The Eilenberg-Moore category and a Beck-type theorem for a Morita context

THE EILENBE R G-MOORE CA TEGOR Y A N D A BECK-TY P E THEOREM F OR A MORIT A CONTEX T TOMASZ BRZEZI ´ NSKI, ADRIAN V AZQUEZ MAR QUEZ, AND JOOST VERCR UYSSE Abstract. The Eilen berg -Mo ore constructions and a Bec k-t yp e theorem for pairs of monads are desc rib ed. More sp eciﬁcally , a notion of a Morita c ontext co mprising of t wo monads, t wo bialgebra functors a nd tw o c o nnecting maps is introduced. It is shown that in many cases equiv alences betw een catego ries of alg e br as are induced by such Morita contexts. The Eilenberg -Mo ore ca tegory of r epresentations of a Morita context is constructed. This construction allows one to ass o ciate t wo pa irs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Mor ita context. It is sho wn tha t, conversely , ev ery Mo rita con text arises fro m a do uble adjunction. The co mparison functor betw een the domain o f right adjoint functors in a double adjunction and the Eilenberg - Mo ore ca tegory of the a s so ciated Mor ita context is deﬁned. The s uﬃcien t a nd neces sary conditions for this compariso n functor to be a n equiv alence (or for the moritability of a pair of functors with a co mmon do main) a re derived. Contents 1. In tro duction 2 2. Double adjunctions and Morita contex ts 3 2.1. Adjunctions and (co)monads 3 2.2. The category of do uble adjunctions 4 2.3. The category of Morit a conte xts 4 3. A Bec k-ty p e theorem fo r a Morita con text 6 3.1. F rom double adjunctions to Morit a contexts 6 3.2. The Eilen b erg- Mo ore category of a Morita context 7 3.3. F rom Morita con texts to double adjunctions 8 3.4. Ev ery Morita context a r ises from a double a djunction 10 3.5. The comparison functor 11 3.6. Moritabilit y 13 4. Morita theory 17 4.1. Preserv ation of co equalisers b y algebras 17 4.2. Morita con texts and equiv alences of categories of a lgebras 18 5. Examples and applicatio ns 24 5.1. Blo wing up o ne a dj unction 24 5.2. Morita theory for rings 27 5.3. Categories with binary copro ducts 29 5.4. F ormal duals 30 5.5. Herds v ersus pretorsors 30 Date : Septem b er 2 009. 2000 Mathematics Su bje ct Classiﬁc ation. 18A40. 1 2 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE 6. Remarks on dualisations and generalisations 32 6.1. Dualisations 32 6.2. Bicategories 32 Ac knowle dgemen ts 33 References 33 1. Intro duction In the last t wo decades, Hopf-Galois theory w en t through a series o f generalisations, ultimately leading t o t he theory of G alois como dules o v er corings, whic h elucidates its relation with we ll-kno wn Bec k’s monadicit y theorem. This ev olut io n has reviv ed t he in t erest a mong Hopf a lgebraists in the theory of ( co) mo na ds. Seve ral asp ects of Hopf(- Galois) theory hav e b een reformulated in the framew ork o f (co)monads so that f urther clariﬁcation of the underlying categorical mec ha nisms of the theory has b een ac hiev ed. This has led in particular to tw o diﬀeren t approac hes to t he deﬁnition of a categorical (functorial) notion of a her d or pr e-torsor , app ear ing almost sim ultaneously in [4] and [3]. The motiv ation for the presen t pap er was to study the connection b et ween these t w o approa ches in more detail. The deﬁnition of a pre-trosor in [3] tak es a pair of adjunctions (with coinciding co domain category f or the left adjoin ts) as a starting p oint. The aim of t his pap er is to show that there is a close relationship b et wee n pairs of adjunctions and f unctorial Morita con texts similar to the corresp ondence b etw een single adjunctions and monads. In this sense the results presen ted here can b e interpreted as a ‘tw o-dimensional’ v ersion of t he latter corresp ondence. A key feat ure of this work is that it links asp ects of the theory that a re of more algebraic nature ( Morit a conte xts) with asp ects that are of more categorical nature (Bec k’s theorem). More precisely , w e pro v e a v ersion of Bec k’s theorem on precise monadicity in this ‘t w o- dimensional’ setting and pro vide a categor ical (monadic) v ersion of classical Morita theory . These are the main results a nd the organisation of the pap er. In Section 2 w e recall from [3] the deﬁnition of the category of double adjunctions on categories X and Y , Adj ( X , Y ), and in tr o duce the catego r y M o r ( X , Y ) of (functorial) Morita con texts. In Section 3 w e describ e functors connecting categories of do uble adjunctions and Morita con texts. More precisely we compare categories Adj ( X , Y ) and Mor ( X , Y ). First we deﬁne a functor Υ : Adj ( X , Y ) → Mo r ( X , Y ). T o construct a functor in the con vers e direction, to eac h Morit a con text T w e asso ciate its Eilenb er g-Mo or e c ate gory ( X , Y ) T . This is v ery reminiscen t of the classical Eilen b erg-Mo ore construction of algebras o f a monad (recalled in Section 2.1), and, in a w a y , is based on doubling o f the latter. Ob jects in ( X , Y ) T are tw o alg ebras, one for each monad in T , together with t w o connecting morphisms. Once ( X , Y ) T is deﬁned, t w o adjunctions, one b etw een ( X , Y ) T and X the ot her b et wee n ( X , Y ) T and Y , are constructed. This construction yields a functor Γ : Mo r ( X , Y ) → Adj ( X , Y ). Next it is shown that the functors (Γ , Υ) form an adjoin t pair, and that Γ is a full and faithful functor. The counit of this adjunction is giv en b y a c omp arison functor K whic h compares the common category Z in a double adjunction Z w ith the Eilen b erg-Mo or e categor y of the asso ciated Morita con text T = Υ( Z ). A necessary and suﬃce n t condition for the comparison functor to b e an equiv alence are deriv ed. This is closely related to the existence of A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 3 colimits of diag r ams of certain t yp e in Z and is a Morita–double adjunction v ersion of t he classical Bec k theorem (on precise mona dicity). In Section 4 w e a na lyse whic h ob jects of Mo r ( X , Y ) describ e equiv alences b et ween categories of algebras of monads. It is also prov en that large classes o f equiv alences b et w een categories o f alg ebras are induced b y Morita con texts. In Section 5 examples and sp ecial cases of the theory deve lop ed in preceding sections are given. In particular, it is sho wn how the main results of Section 3 can b e applied to a single adjunction leading to a new p oint of view on some asp ects of desce n t theory . The Eilen b erg-Mo ore category associated to the mo dule-theoretic Morita con text is iden t iﬁed as the cat ego ry of mo dules of the a sso ciated matrix Morita ring. This pro cedure can b e imiated in t he g eneral case if w e assume the existenc e and preserv ation of binary copro ducts in all categories and b y all functors inv olv ed. W e also sho w that the theory dev elop ed in Sections 2 – 4 is applicable to pre-torsors and herd functors, thus bringing forth means for comparing pre-torsors with balanced herds. The pap er is completed with comments on dual vers ions of constructions presen ted and with an outlo ok. Throughout the pap er, the composition of functors is denoted b y juxtap osition, the sym b ol ◦ is reserv ed for comp o sition of natural transformations and morphisms. The action of a functor on an ob ject or morphism is usually denoted b y juxtap osition of corresp onding sym b ols (no brac k ets are ty pically u sed). Similarly the morphism corresp onding to a natural transformation, sa y α , ev aluated at an ob ject, say X , is denoted b y a juxtap osition, i.e . b y α X . F o r an ob ject X in a category , w e will use the sy m b ol X as well to denote t he iden tit y morphism on X . T ypically , but no t exclusiv ely , ob j ects and functors are denoted by capital Latin letters, morphisms b y small Latin letters and natural tra nsformations by Greek letters. 2. Double adjunctions an d Morit a contexts The aim of this s ection is to recall the standard correspondence b et w een adjoin t functors and (co)monads a nd to in tro duce the main categories studied in the pap er. 2.1. Adjunctions and ( co)monads. It is well-kno wn fro m [6] that there is a close relationship b et w een pairs of adjoint functors ( L : X → Y , R : Y → X ), monads A o n X and c omonads C on Y . Sta rting f r om an adjunction ( L, R ) with u nit η : X → RL and counit ε : LR → Y , the corresp onding monad a nd comonad a re A = ( R L, R εL, η ) and C = ( LR , Lη R, ε ). Starting from a monad A = ( A, m , u ) (where m is the multiplic ation and u is the unit), one ﬁrst deﬁnes the Eilenb er g-Mo or e c ate gory X A of A - algebr as , whose ob jects are pairs ( X , ρ X ), where X is an ob ject in X and ρ X : AX → X is a morphism in X suc h tha t ρ X ◦ Aρ X = ρ X ◦ m X and ρ X ◦ u X = X . There is a pair of adjo in t functors ( F A : X → X A , U A : X A → X ), where U A is a fo r getful functor and F A is the induction o r fr e e algebr a functor , for all ob jects X ∈ X deﬁned by F A X = ( AX , m X ). Similarly , one deﬁnes t he category Y C of c o alg e br as o v er C , and obtains an adjoin t pair ( U C : Y C → Y , F C : Y → Y C ), where F C is the induction or free coalgebra functor and U C is the f orgetful functor. The original and constructed adjunctions are r elated b y the c omp a ri s i o n K : Y → X A (or K ′ : X → Y C in the comona d case). F or an y Y ∈ Y , the comparision functor is given b y K Y = ( RY , RεY ). The functor R is said to b e monadic if K is an equiv alence of categor ies. Similarly L is said to b e c omon adic if K ′ is a n equiv alence 4 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE of categor ies. Be ck’s The or em [2] pro vides o ne with necessary and suﬃcien t conditions for the functors R a nd L to b e monadic or comonadic resp ectiv ely . F or more detailed and comprehensiv e study of matters describ ed in this section w e refer to [1]. 