ject in S et is {∅ } . Hence an o rdinary monoid, i.e. a n a lgebra in S et , has a natur al bialgebra structure in this br a ided category . Mor eo ver any t wisting ma p R : B × A → A × B b et ween t wo monoids ( A, · , 1 A ) and ( B , · , 1 B ) is a t wisting map of bialgebr as in S et . Let ( A, B , ⊲, ⊳ ) b e the corresp onding matc hed pair. One easily shows tha t the second condition in (42) is a lw ays true. By notation, the functions ⊲ and ⊳ ma p ( f , g ) ∈ B × A to f ⊲ g and f ⊳g , r espectively . Hence the equations (4 3) -(46) are equiv alent to the following ones : ( f · f ′ ) ⊳ g = [ f ⊳ ( f ′ ⊲ g )] · ( f ′ ⊳ g ) , f ⊲ ( g · g ′ ) = ( f ⊲ g ) · [( f ⊳ g ) ⊲ g ′ ] , 1 B ⊳ g = 1 B and f ⊲ 1 A = 1 A . Since R ( f , g ) = ( f ⊲ g , f ⊳ g ) the pro duct o f the monoid A ⋊ ⋉ B is deﬁned b y the fo rm ula ( g , f ) · ( g ′ , f ′ ) = ( g ( f ⊲ g ′ ) , ( f ⊳ g ′ ) f ′ ) . In conclus ion, a monoid C factorizes thro ugh A and B if a nd only if C ∼ = A ⋊ ⋉ B . W e ha ve seen tha t the bicr ossed pro duct o f tw o g roupoids is a gr oupoid. Thus, if A and B are groups, then A ⋊ ⋉ B is a group as w ell. This result was proved by T a keuchi who int ro duced the matched pairs of groups in [T ak]. W e now co nsider the braided categ ory K - M o d , who se braiding is the usua l ﬂip ma p. An a lgebra in M , the monoidal categ ory of K -coalgebr as, is a bialgebra o ver the ring K , and conv ersely . Pro ceeding as in the pr evious ca se, one shows that a twisting ma p R : B ⊗ K A → A ⊗ K B of bialgebras is uniquely deter mined b y the coalgebr a maps ⊲ : B ⊗ K A → A and ⊳ : B ⊗ K A → B via the formula R ( f ⊗ g ) = P ( f (1) ⊲ g (1) ) ⊗ ( f (2) ⊳ g (2) ) . Using the Σ- notation, the seco nd equation in (42) is true if a nd only if P ( f (1) ⊳ g (1) ) ⊗ ( f (2) ⊲ g (2) ) = P ( f (2) ⊳ g (2) ) ⊗ ( f (1) ⊲ g (1) ) , for any f ∈ B and g ∈ A. O n the other hand, the equations (43)-(46) ho ld if and only if ( g g ′ ) ⊳ f = P [ g ⊳ ( g ′ (1) ⊲ f (1) )]( g ′ (2) ⊳ f (2) ) , g ⊲ ( f f ′ ) = P ( g (1) ⊲ f (1) )[( g (2) ⊳ f (2) ) ⊲ f ′ ] , f ⊲ 1 A = ε B ( f )1 A , 1 B ⊳ g = ε A ( g )1 B , for any f , f ′ ∈ B and any g , g ′ ∈ A. Thus, we r edisco ver the deﬁnition o f m atch e d p airs of bialgebr as and the for m ula for the m ultiplication o f the double cr oss pr o duct , see [Ma2, Theo rem 7.2.2]. Namely , ( f ⊗ g )( f ′ ⊗ g ′ ) = P f ( g (1) ⊲ f ′ (1) ) ⊗ ( g (2) ⊳ f ′ (2) ) g ′ . 4.8. Twisting systems b et w een thin categories. O ur aim no w is to inv estigate the t wisting systems b et ween t wo thin categ ories B a nd A . Recall tha t a categ ory is thin if there is at most one morphism b etw ee n a ny couple of ob jects. Thus, for a ny x and y in S we hav e that either x A y = { x g y } or x A y is the empty set. Clea rly , if x A y = { x g y } and y A z = { y g z } then x g y ◦ y g z = x g z . The identit y morphism o f x is x g x . Similar ly , if x B y is not empt y then x B y = { x f y } . W e ﬁx a t wis ting system R betw een B and A. It is deﬁned b y a family of maps x R y z : x B y × y A z → ` u ∈ S A u B z that render commutativ e the dia g rams in Figure 3. W e claim that R is simple. W e need a function |· · · | : S 3 → S such that the imag e of x R y z is included in to x A | xy z | B z for a ll ( x, y , z ) ∈ S 3 . Let T ⊆ S 3 denote the set of all triples s uc h that x B y A z = x B y × y A z is not empty . Of course, if F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 23 ( x, y , z ) is not in T then x R y z is the empty function, so we can take | xy z | to be an a rbitrary element in S. F or ( x, y , z ) ∈ T there exists | xy z | ∈ S such that x R y z ( x f y , y g z ) = ( x g | xyz | , | xyz | f z ) . (47) Hence x R y z is a function fro m x B y A z to x A | xy z | B z . Note that x A | xy z | B z is not empty in this case. F or any ( x, y , z ) ∈ S 3 we set x e R y z := x R y z . By P ropos ition 2.5 a nd Co rollary 2.7 it follows that R is simple. W e would like