2.2. The category of double adjunctions. In this pap er, rather than lo oking at one adjunction, we consid er two adjunctions with righ t adjoints op erating on a common category . More precis ely , let X and Y b e t w o categories. F ollo wing [3 ], the c ategory Adj ( X , Y ) is deﬁne d as follo ws. An ob ject in Adj ( X , Y ) is a pentuple (or a triple) Z = ( Z , ( L A , R A ) , ( L B , R B )), where Z is a category and ( L A : X → Z , R A : Z → X ) and ( L B : Y → Z , R B : Z → Y ) are adj unctions whose units and counits are denoted resp ectiv ely b y η A , η B and ε A , ε B . A morphism F : Z = ( Z , ( L A , R A ) , ( L B , R B )) → Z ′ = ( Z ′ , ( L ′ A , R ′ A ) , ( L ′ B , R ′ B )) is a functor F : Z ′ → Z suc h that R A F = R ′ A and R B F = R ′ B . In o ther w o rds, Adj ( X , Y ) is a full sub category of the c ategory of sp ans Span ( X , Y ) w hic h consists of a ll sp ans X ← Z → Y suc h that t he functors Z → X and Z → Y hav e left adjoin ts. T o a morphism F in Adj ( X , Y ), one asso ciates t w o nat ur a l transforma t ions (1) a := ( ε A F L ′ A ) ◦ ( L A η ′ A ) : L A → F L ′ A , b := ( ε B F L ′ B ) ◦ ( L B η ′ B ) : L B → F L ′ B , whic h satisfy the following compatibility conditions ( R A a ) ◦ η A = η ′ A , ( R B b ) ◦ η B = η ′ B , (2) ( F ε ′ A ) ◦ ( aR ′ A ) = ε A F , ( F ε ′ B ) ◦ ( bR ′ B ) = ε B F . (3) 2.3. The category of Morita con text s. Consider t w o categories X and Y . Let A = ( A, m A , u A ) b e a monad on X , B = ( B , m B , u B ) b e a monad on Y . An A - B bialgebr a functor T = ( T , λ, ρ ), is a functor T : Y → X equipped with tw o na tural transformations ρ : T B → T and λ : AT → T suc h that T B B T m B / / ρB   T B ρ   T B ρ / / T , T B ρ / / T T , = O O T u B b b D D D D D D D D (4) AAT m A T / / Aλ   AT λ   AT λ / / T , AT λ / / T T , = O O u A T a a D D D D D D D D AT B λB / / Aρ   T B ρ   AT λ / / T . (5) A bialg ebra morphism φ : T → T ′ b et w een tw o A - B bialgebra functors is a natural transformation that satisﬁes the fo llowing conditions T B ρ / / φB   T φ   T ′ B ρ ′ / / T ′ , AT λ / / Aφ   T φ   AT ′ λ ′ / / T ′ . An A - B bialgebra functor T induces a functor Y → X A , whic h is denoted a g ain b y T and is deﬁned by T Y = ( T Y , λY ), for a ll Y ∈ Y . A Morita c ontext on X and Y is a sextuple T = ( A , B , T , b T , ev , b ev), that consists of a monad A = ( A, m A , u A ) on X , a monad B = ( B , m B , u B ) o n Y , an A - B bialgebra A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 5 functor T , a B - A bialgebra functor b T and natural transformations ev : T b T → A and b ev : b T T → B . These are required to satisfy the follo wing conditions: ev is a n A - A bialgebra morphism, b ev is a B - B bialgebra morphism, and the follo wing diagrams comm ute T b T T T b ev / / ev T   T B ρ   AT λ / / T , b T T b T b ev b T / / b T ev   B b T ˆ λ   b T A ˆ ρ / / b T , (6) b T AT b T λ / / ˆ ρT   b T T b ev   b T T b ev / / B , T B b T ρ b T / / T ˆ λ   T b T ev   T b T ev / / A . (7) Diagrams (7) mean that ev is B -b alanc e d and b ev is A -b alanc e d . A pair of bialgebras T , b T that satisfy all the conditions in this section exc ept for diagrams (7) is termed a p air of formal ly dual bialgebr as or a pr e-Morita c ontext . Th us a Morita con text is a balanced pair of forma lly dual bialgebras. R emark 2.3.1 . Let k A and k B b e unital asso ciative rings. Set X to b e the category of left k A -mo dules and Y the category of left k B -mo dules. F or a k A -ring A (i.e. a ring map k A → A ) with m ultiplication µ A and unit ι A and a k B -ring B with m ultiplicatio n µ B and unit ι B consider monads A = ( A ⊗ k A − , µ A ⊗ k A − , ι A ⊗ k A − ) and B = ( B ⊗ k B − , µ B ⊗ k B − , ι B ⊗ k B − ). Then the deﬁnition of a Morit a con text with monads A and B coincides with the classical deﬁnition of a (ring-theoretic) Morita context b et ween rings A and B ; see Section 2.3 for a more detailed study of this example. Note ho we v er tha t the deﬁnition of a categorical Morita con text in t r o duced in this pap er diﬀers from the notion of a w ide Morita c on text [5], which also giv es a categorical alb eit diﬀeren t in terpretation of classical Morita con texts. The connection b et w een categorical Morita con texts and wide Morita con t exts is discussed in Section 4. A morphism φ : T = ( A , B , T , b T , ev , b ev) → T ′ = ( A ′ , B ′ , T ′ , b T ′ , ev ′ , b ev ′ ) b etw een Morita con texts on X and Y is a quadruple ( φ 1 , φ 2 , φ 3 , φ 4 ) deﬁned a s fo llows. First, φ 1 : A → A ′ and φ 2 : B → B ′ are monad morphisms, i.e. φ 1 : A → A ′ and φ 2 : B → B ′ are natural t r a nsformation suc h t hat AA m A / / φ 1 φ 1   A φ 1   A ′ A ′ m A ′ / / A ′ , 1 1 X u A ′ ' ' N N N N N N N N N N N N N u A / / A φ 1   A ′ , (8) B B m B / / φ 2 φ 2   B φ 2   B ′ B ′ m B ′ / / B ′ , 1 1 Y u B ′ ' ' N N N N N N N N N N N N N u B / / B φ 2   B ′ , (9) 6 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE where the shorthand notation φ 1 φ 1 := A ′ φ 1 ◦ φ 1 A = φ 1 A ′ ◦ Aφ 1 etc. for the Go demen t pro duct of natural transformations is used. These make T ′ an A - B bialgebra functor and b T ′ a B - A bialgebra functor with structures giv en b y T ′ B T ′ φ 2 / / T ′ B ′ ρ ′ / / T ′ , AT ′ φ 1 T ′ / / A ′ T ′ λ ′ / / T ′ , b T ′ A b T ′ φ 1 / / b T ′ A ′ ˆ ρ ′ / / b T ′ , B b T ′ φ 2 b T ′ / / B ′ b T ′ ˆ λ ′ / / b T ′ . Second, φ 3 : T → T ′ is a morphism of A - B bialgebras and φ 4 : b T → b T ′ is a morphism of B - A bialgebras. These morphisms are required to satify the follow ing compatibility conditions (10) T b T ev / / φ 3 φ 4   A φ 1   T ′ b T ′ ev ′ / / A ′ , b T T b ev / / φ 4 φ 3   B φ 2   b T ′ T ′ b ev ′ / / B ′ . This completes the deﬁnition of the category Mo r ( X , Y ) of Morita con texts on X and Y . 3. A Be ck-type theorem for a Morit a context The aim of this section is to analyse the relationship b et w een double adjunctions described in Section 2.2 and Morita con texts deﬁned in Section 2.3. T he same notation and con ven tions as in Section 2 a r e used. 3.1. F rom double adjunctions to Morita con texts. Fix categories X and Y . The aim of this section is to construct a functor Υ from t he category of double adjunctions Adj ( X , Y ) to the category of Morita con texts Mor ( X , Y ) . T ak e an y o b ject Z = ( Z , ( L A , R A ) , ( L B , R B )) in Adj ( X , Y ) and deﬁne an ob ject Υ( Z ) = T := ( A , B , T , b T , ev , b ev) in Mo r ( X , Y ) as f o llo ws. The a dj unction ( L A , R A ) deﬁnes a mo na d A := ( A := R A L A , m A := R A ε A L A , η A ) on X and the a djunction ( L B , R B ) deﬁnes a mo na d B := ( B := R B L B , m B := R A ε B L B , η B ) on Y ; see Section 2.1. Set T := R A L B : Y → X and view it as an A - B bialgebra functor b y R A ε A L B : AT = R A L A R A L B → R A L B = T and R A ε B L B : T B = R A L B R B L B → R A L B = T , i.e. T = ( R A L B , R A ε A L B , R A ε B L B ). T o c heck that T is a bialgebra functor one can pro ceed as follo ws: the asso ciativit y conditions (of left A -action, righ t B -action and mixed asso ciativity) are a straigh tf or- w ard consequence o f naturality of the counits. The unitality of left and righ t a ctions follo ws by the triangular iden tities f or units and counits of adjunctions ( L A , R A ) and ( L B , R B ). The B - A bialg ebra functor b T := ( R B L A , R B ε B L A , R B ε A L A ) is deﬁned similarly . Finally deﬁne natural transformations ev : T b T = R A L B R B L A R A ε B L A / / R A L A = A , b ev : b T T = R B L A R A L B R B ε A L B / / R B L B = B . A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 7 All compatibility diagrams (6)-(7) fo llow (trivially) by the naturality of the counits ε A and ε B . F or a ny morphism F : Z → Z ′ in Adj ( X , Y ), t he corresp onding morphism of Morita con texts, Υ( F ) = ( φ 1 , φ 2 , φ 3 , φ 4 ) : Υ( Z ) → Υ( Z ′ ) , is deﬁned as follows. The monad morphisms a r e: φ 1 : R A L A R A a / / R A F L ′ A = R ′ A L ′ A , φ 2 : R B L B R B b / / R B F L ′ B = R ′ B L ′ B , where a and b are deﬁned by equations (1) in Section 2.2. Equations (2) express exactly that φ 1 and φ 2 satisfy the second diagrams in (8) and (9), resp ectiv ely . That φ 1 and φ 2 satisfy the ﬁrst diag rams in (8) and (9) follow s b y (3) and b y the naturality . The morphisms of bialgebras are: φ 3 : T = R A L B R A b / / R A F L ′ B = R ′ A L ′ B = T ′ , φ 4 : b T = R B L A R B a / / R B F L ′ A = R ′ B L ′ A = b T ′ . That φ 3 and φ 4 are bialgebra morphisms satisfying compatibility conditions (10) f o l- lo ws b y equations (3) a nd b y the naturality (the a rgumen t is v ery similar to the one used for chec king that φ 1 and φ 2 preserv e m ultiplications). 3.2. The Eilen b erg-Mo ore category of a Morita con t ext. Before w e can mak e a conv erse construction for t he f unctor Υ in t ro duced in Section 3 .1, we need to deﬁne a category of re presen tatio ns for a M orita con text. This construction is s imilar to that o f the Eilenberg-Mo ore category of a lg ebras f o r a monad; see Section 2.1. Let T = ( A , B , T , b T , ev , b ev) b e an ob ject in Mor ( X , Y ). The Eilenb e r g -Mo or e c ate- gory asso ciate d to T , ( X , Y ) T , is deﬁned as follow s. Ob jects in ( X , Y ) T are sextuples (or quadruples) X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ), where ( X , ρ X ) ∈ X A is an algebra for the monad A , ( Y , ρ Y ) ∈ Y B is an algebra for the monad B , v : T Y → X is a morphism in X A and w : b T X → Y is a morphism in Y B (i.e. (11) AT Y Av / / λY   AX ρ X   T Y v / / X , B b T X B w / / ˆ λX   B Y ρ Y   b T X w / / Y ) , 8 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE satisfying the follo wing compatibilit y conditions T b T X T w / / ev X   T Y v   AX ρ X / / X , b T T Y b T v / / b ev Y   b T X w   B Y ρ Y / / Y , (12) b T AX b T ρ X / / ˆ ρX   b T X w   b T X w / / Y , T B Y T ρ Y / / ρY   T Y v   T Y v / / X . (13) All these dia grams can b e understo o d as generalised mixed asso ciativit y conditions. A morphism X → X ′ in ( X , Y ) T is a couple ( f , g ), where f : X → X ′ is a morphism in X A and g : Y → Y ′ is a mo r phism in Y B (i.e. (14) AX Af / / ρ X   AX ′ ρ X ′   X f / / X ′ , B Y B g / / ρ Y   B Y ′ ρ Y ′   Y g / / Y ′ ) , suc h tha t (15) b T X b T f / / w   b T X ′ w ′   Y g / / Y ′ , T Y T g / / v   T Y ′ v ′   X f / / X ′ . This completes the construction of the Eilen b erg-Mo o re category of a Morita con text. 3.3. F rom Morita con texts to d ouble adjunctions. In this section a functor Γ from the category of Morita con texts on X and Y , Mor ( X , Y ), to the category of double adjunctions Adj ( X , Y ) is constructed. F or a Morita con text T = ( A , B , T , b T , ev , b ev) ∈ Mor ( X , Y ), the double adjunction Γ( T ) = Z = (( X , Y ) T , ( G A , U A ) , ( G B , U B )) , is deﬁned as follows . ( X , Y ) T is t he Eilen b erg-Mo ore category fo r the Morita con text T as deﬁ ned in Section 3.2. The functors U A : ( X , Y ) T → X and U B : ( X , Y ) T → Y are the forgetful functors (i.e. U A X = X a nd U B X = Y for all ob j ects X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) in ( X , Y ) T ). The deﬁnition of the f unctors G A and G B is sligh tly more inv olv ed. F or an y X ∈ X and Y ∈ Y , deﬁne G A X = (( AX , m A X ) , ( b T X , ˆ λX ) , ev X , ˆ ρX ) , G B Y = (( T Y , λY ) , ( B Y , m B Y ) , ρY , b ev Y ) . ( AX , m A X ) is simply the fr ee A -algebra on X (see Section 2.1), hence it is an ob ject of X A . F ro m the discussion in Section 2.3 we kno w that ( b T X , ˆ λX ) ∈ Y B , with a B - algebra structure induced b y the B - A bialgebra functor b T . T o complete the c heck that G A X is an ob ject of ( X , Y ) T it remains to v erify whether the maps ev X : T b T X → AX A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 9 and ρX : b T AX → b T X satisfy all needed compatibilit y conditions. The left ha nd side of (11) expresse s that ev is left A - linear, the left hand side of (12) that ev is righ t A -linear. The rig h t hand side o f (11) follow s b y the mixed asso ciativit y of the B - A bialgebra b T (see the last diagr am of (5) ) . The righ t hand side of (12) is the second Morita iden tit y of the maps ev and b ev; see the rig h t hand side of (6 ). The left hand side of (13) follows again f r o m t he prop erties of b T as a B - A bialgebra, in particular by the asso ciativit y o f its right A -action; compare with the ﬁrst diagram in (4). Fina lly , the righ t hand side of (1 3) is an application of the f act that ev is B -balanced, whic h is expressed in the r igh t hand side of (7). W e conclude that G A (and, by symmetric argumen t s, also G B ) is w ell-deﬁned on ob jects. F o r a morphism f : X → X ′ in X , deﬁne G A f = ( Af , b T f ) and similarly G B g = ( T g , B g ) for an y morphism g in Y . V eriﬁcation that G A f and G B g are w ell deﬁned is v ery simple a nd left to the reader. Lemma 3.3.1. (( X , Y ) T , ( G A , U A ) , ( G B , U B )) i s a double adjunction. Pro of. W e construct units ν A , ν B and counits ζ A , ζ B of adjunctions. F or any ob jects X ∈ X and Y ∈ Y , ν A , ν B are deﬁned as morphisms in X and Y resp ectiv ely , ν A X = η A X : X → AX, ν B Y = η B Y : Y → B Y . F or an y X ∈ ( X , Y ) T , the counits ζ A , ζ B are giv en b y the follo wing morphisms in ( X , Y ) T , ζ A X = ( f A X , g A X ) : G A U A X → X , (( AX , m A X ) , ( b T X , ˆ λX ) , ev X , ˆ ρX ) → (( X , ρ X ) , ( Y , ρ Y ) , v , w ) , f A X = ρ X : AX → X , g A X = w : b T X → Y , ζ B X = ( f B X , g B X ) : G B U B X → X , (( T Y , λY ) , ( B Y , m B Y ) , ρY , b ev Y ) → (( X , ρ X ) , ( Y , ρ Y ) , v , w ) , f B X = v : T Y → X , g B X = ρ Y : B Y → Y . T o c hec k that ζ A X is a morphism in ( X , Y ) T , one has to v erify that diagr a ms (14) and (1 5) comm ute. The left hand side of (14) holds, since f A X = ρ X is canonically a morphism in X A , the right hand side holds since g A X = w is a morphism in Y B b y deﬁnition. The left hand side of (15) is exactly the left hand side o f (1 3), and the righ t hand s ide of (1 5) is precisely the left hand side of (12). Similarly one che c ks that ζ B X is a morphism in ( X , Y ) T . No w take an y ob ject X ∈ X . The ﬁrst t r ia ngular identit y translates to the f o llo wing diagram G A X ( Aη A X, b T η A X ) ) ) S S S S S S S S S S S S S S S G A U A G A X , ( m A X, ˆ ρX ) u u l l l l l l l l l l l l l l l G A X whic h commutes b y the unit prop erties of the monad A and the bialgebra functor b T (i.e. m A ◦ Aη A = A and ˆ ρ ◦ b T η A = b T ). F or the second triangular iden tit y , tak e any 10 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE X ∈ ( X , Y ) T and consider the dia g ram U A X = X η A X * * U U U U U U U U U U U U U U U U U U A G A U A X = AX , ρ X t t i i i i i i i i i i i i i i i i i U A X = X whic h commute s by the unit prop ert y of the A -algebra ( X , ρ X ). In the same w a y one v eriﬁes that ( G B , U B ) is a n adjoint pair. ⊔ ⊓ Let φ = ( φ 1 , φ 2 , φ 3 , φ 4 ) : T → T ′ b e a mo r phism of Morita contexts . W e need to construct a morphism Γ( φ ) : Γ( T ) → Γ( T ′ ) in Adj ( X , Y ), i.e. a functor F = Γ( φ ) : ( X , Y ) T ′ → ( X , Y ) T suc h that U A F = U ′ A and U B F = U ′ B . It is w ell-known that a morphism of monads φ 1 : A → A ′ induces a f unctor Φ 1 : X A ′ → X A , giv en b y Φ 1 ( X , ρ X ) = ( X , ρ X ◦ φ 1 X ), for all ob jects ( X , ρ X ) ∈ X A ′ , and Φ 1 f = f for all morphisms f in X A ′ (the nat ura lit y of φ 1 implies tha t f is indeed a morphism in X A ). Similarly , a mor phism of monads φ 2 : B → B ′ induces a functor Φ 2 : Y B ′ → Y B . T ak e an y X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) ∈ ( X , Y ) T ′ and deﬁne F X = (( X , ρ X ◦ φ 1 X ) , ( Y , ρ Y ◦ φ 2 Y ) , v ◦ φ 3 Y , w ◦ φ 4 X ) . T o c heck diagrams (11), (12), (13 ), one has to rely on the naturality of φ 1 , φ 2 , φ 3 , φ 4 , and their prop erties a s mor phisms of monads and bialgebras. F or a morphis m ( f , g ) : X → X ′ in ( X , Y ) T ′ , deﬁne F ( f , g ) = ( f , g ). Then, b y construction (or b y natura lit y of φ 1 and φ 2 ), f is a morphism in X A and g is a morphism in Y B . Diagram (15) f o r F ( f , g ) to b e a morphism in ( X , Y ) T follo ws b y the naturalit y o f φ 3 and φ 4 , com bined with the corresp onding diagram for ( f , g ) as a morphism in ( X , Y ) T ′ . Finally , the construction of F immediately implies that, for an y X ∈ ( X , Y ) T ′ , U A F X = U ′ A X = X and U B F X = U ′ B X = Y . This completes the construction of a functor Γ : Mo r ( X , Y ) → Adj ( X , Y ). 3.4. Eve ry Morita con text arises from a double adjunction. Here w e pro v e that start ing with a Morita con text and p erfo r ming subse quen t constructions of a double adjunction and a Morita context give s back the original Morita con text, i.e . w e prov e the following Lemma 3.4.1. The c omp osite functor ΥΓ is the identity f unc tor on M o r ( X , Y ) . Pro of. The computation that, for an y Morita con text T ∈ Mor ( X , Y ), ΥΓ T = T is easy and left to the reader. Consider tw o Morita con texts T = ( A , B , T , b T , ev , b ev), T ′ = ( A ′ , B ′ , T ′ , b T ′ , ev ′ , b ev ′ ) and a morphism φ = ( φ 1 , φ 2 , φ 3 , φ 4 ) : T → T ′ . W rite ( φ 1 Γ( φ ) , φ 2 Γ( φ ) , φ 3 Γ( φ ) , φ 3 Γ( φ ) ) := ΥΓ φ . In view of the deﬁnition of functor Υ in Section 3.1, to compute the φ i Γ( φ ) one ﬁrst needs to compute natural transformations a , b (see equations (1) in Section 2.2) A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 11 corresp onding to double adjunctions Γ T = (( X , Y ) T , ( G A , U A ) , ( G B , U B )) and Γ T ′ = (( X , Y ) T ′ , ( G ′ A , U ′ A ) , ( G ′ B , U ′ B )); see Section 3.3. These are giv en b y a = ζ A Γ( φ ) G ′ A ◦ G A ν ′ A , b = ζ B Γ( φ ) G ′ B ◦ G B ν ′ B where ζ A is the counit of adjunction ( G A , U A ), ζ B is the counit of adjunction ( G B , U B ), ν ′ A is the unit of adjunction ( G ′ A , U ′ A ) and ν ′ B is the unit of adjunction ( G ′ B , U ′ B ). Since, for a ll X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) ∈ ( X , Y ) T ′ , Γ( φ )( X ) = (( X , ρ X ◦ φ 1 X ) , ( Y , ρ Y ◦ φ 2 Y ) , v ◦ φ 3 Y , w ◦ φ 4 X ) , w e o btain ζ A Γ( φ ) G ′ A = ( m A ′ ◦ φ 1 A ′ , ˆ ρ ′ ◦ φ 4 A ′ ) , G A ν ′ A = ( A u A ′ , b T u A ′ ) . Th us φ 1 Γ( φ ) = U A a = m A ′ ◦ φ 1 A ′ ◦ A u A ′ = m A ′ ◦ A ′ u A ′ ◦ φ 1 = φ 1 , where the se cond equalit y follo ws b y the naturalit y of φ 1 , while the third one is a consequenc e of t he unitality of a monad. Similarly , φ 4 Γ( φ ) = U B a = ˆ ρ ′ ◦ φ 4 A ′ ◦ b T u A ′ = ˆ ρ ′ ◦ b T ′ u A ′ ◦ φ 4 = φ 4 . The iden tities φ 2 Γ( φ ) = φ 2 and φ 3 Γ( φ ) = φ 3 are obtained b y symmetric calculations. ⊔ ⊓ The just computed iden tiﬁcation th us deﬁnes a natural transformation (the iden tit y transformation) λ : 1 1 Mor ( X , Y ) → ΥΓ . 3.5. The comparison functor. Consider a double adjunction on catego r ies X a nd Y , i.e. an ob ject Z = ( Z , ( L A , R A ) , ( L B , R B )) in Adj ( X , Y ). Let Υ Z = T be the asso ciated Morita contex t o n X and Y , and consider ( X , Y ) T , the Eilenberg-Mo o re category of represen tations o f T . In this section w e construct a c omp arison functor K : Z → ( X , Y ) T . F or any o b ject Z ∈ Z , deﬁne K ( Z ) = (( R A Z , R A ε A Z ) , ( R B Z , R B ε B Z ) , R A ε B Z , R B ε A Z ) . The ﬁrst tw o comp onen ts in K ( Z ), tha t is, ( R A Z , R A ε A Z ) ∈ X A and ( R B Z , R B ε B Z ) ∈ Y B are an application of the comparison functors K A : Z → X A and K B : Z → Y B , corresp onding to the adjunctions ( L A , R A ) and ( L B , R B ) resp ectiv ely (see Section 2.1). Ob viously , R A ε B Z : R A L B R B Z → R A Z and R B ε A Z : R B L A L B Z → R B Z are w ell-deﬁned. They satisfy conditions (11), (12) and (13) b y the naturalit y of counits. F or a morphism f : Z → Z ′ in Z , deﬁne K ( f ) = ( R A f , R B f ) . In view of the deﬁnition of the comparison functors K A and K B , it is clear that R A f = K A f and R B f = K B f , so R A f and R B f are morphisms in X A and Y B resp ectiv ely . D iagrams (1 5) fo llo w b y the naturalit y of ε A and ε B , respective ly . 12 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Deﬁnition 3.5.1. Let Z = ( Z , ( L A , R A ) , ( L B , R B )) b e an ob ject in Adj ( X , Y ). The pair ( R A , R B ) is said to b e m oritable if and only if the comparison functor K is an equiv alence of categor ies. Prop osition 3.5.2. (Γ , Υ ) is an adjoint p air an d Γ is a ful l and faithful functor. Pro of. Note that the deﬁnition of the comparison functor K imm ediately implies that K is a morphism in Adj ( X , Y ). F urthermore, K can be deﬁned for an y Z ∈ Adj ( X , Y ). W e claim that the assignmen t Z 7→ ( K : Z → ( X , Y ) T ) induces a natural transformation κ : ΓΥ → 1 1 Adj ( X , Y ) . T ak e d ouble adjunctions Z = ( Z , ( L A , R A ) , ( L B , R B )), Z ′ = ( Z ′ , ( L ′ A , R ′ A ) , ( L ′ B , R ′ B )) and a functor F : Z → Z ′ suc h that R A F = R ′ A and R B F = R ′ B (in other w ords, tak e a morphism in Adj ( X , Y )). Let K : Z → ( X , Y ) Υ Z and K ′ : Z ′ → ( X , Y ) Υ Z ′ b e the asso ciated comparison functors. The naturality o f κ is equiv alen t to the commutativit y of t he following diagra m Z ′ F / / K ′   Z K   ( X , Y ) Υ Z ′ ΓΥ( F ) / / ( X , Y ) Υ Z . F or an y o b ject Z of Z ′ , K F Z = (( R A F Z , R A ε A F Z ) , ( R B Z , R B ε B F Z ) , R A ε B F Z , R B ε A F Z ) , and (ΓΥ( F ) K ′ )( Z ) = (( R ′ A Z , R ′ A ε ′ A Z ◦ R A aR ′ A Z ) , ( R ′ B Z , R ′ B ε ′ B Z ◦ R B bR ′ B Z ) , R ′ A ε ′ B Z ◦ R A bR ′ B Z , R ′ B ε ′ A Z ◦ R B aR ′ A Z ) , where a and b are natural transformations (1 ) asso ciated to a morphism of double adjunctions F . The equalities R A F = R ′ A and R B F = R ′ B together with equations (3) yield t he required equalit y K F = ΓΥ( F ) K ′ . Let λ : 1 1 Mor ( X , Y ) → ΥΓ b e the natural (iden tity) transformation described in Sec- tion 3.4. That the comp osite κ Γ ◦ Γ λ is the iden tity natur a l transformatio n Γ → Γ is immediate. T o compute the other compo site Υ κ ◦ λ Υ : Υ → Υ, tak e a double adjunction Z = ( Z , ( L A , R A ) , ( L B , R B )), so that Υ Z = ( R A L A , R B L B , R A L B , R B L A , R A ε B L A , R B ε A L B ) . Then κ Z = K : Z → ( X , Y ) Υ Z is t he comparison functor, and hence Υ κ Z = Υ( K ) = ( φ 1 , φ 2 , φ 3 , φ 4 ) , where φ 1 = U A ζ A K L A ◦ U A G A η A , φ 2 = U B ζ B K L B ◦ U B G B η B , φ 3 = U A ζ B K L B ◦ U A G B η B , φ 4 = U B ζ A K L A ◦ U B G A η A . Here ( G A , U A ), ( G B , U B ) are adjo int pairs give n b y (( X , Y ) Υ Z , ( G A , U A ) , ( G B , U B )) := Γ( R A L A , R B L B , R A L B , R B L A , R A ε B L A , R B ε A L B ) , so G A η A = ( R A L A η A , R B L A η A ) , G B η B = ( R B L B η B , R A L B η B ) . A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 13 F urthermore, using the deﬁnition of the compar ison functor (applied to L A X and L B Y , for an y ob jects X ∈ X , Y ∈ Y ), w e o btain ζ A K L A = ( R A ε A L A , R B ε B L A ) , ζ B K L B = ( R B ε B L B , R A ε A L B ) . Therefore, φ 1 = R A ε A L A ◦ R A L A η A = R A L A , φ 4 = R B ε A L A ◦ R B L A η A = R B L A , since η A is the unit and ε A is the counit of the adjunction ( L A , R A ). Similarly , φ 2 = R B L B and φ 3 = R A L B . Th us Υ κ is the identit y natural transforma t io n, and since also λ Υ is the iden tit y , so is their comp osite Υ κ ◦ λ Υ. This pro v es that λ is a unit and κ is a counit of the adj unction (Γ , Υ) . Since t he unit λ is a natural isomorphism, Γ is a full and faithful functor. ⊔ ⊓ Corollary 3.5.3. (Γ , Υ) is a p air of inverse e quiva lenc es if a nd only if, for al l do uble adjunctions Z = ( Z , ( L A , R A ) , ( L B , R B )) ∈ Adj ( X , Y ) , ( R A , R B ) is a moritable p air. Pro of. The moritability of eac h of ( R A , R B ) is par a moun t to the comparison functor K b eing an equiv alence, fo r all Z ∈ Adj ( X , Y ), i.e. to the natural transfor ma t io n κ in the pro of of Prop osition 3.5.2 b eing an isomorphism. Since the latter is the counit of adjunction (Γ , Υ), the corollary is an immediate consequence o f Prop osition 3.5.2. ⊔ ⊓ 3.6. Moritabilit y. The aim o f this section is to determine, when a pair of functors is mo r it able in the sense of Deﬁnition 3.5.1. W e b egin with the follo wing simple Lemma 3.6.1. L et T b e an obje ct in Mo r ( X , Y ) and let ( f , g ) b e a morphism in ( X , Y ) T . Then ( f , g ) is an is o morphism in ( X , Y ) T if and on ly if f is an i s o morphism in X a nd g is an isomorphism in Y . Pro of. If ( f , g ) is an isomorphism in ( X , Y ) T , then clearly f and g are isomorphisms (as all functors, in particular forgetful functors, preserv e isomorphisms). Conv ersely , let f − 1 b e the in verse (in X ) of f and g − 1 b e the inv erse of g (in Y ). By applying f − 1 , g − 1 to b ot h sides of equalities describ ed b y diagra ms (14) and (15) one immediately obtains that ( f − 1 , g − 1 ) is a morphism in ( X , Y ) T . ⊔ ⊓ F rom now on we ﬁx a double adjunction Z = ( Z , ( L A , R A ) , ( L B , R B )) on X and Y (with counits ε A , ε B and units η A , η B ), and set T := Υ( Z ) = ( R A L A , R B L B , R A L B , R B L A , R A ε B L A , R B ε A L B ) to b e the corresp o nding Morita con text. K : Z → ( X , Y ) T is the comparison functor. The aim of this section is t o determine, when K is an equiv alence of categories. Prop osition 3.6.2. Supp ose that ( R A , R B ) is a mori table p air and let f b e a mo r- phism in Z . If b oth R A f and R B f ar e isomorp h isms, then so is f . Pro of. Note that R A = U A K and R B = U B K , where U A : ( X , Y ) T → X , U B : ( X , Y ) T → Y are forgetful functors. If R A f and R B f a re isomorphisms , then, by Lemma 3.6.1 also K f is an isomorphism. Since an equiv alence of categories r eﬂects isomorphisms, also f is an isomorphism. ⊔ ⊓ 14 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Deﬁnition 3.6.3. The pair ( R A , R B ) is said to r eﬂe ct i s omorphisms if the fa ct that b oth R A f and R B f are isomorphisms for a morphis m f ∈ Z implies that f is an isomorphism. That is, the pair ( R A , R B ) reﬂects isomorphisms if a nd o nly if the induced functor in to the pro duct cat ego ry h R A , R B i : Z → X × Y , Z 7→ ( R A Z , R B Z ), reﬂects isomorphisms. Note that if R A or R B reﬂects isomorphisms, then the pair ( R A , R B ) reﬂects isomor - phisms, but not the ot her w a y ro und. By Prop osition 3.6.2, a morita ble pair ( R A , R B ) reﬂects isomorphisms. T o a na lyse the comparison functor K further w e assume the existence of particular colimits in Z . F or any ob ject X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) ∈ ( X , Y ) T consider the follo wing diagram (16) L A R A L A X L A ρ X   ε A L A X   L B R B L A X ε B L A X y y r r r r r r r r r r r r r r r r r r r r r r L B w ) ) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T L A R A L B Y ε A L B Y % % K K K K K K K K K K K K K K K K K K K K K K L A v u u j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j L B R B L B Y ε B L B Y   L B ρ Y   L A X L B Y . F rom this p oin t un til Theorem 3.6.7 assume that Z has colimits of a ll suc h diagrams, and let ( D X , d A X : L A X → D X , d B X : L B Y → D X ) , b e the colimit of (16). An y morphism ( f , g ) : X → X ′ in ( X , Y ) T determines a morphism of diagrams (16) (i.e. it induces a natural transformation b etw een f unctor s from a six-ob ject category to Z that deﬁne diagrams (16)) by a pplying suitable com- binations of the L and R to f and g . There fore, b y the univ ersalit y of colimits, there is a unique morphism D ( f , g ) : D X → D X ′ in Z whic h satisﬁes the follo wing identie s (17) d A X ′ ◦ L A f = D ( f , g ) ◦ d A X , d B X ′ ◦ L B f = D ( f , g ) ◦ d B X . This construction yields a functor D : ( X , Y ) T → Z , X 7→ D X , ( f , g ) 7→ D ( f , g ) . Prop osition 3.6.4. The functor D is the left adjoin t of the c omp ariso n functor K . Pro of. F or any ob ject Z in Z the re is a co cone L A R A L A R A Z L A R A ε A Z   ε A L A R A Z   L B R B L A R A Z ε B L A R A Z x x p p p p p p p p p p p p p p p p p p p p p p p L B R B ε A Z + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W L A R A L B R B Z ε A L B R B Z & & N N N N N N N N N N N N N N N N N N N N N N N L A R A ε B Z s s g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g L B R B L B R B Z ε B L B R B Z   L B R B ε B Z   L A R A Z ε A Z , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y L B R B Z ε B Z r r e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Z. This is a co cone under the diagra m of type (16) corresp onding to the ob ject K Z . By the univ ersal prop erty of colimits there is a unique morphism εZ : D K Z → Z , suc h A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 15 that (18) εZ ◦ d A K Z = ε A Z , εZ ◦ d B K Z = ε B Z . This construction deﬁnes a natural transformation ε from the functor D K to the iden tity functor o n Z . F or all ob jects X in ( X , Y ) T , deﬁne η A X : X → R A D X and η B X : Y → R B D X as comp osites η A X = R A d A X ◦ η A X , η B X = R B d B X ◦ η B Y . Using the natura lit y of ε A , ε B , η A and η B , t he triangular iden tities fo r units and counits of adj unctions, and the deﬁnition of ( D X , d A X , d B X ) as a co cone under the diagram (16), one can v erify that the pair η X := ( η A X , η B X ) is a morphism in ( X , Y ) T . The assignmen t X 7→ η X deﬁnes a natural transformation η from the iden tity functor on ( X , Y ) T to K D . W e now pro ve the triangular iden tities for ε a nd η . T ak e any ob ject Z in Z and compute R A εZ ◦ η A K Z = R A εZ ◦ R A d A K Z ◦ η A R A Z = R A ε A Z ◦ η A R A Z = R A Z , where t he second equalit y follows b y (18) and the thir d by the triangular iden tities for ε A and η A . Similarly , R B εZ ◦ η B K Z = R B Z . This prov es that the comp osite K ε ◦ η K is the iden tit y natural transformation on K . Next ta k e an y ob ject X in ( X , Y ) T . Since D η X = D ( η A X , η B X ), Dη X s atisﬁes equalities (17). In particular D η X ◦ d A X = d A K D X ◦ L A R A d A X ◦ L A η A X . Therefore, εD X ◦ D η X ◦ d A X = εD X ◦ d A K D X ◦ L A R A d A X ◦ L A η A X = ε A D X ◦ L A R A d A X ◦ L A η A X = d A X ◦ ε A L A X ◦ L A η A X = d A X , where the second equalit y follo ws b y (18), the third one by the naturalit y of ε A , and the ﬁnal o ne is one of the triangular iden tities for ε A and η A . Similarly , εD X ◦ D η X ◦ d B X = d B X . The unive rsalit y of colimits no w yields εD X ◦ D η X = D X , i.e. the second triangular iden tity for ε and η . ⊔ ⊓ Deﬁnition 3.6.5. The pa ir ( R A , R B ) is said to c onvert c olimits into c o e qualise rs if, for all ob jects X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) ∈ ( X , Y ) T , the dia g rams R A L A R A L A X R A L A ρ X / / R A ε A L A X / / R A L A X R A d A X / / R A D X and R B L B R B L B Y R B L B ρ Y / / R B ε B L B Y / / R B L B Y R B d B X / / R B D X are co equalisers in X and Y resp ectiv ely . 16 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Lemma 3.6.6. Th e functor D is ful ly faithful if and onl y if the p air ( R A , R B ) c onverts c olimits into c o e qualisers. Pro of. F or a ll ob jects X = (( X , ρ X ) , ( Y , ρ Y ) , v , w ) ∈ ( X , Y ) T , consider the following diagram R A L A R A L A X R A L A ρ X / / R A ε A L A X / / R A L A X R A d A X & & M M M M M M M M M M ρ X / / X R A D X . Since the m ultiplication in monad R A L A is give n b y R A ε A L A , the top row is a co- equaliser; see [1, Prop osition 4, Section 3.3 ]. Th us there exists a morphism X → R A D X , which , by its uniqueness, m ust coincide with η A X . By considering a similar diagram for the other adjoin t pair, one ﬁts into it the other comp onen t of the unit of adjunction, η B X . If ( R A , R B ) con verts colimits into co equalisers, then (b y the unique- ness of co equalisers) b oth η A X and η B X are isomorphisms. Hence, b y Lemma 3.6.1, η is an isomorphism, i.e. D is fully faithful. Conv ersely , if η is an isomorphism, then b oth ( R A D X , R A d A X ) and ( R B D X , R B d B X ) are (isomorphic to) co equalisers, i.e. the pair ( R A , R B ) con verts colimits into co equalisers. ⊔ ⊓ The main result of this section is contained in the follo wing precise moritability theorem (Bec k’s theorem for do uble a djunctions). F or this theorem the existence of colimits of diagrams (16) needs not to b e assumed a priori . Theorem 3 .6.7. L et Z = ( Z , ( L A , R A ) , ( L B , R B )) b e a double a d junction. Then the p air ( R A , R B ) is moritable i f and o n ly if Z has c o limits o f al l the diagr ams (16) and the p a i r ( R A , R B ) r eﬂe c ts isomorph i s m s and c onve rts c olim its into c o e qualisers. Pro of. Assume that K is an equiv alence of categories. Consider a diagram of the form ( 16) in Z . W e can c ho ose X = K Z for some Z ∈ Z and w e kno w that ( Z , ε A , ε B ) is a co cone fo r this diagra m. Now apply the f unctor K t o diagram (16), then w e claim that ( K Z , K ε A , K ε B ) is a colimit for the new diagram in ( X , Y ) T . T o chec k this, consider an y co cone ( H , ( f A , g A ) , ( f B , g B )) on the diagram. Deﬁne ( h, k ) : K Z → H b y putting h = f A ◦ η A R A Z and k = g B ◦ η B R B Z . One can ve rify that ( h, k ) is indeed a morphism in ( X , Y ) T (apply the fact that H is a co cone and that ( f A , g A ) and ( f B , g B ) are morphisms in ( X , Y ) T , tog ether with the adjunction prop erties of ( L A , R A ) and ( L B , R B )). Since K is an equiv alence of categories, it reﬂects colimits and therefore ( Z , ε A , ε B ) is the colimit of the original dia gram (1 6) in Z . F urthermore, ( R A , R B ) reﬂects isomorphisms by Prop osition 3.6.2 and it con v erts colimits in to co equalisers b y Lemma 3.6.6. In t he conv erse direction, the counit η is an isomorphism b y Lemma 3.6.6. Applyin g R A to the ﬁrst column of the co cone deﬁning ε Z w e obtain the follo wing dia gram R A L A R A L A R A Z R A L A R A ε A Z   R A ε A L A R A Z   R A L A R A L A R A Z R A L A R A ε A Z   R A ε A L A R A Z   R A L A R A Z R A ε A Z   R A L A R A Z R A d A K Z   R A Z R A D K Z . R A εZ o o A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 17 The ﬁrst column is a (contractible) coequaliser, the second is a coequaliser b y the assumption that ( R A , R B ) conv erts colimits in to co equalisers. Th us R A εZ is an iso- morphism. Similarly , R B εZ is an isomorphism. Since the pair ( R A , R B ) reﬂects isomorphisms, also εZ is an isomorphism in Z . ⊔ ⊓ 4. Morit a theor y In this section w e study when, give n a Morita contex t T = ( A , B , T , b T , ev , b ev) on X and Y , the bialgebra functors T a nd b T induce an equiv alence of categories b etw een the categories X A and Y B . In particular, w e prov e that if X and Y hav e coequalisers and U A and U B preserv e co equalisers, the n an y equiv a lence bet w een t wo categor ies X A and Y B of algebras of monads A a nd B is induced b y a Morita contex t in Mor ( X , Y ). 