now to re w r ite the co nditions fr om the deﬁnition of simple twisting systems in an equiv alent form, that only in volv es pr operties of T and |· · · | . F or instance let us sho w that the ﬁrst c ondition from Cor ollary 2.7 is equiv alent to: (i) If ( y, z , t ) ∈ T and ( x, y , | y z t | ) ∈ T then | xy | y z t || = | xz t | . Indeed, if x B y B z A t is not empty then y B z A t 6 = ∅ , s o ( y , z , t ) ∈ T . W e hav e alrea dy noticed tha t y A | y zt | B t is not empt y , provided that y B z A t is so . Since x B y and y A | y zt | are not empty it follows that x B y A | y zt | has the sa me pr operty , that is ( x, y , | y z t | ) ∈ T . There fore, if x B y B z A t is not empty then ( y , z , t ) ∈ T and ( x, y , | y z t | ) ∈ T . It is easy to see that the reversed implication is also true. Thu s it remains to prove that the eq uation (1 3) holds in the case when x B y B z A t is not empty . But this is obvious, as x R z t ◦ x b y z A t and ( x A | xy | yz t || b | y zt | t ) ◦ x R y | y zt | B t ◦ x B y R z t hav e the same sour ce x B y B z A t and the s a me target x A | xz t | B t = x A | xy | xy z || B t . Both sets are singleto ns , so the ab o ve t wo morphisms must b e equa l. Pro ceeding in a similar w ay , we can prov e that the other three conditions from Corollar y 2 .7 are re spectively equiv alent to: (ii) If ( x , y , z ) ∈ T and ( | xy z | , z , t ) ∈ T then || xy z | z t | = | xy t | . (iii) If ( x, x, y ) ∈ T then | xxy | = y . (iv) If ( x, y , y ) ∈ T then | xyy | = x. The last condition in the deﬁnition of simple t wisting systems is equiv ale n t to: (v) If ( x, y , z ) ∈ T then x A | xy z | B z is not empt y . Hence for a twisting sys tem R the function |· · · | satis ﬁe s the a bov e ﬁve conditions. Co nversely , let |· · · | : S 3 → S denote a function such that the ab ov e ﬁve co nditions hold. Let x R y z be the empt y function, if ( x, y , z ) is not in T . Otherwise we deﬁne x R y z by the formula (47). In view of the foreg oing rema rks it is not diﬃcult to see that R := { x R y z } x,y ,z ∈ S is a simple t wisting system. Clearly tw o functions |· · · | and |· · · | ′ induce the same twisting system if and only if their r e- striction to T are eq ual. Summar izing, we have just proved the theore m below. Theorem 4.9 . Let A a nd B b e thin categories. Let T denote the set of a ll triples ( x, y , z ) ∈ S 3 such that x B y A z is not empty . If R is a twisting system b et ween B and A then there ex ists a function |· · · | : S 3 → S such that the co nditions (i)-(v) from the previous subsection hold, and conv ersely . Two functions |· · · | and |· · · | ′ induce the same t wisting system R if a nd only if their restriction to T are equa l. 4.10. The twisted tensor pro duct of thin categories. Let R b e a t wisting system be tw een tw o thin ca teg ories B and A. By the preceding theorem, R is simple a nd there are T and | · · · | : S 3 → S such that the conditions (i)-(v) ho ld. In par ticular, the twisted tensor pro duct of these categories exists. By deﬁnition, we have x ( A ⊗ R B ) y = ` u ∈ S ( x A u × u B z ) . W e can identify this s et with x S y := { u ∈ S | x A u B y 6 = ∅ } . F or u ∈ x S y and v ∈ y S z we hav e ( u, y , v ) ∈ T . Thus u R y v ( u f y , y g v ) = ( u g | uy v | , | uy v | f v ) , so the comp osition in A ⊗ R B is giv en by ( x g u , u f y ) ◦ ( y g v , v f z ) = ( x g u ◦ u g | uyv | , | uy v | f v ◦ v f z ) = ( x g | uy v | , | uy v | f z ) . Let C ( S, T , |· · · | ) b e the ca teg ory who s e ob jects are the elements of S. By deﬁnition, the hom-set x C ( S, T , |· · · | ) y is x S y , the identit y ma p o f x ∈ S is x itself and the composition is given by ◦ : x S y × y S z → x S z , u ◦ v = | uy v | . Therefore, we hav e just proved that A ⊗ R B a nd C ( S, T , | · · · | ) are isomo rphic. 