4.1. Preserv ation of c o equalisers b y algebras. Consider a monad A = ( A, m A , u A ) on a category X and a functor S : X → Y . A pair ( S, σ ) is called a right A -algebr a functor if ( S, S, σ ) is a Y - A bialgebra functor, where Y is the trivial monad on Y . Th us ( S, σ ) is a righ t A -algebra f unctor if and only if σ : S A → S is a nat ura l trans- formation such that σ ◦ S m A = σ ◦ σ A and S = σ ◦ S u A . Similarly , one in tro duces the notion of a left A -algebra functor. Note that if T = ( T , λ, ρ ) is an A - B bialg ebra functor, then ( T , λ ) is a left A -algebra f unctor and ( T , ρ ) is a righ t B -alg ebra functor. Lemma 4.1.1. L et A = ( A, m A , u A ) b e a m onad on X an d S = ( S, σ ) a right A - algebr a functor. A ny c o e q uali s e r pr eserv e d by S A is also pr ese rv e d by S . I f ( S ′ , σ ′ ) is a left A -algebr a functor, then any c o e qualiser pr eserve d by AS ′ is als o p r eserve d by S ′ . Pro of. Conside r a co equaliser X f / / g / / Y z / / Z in X and a ssume that it is preserv ed b y S A . Applying the f unctor s S and S A to this co equaliser, one obtains the fo llo wing dia g ram in Y S X S u A X   S f / / S g / / S Y S u A Y   S z / / S Z S u A Z   S AX σX O O S Af / / S Ag / / S AY σY O O S Az / / S AZ . σZ O O By assumption the low er row is a co equaliser. Suppose that there exists a pair ( h, H ), where H is an ob ject in Y and h : S Y → H is a morphism in Y suc h that h ◦ S f = h ◦ S g . Since σ is a natural transformat io n, h ◦ σ Y ◦ S Af = h ◦ σ Y ◦ S Ag . By the univ eral prop erty of the co equaliser ( S AZ , S Az ), there is a unique morphism k ′ : S AZ → H suc h that k ′ ◦ S Az = h ◦ σ Y . This, together with the nat ur a lit y of u A and the unitality of the right A - algebra S imply that k ′ ◦ S u A Z ◦ S z = h . Then k = k ′ ◦ S u A Z : S Z → H is a unique morphsim suc h that h = k ◦ S z . The second statemen t is v eriﬁed b y a similar computation. ⊔ ⊓ 18 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Lemma 4.1.2. L et f , g : X → Y b e morphism s in X A . Supp ose that the c o e q uali s e r ( E , e ) of ( U A ( f ) , U A ( g )) ex i sts in X . If AA pr ese rves this c o e q uali s e r, then the c o - e qualiser ( E , ǫ ) of the p air ( f , g ) exists in X A and U A ( E , ǫ ) = ( E , e ) . Pro of. By Lemma 4.1.1, A preserv es the co equaliser ( E , e ). The univ ersal prop ert y of co equalisers implies the existence of a unique map ρ E : AE → E suc h that ρ E ◦ Ae = e ◦ ρ Y and ρ E ◦ u A E = E . Using the fact that the co equaliser ( E , e ) is preserv ed b y AA , one c hec ks that ρ E deﬁnes a (asso ciativ e) left action of A on E , i.e. ( E , ρ E ) is an ob ject of X A . ⊔ ⊓ The following standard lemma relates v arious preserv ation prop erties for co equalis- ers. W e include a brief pro of fo r completeness. Lemma 4.1.3. If X has (al l) c o e qualisers, then the fol lowing statements ar e e quiva- lent: (i) A : X → X pr es e rves c o e qualis e rs; (ii) AA : X → X pr es e rves c o e qualis e rs; (iii) X A has (al l) c o e qualise rs and they ar e pr eserve d by U A : X A → X ; (iv) U A : X A → X pr eserves c o e qualisers. Pro of. Implications ( i ) ⇒ ( ii ) and ( iii ) ⇒ ( iv ) ar e ob vious. ( ii ) ⇒ ( iii ) . F ollows b y Lemma 4.1.2. ( iv ) ⇒ ( i ). Th e free alg ebra functor F A : X → X A preserv es co equalisers, since it is a left adjoint functor. By assumption U A preserv es co equalisers as we ll and A = U A F A , so the statemen t ( i ) follo ws. ⊔ ⊓ 4.2. Morita con text s and equiv alences of categories of algebras. Let B = ( B , m B , u B ) b e a monad on Y and let ( T : Y → X , ρ ) b e a right B -algebra functor. F or an y ( Y , ρ Y ) in Y B , let ( T B Y , τ Y ) b e the follo wing co equaliser in X (if it exists) (19) T B Y ρY / / T ρ Y / / T Y τ Y / / T B Y . Similarly , giv en a monad A = ( A, m A , u A ) on X and a righ t A -alg ebra functor ( b T : X → Y , ˆ ρ ), consider an o b ject ( X , ρ X ) in X A , and set ( b T A X , ˆ τ X ) to b e the follo wing co equaliser in Y (if it exists) (20) b T AX ˆ ρX / / b T ρ X / / b T X ˆ τ X / / b T A X . Finally , recall tha t for any B - algebra ( Y , ρ Y ), the following diagram is a contractible co equaliser in Y and a (usual) co equaliser in Y B , (21) B B Y m B Y / / B ρ Y / / B Y ρ Y / / Y . As in Section 2.1, t he free–forg etful adj unctions f o r A and B a re denoted b y ( F A , U A ), ( F B , U B ), respective ly . The counits a re denoted b y ¯ ε A , ¯ ε B . W e are no w ready to state the follo wing lif ting theorem. A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 19 Prop osition 4.2.1. L et A b e a monad on X and B a monad on Y . The r e is a bije ctive c orr e s p ondenc e b etwe en the fol lowing da ta: (i) A - B bialgebr a functors T , such that c o e qualisers of the form (19) exist in X for any Y ∈ Y , an d they ar e pr es e rv e d by AA ; (ii) functors T B : Y B → X A such that AAU A T B : Y B → X pr eserves c o e qualisers of the form (2 1) for al l ( Y , ρ Y ) ∈ Y B . L et T b e a b ialgebr a functor as in ( i ) and T B the c orr esp on ding functor of ( ii ) , then given a functor P : X → W , the functor P pr eserves c o e qualise rs of the form (19) if and only if the functor P U A T B : Y B → W pr eserves c o e qualisers of the form (21) . Pro of. ( i ) ⇒ ( ii ) . Giv en a B -algebra ( Y , ρ Y ), deﬁne T B Y b y (19). Then it follo ws b y Lemma 4.1 .2 that there is a morphism ρ T B Y : AT B Y → T B Y suc h tha t ( T B Y , ρ T B Y ) is a n ob ject in X A . F or a morphism f : Y → Y ′ in Y B , t he univ ersality of the co equaliser induces a morphism T B f : T B Y → T B Y ′ suc h tha t T B f ◦ τ Y = τ Y ′ ◦ T f . T o c hec k that T B f is a morphism in X A , o ne can pro ceed as follo ws. By Lemma 4 .1 .1, A preserv es co equalisers of the f orm (19), so in particular, Aτ Y is an epimorphism. Therefore, it is enough to v erify that T B f ◦ ρ T B Y ◦ Aτ Y = ρ T B Y ′ ◦ AT B f ◦ Aτ Y , whic h follo ws f r om the deﬁning prop erties of ρ T B Y , ρ T B Y ′ and T B f , com bined with t he naturalit y of the righ t actio n ρ of T . Th us there is a w ell-deﬁned functor T B : Y B → X A . Consider now a functor P : X → W whic h preserv es all the co equalisers of t he form (1 9). F or any ( Y , ρ Y ) ∈ Y B , construct the follo wing dia g ram in X P T B B B Y P T B m B Y / / P T B B ρ Y / / P T m B B Y   P ρB B Y   P T B B Y P T B ρ Y / / P T m B Y   P ρB Y   P T B Y P T ρ Y   P ρY   P T B B Y P T m B Y / / P T B ρ Y / / P τ B B Y   P T B Y P T ρ Y / / P τ B Y   P T Y P τ Y   P U A T B B B Y P T B m B Y / / P T B B ρ Y / / P U A T B B Y P T B ρ Y / / P U A T B Y . The ﬁrst and second r ows of t his diagram are co equalisers, as they a r e obtained b y applying functors to the c ontr actible co equaliser (21). All three columns are co equalis- ers, since they are co equalisers of t yp e ( 1 9) to whic h t he functor P is applied. The diagram c ha sing argumen ts then yield that the low er row is a co equaliser to o. The ﬁrst part of the pro of is then completed b y setting P = AA . ( ii ) ⇒ ( i ) . Deﬁne T = U A T B F B , and natural transformations ρ = U A T B ¯ ε B F B = U A T B m B : T B → T and λ = U A ¯ ε A T B F B : AT → T . Then ( T , λ, ρ ) is an A - B bialgebra functor. Note that U A T B is also a left A - algebra functor with action U A ¯ ε A T B . Lemma 4.1.1 implies that U A T B preserv es the co equaliser (21). This means that, for an y Y = ( Y , ρ Y ) ∈ Y B , the following diagram is a co equaliser in X (22) U A T B F B B Y U A T B m B Y / / U A T B F B ρ Y / / U A T B F B Y U A T B ρ Y / / U A T B Y . This diagram is exactly the co equaliser (19) in this situation. Applying the f unctor AA (res p. an y functor P : X → W ) to the co equaliser (22) yields the same res ult 20 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE as applying the f unctor AAU A T B (resp. P U A T B ) to (21). Therefore, the functor AA (resp. P ) preserv es co equalisers (19). ⊔ ⊓ In the follo wing the notation intro duced in Propo sition 4.2.1 is used. F or an A - B bialgebra f unctor T , T B denotes the functor deﬁned b y co equalisers (19), and T = U A T B F B . Lemma 4.2.2. Supp ose that the c o e q ualis ers of the form (19) and (20) exist and they ar e pr e serve d by AA , b T A and by B B r esp e ctively, then the fu nctor b T A : X A → Y B pr eserves c o e qualisers of the form (19 ) . Pro of. Note that the f unctor b T A is w ell-deﬁned by (the dual v ersion o f ) Prop o si- tion 4.2.1. W e can no w consider t he following diagram in Y : (23) b T AT B Y b T AρY / / b T AT ρ Y / / ˆ ρT B Y   b T λB Y   b T AT Y b T Aτ Y / / ˆ ρT Y   b T λY   b T AT B Y ˆ ρT B Y   b T ρ T B Y   b T T B Y b T ρY / / b T T ρ Y / / ˆ τ T B Y   b T T Y b T τ Y / / ˆ τ T Y   b T T B Y ˆ τ T B Y   b T A T B Y b T A ρY / / b T A ρ Y / / b T A T Y b T A τ Y / / b T A T B Y . By assumption, the ﬁrst row is a co equaliser; b y Lem ma 4.1.1, the second row is a co equaliser to o . All three columns a r e co equalisers by deﬁnition. Therefore, the low er ro w is an equaliser as well. If we a pply the functor B B to this diagram, the same reasoning yields that B B preserv es the equaliser in the low er ro w of (23), therefore it is a n equaliser in Y B b y Lemma 4.1 .2. ⊔ ⊓ Recall that a wide Morita c ontext ( F , G, µ, τ ) b et wee n categor ies C and D , consists of t w o functors F : C → D and G : D → C , and tw o natural transformations µ : GF → C and τ : F G → D , satisfying F µ = τ F and µG = Gτ . In [5], left ( r esp. righ t ) wide Morita con texts are studied. In this setting, C and D are ab elian ( or Grothendiec k) categories and F and G are left (resp. right) exact functors. Left and right wide Morita con texts are used to c haracterise equiv alences betw een the categories C a nd D . In the remainder of this section w e study wide Morita con t exts b etw een categories o f algebras o v er monads. This allo ws us to w eak en the assumptions made in [5]. Moreov er, we study the r elat io nship b et w een wide Morita con texts and Morita con texts. Consider monads A on X and B on Y . A pair of functors b T A : X A → Y B and T B : Y B → X A is said to b e a lgebr aic if the functors AAU A T B and U B b T A AAU A T B preserv e co equalisers of the for m (21) for all ( Y , ρ Y ) ∈ Y B , and the functors B B U B b T A and U A T B B B U B b T A preserv e co equalisers of the form (24) AAX m A X / / Aρ X / / AX ρ X / / X , for all ( X , ρ X ) ∈ X A . By an algebr aic wide Morita con text w e mean a wide Morita con text ( T B , b T A , ˆ ω , ω ) b et wee n categories of algebras Y B and X A , suc h t ha t the func- tors T B and b T A form an a lg ebraic pair of functors. A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 21 R emark 4.2.3 . If the fo rgetful functors U B and U A preserv e co equalisers and the cat- egories Y B and X A ha v e all co equalisers ( e.g. Y B and X A are ab elian categories, as in [5]) or equiv alen tly the categories Y and X hav e all co equalisers, see Lemma 4.1.3, then the situation simpliﬁes signiﬁcan tly . In this case, for an y bialgebra functors T and b T , all co equalisers of the form (19) a nd (20) exis t and, by Lemma 4.1.3, are preserv ed b y AA and B B resp ectiv ely . F urthermore, in this situation, ( T B , b T A ) is a n algebraic pair if and only if the functors T B and b T A preserv e co equalisers (of the form (21) and (2 4) resp ectiv ely). In particular, it is an adjo in t a lg ebraic pair if b T A preserv es co equalisers (as a n y left adjoin t preserv es all co equalisers). Also, if Y B and X A are ab elian, then a right wide Morita con text b et w een Y B and X A is a lw ays algebraic. Theorem 4.2.4. L et A b e a monad on X and B a monad on Y . Ther e is a bije c tive c orr esp ondenc e b etwe en the fol lowing data: (i) Morita c ontexts T = ( A , B , T , b T , ev , b ev) on X an d Y , such that al l c o e qualis e rs of the form (19) and (20) exis t, and they ar e pr es e rve d by AA , b T A and B B , T B r esp e ctively; (ii) functors T B : Y B → X A and b T A : X A → Y B , and natur al tr ansformations ˆ ω : b T A T B → Y B and ω : T B b T A → X A such that ( T B , b T A , ˆ ω , ω ) is an algebr aic wide Morita c ontext b etwe en Y B and X A . Pro of. ( i ) ⇒ ( ii ) . The existence of the functors b T A and T B , forming an algebraic pair, fo llo ws from Prop osition 4.2.1. Moreov er, in lig h t of Lemma 4.2.2, b T A preserv es co equalisers of the fo rm (1 9) and T B preserv es co equalisers of the form (20). Consider again diagram (23) in whic h no w all the ro ws and columns ar e co equalisers. The natural tra nsformation b ev induces a map ρ Y ◦ b ev Y : b T T Y → Y that equalises b oth the horizon tal and v ertical arrows . A diagram chas ing argumen t aﬃrms the existence of a map ˆ ω Y : b T A T B Y → Y . The naturalit y of b ev implies t ha t ˆ ω is a natural transformation as w ell. Similarly , one deﬁnes ω : T B b T A → X A . T o c heck that the compatibility conditions b et w een ˆ ω and ω hold, one starts with ob j ects T B b T AT B Y and b T AT B b T AX and constructs co equaliser diagrams resulting in T B b T A T B Y and b T A T B b T A X , respectiv ely . The compatibilit y conditions b et w een ev and b ev (diagrams (6)-(7)), as w ell as the facts that they a re bialgebra morphisms and tha t T and b T are bialgebra functors, induce the needed compatibility conditions f or ˆ ω and ω . ( ii ) ⇒ ( i ) . By Prop osition 4.2.1 there exist bialgebra functors T and b T , where T = U A T B F B and b T = U B b T A F A , suc h that a ll co equalisers of the form (19) and (20) exist, and they are preserv ed b y AA , b T A and B B , T B resp ectiv ely . Deﬁne ev : T b T = U A T B F B U B b T A F A U A T B ¯ ε B b T A F A / / U A T B b T A F A U A ω F A / / U A F A = A , b ev : b T T = U B b T A F A U A T B F B U B b T A ¯ ε A T B F B / / U B b T A T B F B U B ˆ ω F B / / U B F B = B , where ¯ ε B and ¯ ε A are the counits of the adjunctions ( F B , U B ) and ( F A , U A ). Then ( A , B , T , b T , ev , b ev) is a Morita contex t. ⊔ ⊓ 22 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Consider a Morita con text ( A , B , T , b T , ev , b ev) on X and Y . Supp ose that the co equalisers of the form (19) and (20) exist a nd are preserv ed b y AA , b T A and B B , T B resp ectiv ely . Because ev and b ev a re resp ectiv ely B and A -balanced, the unive rsal prop ert y of the co equaliser implies the existence of natura l transformations π : T B b T → A and ˆ π : b T A T → B suc h tha t, for a ll ob j ects X ∈ X and Y ∈ Y , ev X = π X ◦ τ b T X and b ev Y = ˆ π Y ◦ ˆ τ T Y . Since the co equalisers (19) and (20) are preserv ed b y b T A and T B , resp ectiv ely ( see Prop osition 4.2.1), for all ob jects ( X, ρ X ) ∈ X A , ( Y , ρ Y ) ∈ Y B , (25) ρ X ◦ π X = ω X ◦ T B ˆ τ X and ρ Y ◦ ˆ π Y = ˆ ω Y ◦ b T A τ Y , where ω : T B b T A → X A and ˆ ω : b T A T B → Y B are deﬁned in Theorem 4.2.4. With this notation, w e hav e the follow ing Theorem 4.2.5. L et A b e a m onad on X and B a monad on Y . (1) Ther e is a bije ctive c orr esp ondenc e b etwe en the f o l lowing data: (i) p airs of adjo i nt functors ( T B : Y B → X A , b T A : X A → Y B ) such that ( T B , b T A ) is an algebr aic p ai r and T B is f ul ly faithful; (ii) a lgebr aic wide Morita c ontexts ( T B , b T A , ω , ˆ ω ) such that ˆ ω is a natur al isomorphism; (iii) M orita c o ntexts ( A , B , T , b T , ev , b ev) such that a l l c o e qualis e rs of the form (19) and (20 ) exist and ar e p r e s erve d by AA , b T A an d B B , T B , r esp e c- tively, and ther e exi s ts a natur al tr ansformation ˆ χ : Y → b T A T such that ˆ π ◦ ˆ χ = u B . (2) Ther e is a bije ctive c orr esp ondenc e b etwe en the f o l lowing data: (i) inverse p airs of e quivale nc es ( T B : Y B → X A , b T A : X A → Y B ) such that ( T B , b T A ) is an algebr aic p ai r; (ii) a lgebr aic wide Morita c ontexts ( T B , b T A , ω , ˆ ω ) , such that ˆ ω and ω ar e nat- ur al isomorphisms; (iii) M orita c o ntexts ( A , B , T , b T , ev , b ev) such that a l l c o e qualis e rs of the form (19) and (20 ) exist and ar e pr eserve d by AA , b T A and B B , T B r e- sp e ctively, and ther e exist natur al tr ansformations ˆ χ : Y → b T A T and χ : X → T B b T such that ˆ π ◦ ˆ χ = u B and π ◦ χ = u A . Pro of. W e only pro v e part (1), as (2) follow s b y com bining (1) with its X - Y symmetric v ersion. ( ii ) ⇔ ( i ) . Let ( T B , b T A , ω , ˆ ω ) b e a n algebraic wide Morita context and let ˆ  denote the in ve rse natural tra nsformation o f ˆ ω . F or any X ∈ X A , b T A ω X ◦ ˆ  b T A X = ˆ ω b T A X ◦ ˆ  b T A X = b T A X . The ﬁrst equalit y is the compatibilit y condition b etw een ω and ˆ ω in the wide Morita con text (see Theorem 4.2.4). Similarly , T B ˆ ω Y ◦ ˆ  T B Y = T B ω Y ◦ T B ˆ  Y = T B Y , for all Y ∈ Y B . Hence ( b T A , T B ) is an adjoint pair with unit ˆ  and counit ω . T B is fully faithful since the unit of adjunction is a natur a l isomorphism. Conv ersely , if ( b T A , T B ) is an algebraic adjo in t pair and T B is fully faithf ul, then similar computatio n conﬁrms A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 23 that ( T B , b T A , ω , ˆ ω ) is a n algebraic wide Morita con text, where ω is the counit and ˆ ω is t he inv erse of the unit o f the adjunction. ( ii ) ⇔ ( iii ) . Assume ﬁrst that the statemen t ( iii ) holds. In light of Theorem 4.2.4 suﬃces it to construct the natural in v erse ˆ  : Y B → b T A T B of ˆ ω . F or any algebra Y = ( Y , ρ Y ) ∈ Y B , set ˆ  Y = b T A τ Y ◦ ˆ χY . The naturality of ˆ  follows by the preserv ation of co equalisers b y b T A and T B (see diagr am (23)). T ak e any ( Y , ρ Y ) ∈ Y B and compute ˆ ω Y ◦ ˆ  Y = ˆ ω Y ◦ b T A τ Y ◦ ˆ χY = ρ Y ◦ ˆ π Y ◦ ˆ χY = ρ Y ◦ u B Y = Y . The second equalit y fo llo ws by (25). F urthermore, for an y ( Y , ρ Y ) ∈ Y B , ˆ  Y ◦ ˆ ω Y ◦ b T A τ Y ◦ ˆ τ T Y = b T A τ Y ◦ ˆ χY ◦ ρ Y ◦ ˆ π Y ◦ ˆ τ T Y = b T A τ Y ◦ ˆ χY ◦ ρ Y ◦ b ev Y = b T A τ Y ◦ b T A T ρ Y ◦ ˆ χB Y ◦ b ev Y = b T A τ Y ◦ b T A ρY ◦ b T A T b ev Y ◦ ˆ χ b T T Y = b T A τ Y ◦ b T A λY ◦ b T A ev T Y ◦ ˆ χ b T T Y . The ﬁrst t wo equalities follo w b y the deﬁnitions of ˆ  and ˆ π , and by(25). The third step is a consequenc e of the naturalit y o f ˆ χ . The equalising prop erty of the maps b T τ Y and ˆ τ T Y , as w ell as t he equiv alen t expressions for the Go demen t pro duct are used in the fourth step. The ﬁnal equalit y follo ws b y the ﬁrst of diag rams (6 ) that express compatibilit y b et w een ev and b ev in the Morita con text. On the other hand, b T A λY ◦ b T A ev T Y ◦ ˆ τ T b T T Y = ˆ τ T Y ◦ b T λY ◦ b T ev T Y = ˆ τ T Y ◦ ˆ ρT Y ◦ b T ev T Y = ˆ τ T Y ◦ ˆ λT Y ◦ b ev b T T Y = ˆ τ T Y ◦ ˆ λT Y ◦ ˆ π b T T Y ◦ ˆ τ T b T T Y , where the ﬁrst equalit y follows b y the naturalit y of ˆ τ , the second is the equalising prop ert y of ˆ τ T Y (r ecall from Section 2.3 that ρ T Y = λY ) , the third equalit y follow s b y the second of diag r ams (6). The ﬁnal equality is the deﬁning prop erty of ˆ π Y . Since ˆ τ T b T T Y is an epimorphism, w e conclude that b T A λY ◦ b T A ev T Y = ˆ τ T Y ◦ ˆ λT Y ◦ ˆ π b T T Y , so ˆ  Y ◦ ˆ ω Y ◦ b T A τ Y ◦ ˆ τ T Y = b T A τ Y ◦ ˆ τ T Y ◦ ˆ λT Y ◦ ˆ π b T T Y ◦ ˆ χ b T T Y = b T A τ Y ◦ ˆ τ T Y ◦ ˆ λT Y ◦ u B b T T Y = b T A τ Y ◦ ˆ τ T Y . Since b T A τ Y ◦ ˆ τ T Y is an epimorphism, ˆ  ◦ ˆ ω is the iden tity natural transformation on b T A T B , and hence ˆ ω is a natural isomorphism with inv erse ˆ  . In the con v erse direction, apply Theorem 4.2.4 to obtain a Morita con text. Let ˆ  : Y B → b T A T B b e t he inv erse of ˆ ω and put ˆ χ = ˆ  ◦ u B . It follows from naturalit y that ˆ χ satisﬁes the r equired conditions. ⊔ ⊓ R emark 4 .2 .6 . A suﬃcien t condition for the existence of a natural transforma t io n ˆ χ as in part (1)(iii) of Theorem 4.2.5 is t he existence of a natural transformation b v e : Y → b T T suc h that b ev ◦ b v e = u B . In the case of a r ing-theoretic Morita con text (see Remark 2.3.1) t his condition expresses exactly that the Morita map b ev is surjectiv e. R emark 4.2.7 . In case the forgetful functors U A and U B preserv e co equalisers and X A and Y B (or equiv a lently X and Y ) hav e all co equalisers, Theorem 4.2.5 has the 24 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE follo wing sligh tly stronger formulation, whic h follo ws directly from the observ ations made in Remark 4.2.3 . Ther e is a bije ctive c orr e s p ondenc e b etwe en the fol lowing data: (i) p airs of adjoint functors (r esp. e quivalenc e s of c ate gories) ( T B : Y B → X A , b T A : X A → Y B ) such that b T A pr eserves c o e qualisers; (ii) right w ide Morita c ontexts ( T B , b T A , ω , ˆ ω ) su ch that ˆ ω is a natur al isomorphism (r esp. ˆ ω and ω ar e natur al isomorphisms) ; (iii) Morita c ontexts ( A , B , T , b T , ev , b ev) such that b T and T pr eserve c o e qualisers and ther e exist a natur al tr ans f o rmation ˆ χ : Y → b T A T such that ˆ π ◦ ˆ χ = u B (r esp. ther e exi s ts as wel l a natur al tr ansformation χ : X → T B b T such that π ◦ χ = u A ). 5. Examples and app lica tions In this section w e apply the criterion for moritability to sp eciﬁc situations of one adjunction, ring-theoretic Morita con texts, and herds and pre-torsors. 