24 AUR A B ˆ ARDES ¸ AND DRA GOS ¸ S ¸ TEF AN R emark 4.11 . Let C b e a small categor y . Let S denote the s et o f ob jects in C . The catego ry C factorizes through tw o thin ca tegories if and only if there are T ⊆ S and |· · · | : S 3 → S a s in the previous subsection suc h that C is isomorphic to C ( S, T , |· · · | ) . 4.12. Twisting system s be t ween p osets. An y p oset is a thin ca tegory , so we can apply Theorem 4.9 to character ize a twisting system R betw een tw o po sets B := ( S,  ) and A := ( S, ≤ ). In this setting the co rrespo nding set T contains a ll ( x, y , z ) ∈ S 3 such that x  y and y ≤ z . F or simplicity , we shall wr ite this condition as x  y ≤ z . A similar no tation will b e used for arbitrarily long sequences of elements in S. F or instance, x ≤ y  z  t ≤ u means that x ≤ y , y  z , z  t and t ≤ u . The function |· · ·| must s atisﬁes the following conditions: (i) If x  y ≤ z then x ≤ | xy z |  z . (ii) If x  y  z ≤ t then | xy | y z t || = | xz t | . (iii) If x  y ≤ z ≤ t then || xyz | z t | = | xy t | . (iv) If x ≤ y then | xxy | = y . (v) If x  y then | xyy | = x. In the case when the p osets ≤ and  are identical, an ex ample of function |· · ·| : S 3 → S that satisﬁes the above conditions is given b y | xyz | = z , if y 6 = z , and | xyz | = x, otherwise. 4.13. E xa mple of t wisting map b et ween tw o group oids. Let A b e a group oid with tw o ob jects, S = { 1 , 2 } . The hom-se ts of A are the fo llowing: 1 A 2 = { u } , 2 A 1 = { u − 1 } , 1 A 1 = { I d 1 } , 2 A 2 = { I d 2 } . Note that A is thin. W e set B := A and w e take R to b e a twisting system betw een B a nd A . By Theorem 4.9 there ar e T and |· · ·| : S 3 → S that satisﬁes the conditions (i)-(v) in § 4 .8. Since all sets x B y A z = x B y × y A z are nonempt y it follows that T = S. Thus | xxy | = y and | xy y | = x , for all x, y ∈ S . There are t wo triples ( x, y , z ) ∈ S 3 such that x 6 = y and y 6 = z , namely (1 , 2 , 1 ) and (2 , 1 , 2) . Hence w e have to compute | 121 | and | 212 | . If w e assume that | 121 | = 1 , then 1 = | 221 | = | 21 | 121 || = | 211 | = 2 , so w e get a contradiction. Th us | 121 | = 2 , and pro ceeding in a simila r wa y o ne proves that | 21 2 | = 1 . It is easy to check that |· · ·| sa tisﬁes the required co nditions, so there is only one twisting map R betw een A and itself. Since A is a g roupoid, the cor responding bicr ossed pro duct C := A ⋊ ⋉ A is a gro upoid as w ell, see the subsection (4.3). By deﬁnition, 1 C 1 = ` x ∈{ 1 , 2 } 1 A x × x A 1 = { ( I d 1 , I d 1 ) , ( u, u − 1 ) } . Analogously one sho ws that 1 C 2 = { ( I d 1 , u ) , ( u, I d 2 ) } , 2 C 1 = { ( I d 2 , u − 1 ) , ( u − 1 , I d 1 ) } and 2 C 2 = { ( I d 2 , I d 2 ) , ( u − 1 , u ) } . By construction of the twisting map R w e get 1 R 2 1 ( u, u − 1 ) ∈ 1 A | 121 | × | 121 | A 1 = { ( u, u − 1 ) } . The other maps x R y z can b e deter mined analogous ly . The complete structur e of this g roupoid is given in the picture b elow, where we used the notation f := ( u, u − 1 ) and g := ( I d 1 , u ). 1 2 Id 1   f Y Y Id 2   g ◦ f ◦ g − 1 Y Y   Y Y g + + g ◦ f " " g − 1 k k f − 1 ◦ g − 1 b b Note that f 2 = I d 1 and g − 1 = ( I d 2 , u − 1 ). Now we can say easily which arrow cor respo nds to a given mor phism in C , a s in eac h home-s et we have identiﬁed a t least one element. Ac knowledgmen ts. The ﬁr st named a utho r was ﬁnancially suppo rted by the funds o f the Con- tract POSDRU/6/1.5/S/1 2. The seco nd named author was ﬁnancially s upported by CNCSIS, Contract 560 /2009 (CNCSIS code ID 69). F ACTORIZAB LE E NRICHED CA TEGORIES AND APPLICA TIONS 25 References [Be] J. Beck, Distri b utive laws , Lecture Notes in Mathematics 80 (1969), 119-140. [BD] Y. Bespalov and B. Drabant, Cr oss pr o duct bialgebr as , J. Algebra 2 40 (2001), No.2, 445-504. [BV] A. Bruguieres and A. Virelizier, The double of a Hopf monad , arXiv:0812.24 4. 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Factorizable enriched categories and applications

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