5.1. Blo wing up one adjunction. Consider an adjunction ( L : X → Z , R : Z → X ) with unit η and counit ε . Let C = ( LR, Lη R , ε ) b e the asso ciated comonad on Z and A = ( RL, RεL, η ) the asso ciated monad o n X ; see Section 2.1. Associated to the Eilen b erg- Mo ore category of C -coalgebras Z C , there is a second a djunction ( U C : Z C → Z , F C : Z → Z C ) with unit ν and counit ζ , and hence there is an ob ject ( Z , ( L, R ) , ( U C , F C )) in the catego ry Adj ( X , Z C ). The adjunction ( U C , F C ) induces a monad B = ( F C U C , F C ζ U C , ν ) = ( LR, LRε, ν ) on Z C . Therefore, there is a Morita con text (26) T = ( A , B , T , b T , ev = R εL, b ev = LR ε ) , where T = R and b T = LRL , and the corresponding Eilenberg-Mo ore category ( X , Z C ) T can b e constructed. This leads to the follo wing diagram of functors. X L   @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ b T   F A / / X A U A o o Λ   H { { v v v v v v v v v v v v v v v v v v v Z R _ _ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ F C                    K / / K A 3 3 K B * * ( X , Z C ) T V A ; ; v v v v v v v v v v v v v v v v v v v V B # # G G G G G G G G G G G G G G G G G G G Z C U C ? ?                  T O O F B / / ( Z C ) B U B o o The adjunctions ( F A , U A ) and ( F B , U B ) are the us ual free algebra–for g etful a djunc- tions asso ciated to a monad (see Section 2.1). The category ( Z C ) B is the category of dual descen t data and consists of triples ( Y , ρ Y , τ Y ), where ( Y , ρ Y : Y → LRY ) is an ob ject in Z C , and τ Y : LRY → Y is a morphism in Z C that satisﬁes the f o llo wing con- ditions τ Y ◦ ρ Y = Y and τ Y ◦ LRτ Y = τ Y ◦ LRεY . Finally , the functor Λ : X A → ( Z C ) B is deﬁned as fo llo ws: for all ob jects ( X , ρ X ) in X A , set Λ( X , ρ X ) := ( LX , Lη X , Lρ X ). A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 25 Here V A and V B denote the obvious forgetful func tors. The functor H can no w b e deﬁned as f ollo ws: H ( X , ρ X ) = (( X , ρ X ) , Λ( X, ρ X ) , ρ X , Lρ X ) . Lemma 5.1.1. ( H , V A ) is a p air of adjoint functors, and H i s ful ly faithful. F ur- thermor e, ther e is a natur al tr ansforma tion γ : Λ V A → V B , for al l obje c ts X = (( X , ρ X ) , ( Y , ρ Y , τ Y ) , v , w ) in ( X , Z C ) T , given by γ X = w ◦ Lη X : LX → Y . Pro of. W e construct the unit α and t he counit β of this adjunction. T ake a n ob ject ( X , ρ X ) in X A and deﬁne αX : ( X , ρ X ) → ( X, ρ X ) as t he iden tit y . F or all ob jects X ∈ ( X , Z C ) T , deﬁne β X = ( f X , g X ) : H V A ( X ) → X , (( X , ρ X ) , ( LX, Lη X , Lρ X ) , Lρ X , ρ X ) → (( X , ρ X ) , ( Y , ρ Y , τ Y ) , v , w ) , f X = X : X → X , g X = w ◦ Lη X : LX → Y . Com bining natura lit y and the adjunction prop erties of the unit η and counit ε with the diagrams (1 2 ) − ( 1 3), one can c hec k that f and g are indeed mo r phisms in ( X , Z C ) T . No w deﬁne γ b y putting γ X = g X , for all X ∈ ( X , Z C ) T . ⊔ ⊓ Prop osition 5.1.2. With notation as ab ove, K is an e quivalenc e o f c ate gories if and only if K A is an e quivalenc e of c ate gories and γ : Λ V A → V B is a natur al isomorphism. Pro of. Assume that K is an equiv alence. Up to an isomorphism, any X ∈ ( X , Z C ) T has the fo r m X = K Z , for some Z ∈ Z , i.e. X = (( RZ , RεZ ) , ( LRZ , Lη RZ , LRεZ ) , RεZ , LRεZ ) . The natural t r ansformation γ : LV A → V B described in Lemma 5.1.1 comes out as γ X : LRZ LηRZ / / LRLRZ LRεZ / / LRZ . Since ( L, R ) is an adjoin t p air, γ X is an isomorphism. By the construction of the comparison functor in Section 3.5, K B = V B K and K A = V A K . Denote the in v erse functor of K by D . In view of Prop osition 3.6.4 , for a n ob ject X in ( X , Z C ) T , D X is the follo wing colimit in Z , LRLX εLX   Lρ X   w ' ' LRY Lv s s εY   τ Y   LX d A X ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Y d B X v v m m m m m m m m m m m m m m D X . 26 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE Since γ is a natural isomorphism, this colimit r educes to (eac h) one o f the following isomorphic co equalisers LRLX εLX   Lρ X   LRγ X / / LRY εY   τ Y   LX γ X / /   Y   D A X ∼ = / / D B Y . Therefore, there a r e func tors D A : X A → Z and D B : ( Z C ) B → Z suc h that D ≃ D A V A ≃ D B V B and D A ≃ D H . These yield nat ur a l isomorphisms D A K A ≃ D A V A K ≃ D K ≃ Z , (27) X A ≃ V A H ≃ V A K D H ≃ K A D A . (28) The fact that K is an equiv alence of categories is use d in the last isomorphism of (27) and in the second isomorphism of (28). The ﬁrst is omorphism of ( 28) f ollo ws b y Lemma 5.1.1. These natural isomorphisms are exactly the unit and counit of the adjunction ( D A , K A ), hence K A is a n equiv alence of categories. Con vers ely , if γ is a natural isomorphism, then, in ligh t of its construction in the pro of of Lemma 5.1.1, the counit of the adjunction ( H , V A ) is a natural isomorphism, hence V A is a n eq uiv alence of categories. Sinc e K = V A K A , w e infer that K is an equiv alence of categor ies as w ell. ⊔ ⊓ Next, w e apply the results of Section 4 t o the pair of monads desc rib ed at the b eginning of this section. T of (26) is a Mor it a context b et w een X a nd Z C . F or an y ( X , ρ X ) ∈ X A , the co equaliser (20) is a con tr actible co equaliser in Z C : LRLRLX LRεLX / / LRLρ X / / LRLX Lρ X / / LηRLX e e LX. LηX d d Hence the functor b T A exists and, furthermore, b T A = Λ. F or a ny ( Y , ρ Y , τ Y ) ∈ ( Z C ) B , consider the f ollo wing pair in X ( o r in X A ) (29) RLRY RεY / / Rτ Y / / RY . Then Rρ Y is a common right inv erse for RεY and Rτ Y . F urthermore, the follow ing is a contractible co equaliser in ( Z C ) B LRLRY LRεY / / LRτ Y / / LRY τ Y / / LηRY d d Y . ρ Y c c Hence, (29) is a r eﬂexiv e L -con tractible co equaliser pair . Th us if the co equaliser of the pair (29) exists (in X ), it is exactly the co equaliser (19). A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 27 Prop osition 5.1.3. Consider the fol lowing statements. (i) K B is a n e q uivalenc e of c ate gories; (ii) X c ontain s c o e qualisers of p a irs (29) (i.e. r eﬂexive L -c ontr actible c o e qualise r p airs) and AA pr eserves them; (iii) The Morita c ontext T of ( 2 6) induc es an e quivalenc e of c ate gories X A and ( Z C ) B . Then ( i ) implies ( ii ) implie s ( iii ) . Pro of. ( i ) ⇒ ( ii ) . Since K B is an equiv alence, up to an isomorphism, o b jects in ( Z C ) B are of the form K B ( Z ) = ( LRZ , Lη R Z , LRεZ ) for some Z ∈ Z . Conseq uen t ly the pair (29) results in the follo wing con tractible co equaliser in Z C RLRLRZ RεLRZ / / RLRεZ / / RLRZ RεZ / / ηRLRZ e e RZ . ηRZ d d ( ii ) ⇒ ( iii ). This f ollo ws immediately fro m Theorem 4.2.5 and the ab o v e observ ations. ⊔ ⊓ Corollary 5.1.4. K B is a n e quivalenc e if and only if K A is a n e quivalenc e and c on - diton ( ii ) of Pr op osition 5.1.3 holds. Pro of. This follows fro m Prop o sition 5.1.3 and the equalit y K B = Λ K A (= b T A K A ). ⊔ ⊓ R emark 5.1 .5 . Prop osition 5.1.3 a nd Corollary 5.1.4 provide an alternative pro of of (the dual v ersion of ) [8, Theorem 2.19, Theorem 2.20]. 5.2. Morita theory for rings. In ring and mo dule theory , a Morita c on text is a sextuple ( A, B , M , N , σ , ˆ σ ), where A and B are rings, M is an A - B bimo dule, N is a B - A bimo dule, σ : M ⊗ B N → A is a n A - A bimo dule map, and ˆ σ : N ⊗ A M → B is a B - B bimo dule map rendering commutativ e the following diagra ms (30) M ⊗ B N ⊗ A M σ ⊗ A M / / M ⊗ B ˆ σ   A ⊗ A M   M ⊗ B B / / M , N ⊗ A M ⊗ B N ˆ σ ⊗ B N / / N ⊗ A σ   B ⊗ B N   N ⊗ A A / / N . The unmarked arrows are canonical isomorphisms (induced b y actions). With eve ry Morita con text one asso ciates a matrix-type Morita ring Q =  A M N B  := {  a m n b  | a ∈ A, b ∈ B , m ∈ M , n ∈ N } , with the pro duct giv en by  a m n b   a ′ m ′ n ′ b ′  =  aa ′ + σ ( m ⊗ A n ′ ) am ′ + mb ′ na ′ + bn ′ bb ′ + ˆ σ ( n ⊗ B m ′ )  . Since Q has tw o orthogonal idemp oten ts summing up to the iden tit y , its left mo d- ules split into direct sums. More precisely , left Q -mo dules corresp ond to quadruples ( X , Y , ¯ v , ¯ w ), where X is a left A -mo dule, Y is a left B -mo dule, ¯ v : M ⊗ B Y → X is 28 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE a left A -mo dule map a nd ¯ w : N ⊗ A X → Y is a left B -mo dule map suc h that the follo wing diagrams (31) M ⊗ B N ⊗ A X σ ⊗ A X / / M ⊗ B ¯ w   A ⊗ A X   M ⊗ B Y ¯ v / / X , N ⊗ A M ⊗ B Y ˆ σ ⊗ B Y / / N ⊗ A ¯ v   B ⊗ B Y   N ⊗ A X ¯ w / / Y , comm ute. The left action of Q on X ⊕ Y is giv en by  a m n b   x y  =  ax + ¯ v ( m ⊗ B y ) by + ¯ w ( n ⊗ A x )  . T ak e ab elian groups ( Z - mo dules) A and B . The asso ciated tensor functors A ⊗ = A ⊗ − : Ab → Ab , B ⊗ = B ⊗ − : Ab → Ab are monads on the cat ego ry of ab elian groups if and only if A and B are rings. F urthermore Ab A ⊗ = A M (the category of left A - mo dules) and Ab B ⊗ = B M . T ak e a b elian groups M and N . T he tensor functor T = M ⊗ − : Ab → Ab is an A ⊗ ( − )- B ⊗ ( − ) bialg ebra if and only if M is an A - B bimo dule. The left action λ : A ⊗ T = A ⊗ M ⊗ − → M ⊗ − is, for all ab elian groups X , λX : λ Z ⊗ X , where λ Z : A ⊗ M → M is a left action of A on M . Similarly , the righ t action of B ⊗ on T corr espo nds to a righ t action of B on M . Symmetrically , b T = N ⊗ − is a B ⊗ - A ⊗ bialgebra if and only if N is a B - A bimo dule. An y A ⊗ - A ⊗ bialgebra map ev : T b T = M ⊗ N ⊗ − → A ⊗ − = A ⊗ is fully determined b y its v alue at Z b y ev X = ev Z ⊗ X ; the map ev Z : M ⊗ N → A is an A - A bimo dule map. If the map ev is required to satisfy the second o f the diagrams in (7), then the univ ersalit y of t ensor pro ducts yields a unique A -bimo dule map σ : M ⊗ B N → A suc h that ev Z : M ⊗ N / / M ⊗ B N σ / / A . Con vers ely , an y A - A bilinear map σ : M ⊗ B N → A determines a B ⊗ -balanced A ⊗ - bialgebra natural map ev : T b T = M ⊗ N ⊗ − → A ⊗ − = A ⊗ . In a symmetric w ay there is a bijectiv e correspondence betw een A ⊗ -balanced B ⊗ -bialgebra maps b ev : b T T → B ⊗ and B - B bilinear maps ˆ σ : N ⊗ A M → B . The natural transformat ions ev , b ev satisfy conditions (6) if and only if the corresp onding maps σ , ˆ σ satisfy c onditions (30). These observ ations establish bijectiv e corresp ondence (in fact, an isomorphism of categories) b et w een mo dule theoretic Morita con t exts ( A, B , M , N , σ, ˆ σ ) and ob jects in Mo r ( A ⊗ − , B ⊗ − ). Once an ob ject T in M o r ( A ⊗ − , B ⊗ − ) is iden tiﬁed with a mo dule-theoretic Mor it a con text ( A, B , M , N , σ , ˆ σ ) one can compute the corr espo nding Eilen b erg-Mo ore cate- gory ( Ab , Ab ) T . An ob ject in ( Ab , Ab ) T consists of a left A -mo dule X (an algebra of the monad A ⊗ − ) and a left B -mo dule Y (an algebra of the monad B ⊗ − ), and tw o mo dule maps v : M ⊗ Y → X a nd w : N ⊗ X → Y . The comm uta t ivity of diagrams (13) yields unique mo dule maps ¯ v : M ⊗ B Y → X and ¯ w : N ⊗ A X → Y suc h that the follo wing dia grams M ⊗ Y v / / & & M M M M M M M M M M X M ⊗ B Y , ¯ v : : u u u u u u u u u u N ⊗ X w / / & & M M M M M M M M M M Y N ⊗ A X ¯ w : : v v v v v v v v v v A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 29 comm ute. Diagrams (12) for v and w a r e equiv alent to diagrams (31) for ¯ v and ¯ w . This establishes a n iden tiﬁcation (isomorphism) of the Eilenberg-Mo ore cat ego ry ( Ab , Ab ) T with the category o f left mo dules of the corresp onding matrix Morita ring Q . With the in terpretation of ( Ab , Ab ) T as left Q - mo dules Q M , the functors U A ⊗ , U B ⊗ constructed in Section 3.3 are forgetful functors Q M → Ab , U A ⊗ ( X ⊕ Y ) = X , U B ⊗ ( X ⊕ Y ) = Y . The func tors G A ⊗ , G B ⊗ send an ab elian group X to its tens or pro duct with resp ectiv e columns in Q . More precisely , G A ⊗ ( X ) = A ⊗ X ⊕ N ⊗ X with the multiplication by Q ,  a m n b   a ′ ⊗ x n ′ ⊗ y  =  aa ′ ⊗ x + σ ( m ⊗ A n ′ ) ⊗ y na ′ ⊗ x + bn ′ ⊗ y  , while G B ⊗ ( X ) = M ⊗ X ⊕ B ⊗ X with the action of Q ,  a m n b   m ′ ⊗ x b ′ ⊗ y  =  am ′ ⊗ x + mb ′ ⊗ y ˆ σ ( n ⊗ B m ′ ) ⊗ x + bb ′ ⊗ y  , for all m, m ′ ∈ M , n, n ′ ∈ N , a, a ′ ∈ A , b, b ′ ∈ B and x, y ∈ X . The construction presen ted in this section can b e rep eated with X c hosen to b e t he category k A M of left mo dules o v er a r ing k A and Y the category of left mo dules o ver a r ing k B . All the functors A ⊗ , B ⊗ , T , b T can b e c hosen as tensor functors with the tensor pro duct o v er resp ectiv e ring s k A or k B . F o r example, tak e a k A - k A bimo dule A a nd deﬁne A ⊗ as a functor − ⊗ k A A : k A M → k A M . A ⊗ is a monad if and only if A is a k A -ring. Similar ly c ho ose B ⊗ = − ⊗ k B B for a k B -ring B . Since mo dules o v er a k A -ring A coincide with mo dules of the ring A , one can c ho ose further a n A - B bimo dule M and B - A bimo dule N and pro ceed as ab ov e, taking care t o decorate suitably tensor pro ducts with k A and k B . 5.3. Categories with binary copro ducts. The characterisation of the Eilenberg- Mo ore category of a mo dule-theoretic Morita con text described in Section 5 .2 can b e seen as a special case of the follow ing situation. As sume that categories X a nd Y ha v e binary copro ducts. T ak e a Morita con text T = ( A , B , T , b T , ev , b ev) on X a nd Y whic h functors A , B , T and b T preserv e (binary) copro ducts. These data lead to the follo wing monad Q = ( Q, m , u ) on the pro duct category X × Y . The f unctor Q is deﬁned as Q : ( X , Y ) 7→ ( AX + T Y , B Y + b T X ). The unit u of Q is the comp osite: u ( X , Y ) : ( X , Y ) ( u A X, u B Y ) / / ( AX , B Y ) / / ( AX + T Y , B Y + b T X ) , where the second morphism consists of the canonical injections AX → AX + T Y and B Y → B Y + b T X . The m ultiplicatio n, m ( X , Y ), ( AAX + AT Y + T B Y + T b T X , B B Y + B b T X + b T AX + b T T Y ) → ( AX + T Y , B Y + b T X ) , com bines m ultiplications in A (and B ) with actions of monads on T (a nd b T ) and with ev (and b ev). That is AAX + T b T X + AT Y + T B Y → AX + T Y is giv en as a sum ( m A + ev) X + ( λ + ρ ) Y , where ( m A + ev) X : AAX + T b T X → AX a nd 30 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE ( λ + ρ ) Y : AT Y + T B Y → T Y are the unique ﬁllers in the diagrams AAX + T b T X AAX 8 8 q q q q q q q q q q q q m A X ' ' N N N N N N N N N N N N T b T X , f f N N N N N N N N N N N ev X w w p p p p p p p p p p p p AX AT Y + T B Y AT Y 7 7 o o o o o o o o o o o λY ' ' O O O O O O O O O O O O T B Y g g O O O O O O O O O O O ρY w w o o o o o o o o o o o o T Y . The second comp onent of m ( X, Y ) is deﬁned similarly . In this case, the Eilen b erg- Mo ore category for the Mor it a con text T is isomorphic to the Eilen b erg-Mo ore cate- gory of a lgebras of the monad Q , i.e. ( X , Y ) T ∼ = ( X × Y ) Q . Giv en a double adjunction Z = ( Z , ( L A , R A ) , ( L B , R B )) ∈ Adj ( X , Y ), in whic h also Z has binary copro ducts whic h are preserv ed b y R A , R B , the functor h R A , R B i : Z → X × Y , Z 7→ ( R A Z , R B Z ) has a left adjoint [ L A , L B ] : ( X , Y ) 7→ L A X + L B Y . The monad deﬁned b y the adjunction [ L A , L B ] ⊣ h R A , R B i is simply the monad Q corresp onding to the Morita con text Υ( Z ) (see Section 3.1). In view of the ab ov e iden tiﬁcation of Eilenberg-Mo ore categories, the moritability of ( R A , R B ) is equiv alen t to the monadicity of h R A , R B i a nd thus is decided b y the classical Bec k theorem. 5.4. F ormal duals. As recalled in Section 2 .3, giv en a monad A = ( A, m A , u A ) on X a nd a mo na d B = ( B , m B , u B ) on Y a pair of formally dual bialgebras is a pair of bialgebra f unctor s T : Y → X , b T : X → Y equipped with natural bialgebra transformations ev : T b T → A , b ev : b T T → B tha t satisfy compatibilit y conditions expresse d b y diagr a ms (6). In o ther w o r ds, a pa ir of formally dual bialgebra f unctors is the same as an unb alanc e d Morita c ontext . Morphisms betw een pairs of forma lly dual bialgebras are quadruples consisting of tw o monad mor phism and t wo bialgebra morphisms whic h satisfy the same conditions a s morphisms b et wee n Morita con texts. This deﬁnes a category Dual ( X , Y ) o f whic h Mor ( X , Y ) is a full sub category . Th us the functor Υ : Adj ( X , Y ) → Mo r ( X , Y ) describ ed in Section 3.1 extends to the functor Adj ( X , Y ) → Dual ( X , Y ) . 5.5. Herds versus pretorsors. F ollowing [4, App endix], a he r d functor is a pa ir o f formally dual bialgebra functors T = ( A , B , T , b T , ev , b ev) (i.e. an ob j ect of Dua l ( X , Y )) together with a natural transformatio n γ : T → T b T T r endering comm utative the follo wing diagrams (32) T γ   T u B { { w w w w w w w w w u A T # # H H H H H H H H H T B AT , T b T T T b ev b b E E E E E E E E E ev T ; ; w w w w w w w w T γ / / γ   T b T T T b T γ   T b T T γ b T T / / T b T T b T T . The ma p γ is called a shepher d . A morphism b etw een herds φ : ( A , B , T , b T , ev , b ev , γ ) → ( A ′ , B ′ , T ′ , b T ′ , ev ′ , b ev ′ , γ ′ ) is a morphism φ = ( φ 1 , φ 2 , φ 3 , φ 4 ) in Dual ( X , Y ) compatible A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 31 with shepherds γ and γ ′ , i.e. whose third and fo urt h comp onen ts make the fo llo wing diagram (33) T γ / / φ 3   T b T T φ 3 φ 4 φ 3   T ′ γ ′ / / T ′ b T ′ T ′ comm ute, where φ 3 φ 4 φ 3 = φ 3 b T ′ T ′ ◦ T φ 4 T ′ ◦ T b T φ 3 denotes the Go demen t pro duct. The category of herd functors b et w een categories X a nd Y is denoted b y Herd ( X , Y ). This category con tains a full sub category Herd ( X , Y ) of b a l a nc e d her d s with ob jects those herds, whose underlying formally dual pair is a Morita cotext (i.e. c haracterised b y the fo rgetful functor Herd ( X , Y ) → M o r ( X , Y )). F ollowin g [3, Deﬁnition 4.1], a pr e-torsor is an ob ject Z = ( Z , ( L A , R A ) , ( L B , R B )) of Adj ( X , Y ) together with a natural transformation τ : R A L B → R A L B R B L A R A L B rendering comm uta tiv e the following diagr a ms (34) R A L B τ   R A L B η B u u k k k k k k k k k k k k k k k η A R A L B ) ) S S S S S S S S S S S S S S S R A L B R B L B R A L A R A L B , R A L B R B L A R A L B R A L B R B ε A L B i i S S S S S S S S S S S S S S R A ε B L A R A L B 5 5 k k k k k k k k k k k k k k (35) R A L B τ / / τ   R A L B R B L A R A L B R A L B R B L A τ   R A L B R B L A R A L B τ R B L A R A L B / / R A L B R B L A R A L B R B L A R A L B . A morphism from a pre-torsor ( Z , ( L A , R A ) , ( L B , R B ) , τ ) to ( Z ′ , ( L ′ A , R ′ A ) , ( L ′ B , R ′ B ) , τ ′ ) is a morphism F in Adj ( X , Y ) t ha t is compatible with the structure maps τ and τ ′ . The compatibilit y condition is expressed as the equalit y (36) ( R A bR B aR A b ) ◦ τ = τ ′ ◦ R A b, where a and b are deﬁned by (1) and the shorthand notation for the Go demen t pro duct is used. T he category of pre-torsors on catego ries X and Y is denoted b y PreT or ( X , Y ). Ev en the most p erfunctory comparison of dia grams (32) with (34)-(35) rev eals that the functor Υ applied to the do uble a djunction Z underlying a pre-to r sor ( Z , τ ) yields a (balanced) herd with a shepherd τ , i.e. ( Υ Z , τ ) is a herd. The deﬁnition of Υ on morphisms immediately a ﬃrms that the condition (36) f or F implies condition (33) for Υ F . Th us Υ yields the functor Υ : PreT or ( X , Y ) → Herd ( X , Y ) ⊆ Herd ( X , Y ) , ( Z , τ ) 7→ (Υ Z , τ ) . In the conv erse direction, the functor Γ constructed in Section 3.3 yields the f unctor Γ : Herd ( X , Y ) → PreT o r ( X , Y ) , ( T , γ ) 7→ (Γ T , γ ) . 32 TOMASZ BRZEZI ´ NSKI, ADRIA N V A Z Q UEZ MARQUEZ, AND JOOST V ERCRUYSSE The k ey observ ation is tha t T = U A G B and b T = U B G A , hence the s hepherd γ of the herd T b ecomes the natural transforma t io n τ f o r the correspo nding pre-torsor (( X , Y ) T , ( G A , U A ) , ( G B , U B )); see the deﬁnition of U A , G A , U B , G B in Section 3.3 . As (implicitly) calculated in the pro o f of Prop osition 3.5.2, the natural trasforma- tions a a nd b corresp onding to the comparison functor K : Z → ( X , Y ) T are identit y maps, hence the condition (36) is trivially satisﬁed, so, fo r an y pre-torsor, K is a morphism of pre-torsors. Th us Prop osition 3.5.2 and Coro lla ry 3.5.3 immediately imply Corollary 5.5.1. ( Γ , Υ) is an adjoint p air a n d Γ is a ful l and faithful functor. F ur- thermor e, ( Γ , Υ) is a p air of inverse e quivalenc e s b e twe e n c ate gories of pr e-torsors and b alanc e d her d functors if and only if, for al l pr e-torsors ( Z , ( L A , R A ) , ( L B , R B ) , τ ) ∈ PreT or ( X , Y ) , ( R A , R B ) is a moritable p air. In p articular, if (Γ , Υ) is a p air of invers e e quivalenc es, then so is ( Γ , Υ) . 6. Remarks on dualisa tions and ge neralisa tions 6.1. Dualisations. There are v arious w ays in whic h the categories studied in pre- ceding sections and thus results describ ed there can b e (semi-)dualised. One can deﬁn e the category Adj o ( X , Y ), whose ob j ects are p en t uples (or tr iples) ( Z , ( L A , R A ) , ( L B , R B )), where Z is a category and ( L A : X → Z , R A : Z → X ) and ( L B : Z → Y , R B : Y → Z ) are adjunctions. Morphisms a re deﬁned in the natural w ay . The category Adj c ( X , Y ) is deﬁned by taking ob jects ( Z , ( L A , R A ) , ( L B , R B )), where Z is a category and ( L A : Z → X , R A : X → Z ) a nd ( L B : Y → Z , R B : Z → Y ) are adjunctions. Finally , the category Adj o,c ( X , Y ) has ob jects ( Z , ( L A , R A ) , ( L B , R B )), where Z is a category and ( L A : Z → X , R A : X → Z ) and ( L B : Z → Y , R B : Y → Z ) are adjunctions. On the other hand, one can consider the category of Morita-T ake uc hi contexts , whose ob j ects are sextuples ( C , D , P , b P , co v , co v ) consisting of tw o comonads, t w o bicoalgebra functors, and tw o bicolinear cobalanced natural transformations satisfying compatibilit y conditions dual t o those in Section 2.3. Also, one can consider an in t ermediate v ersion, where the ﬁrst tw o ob jects in the sextuple are a monad and a comonad resp ective ly . By (semi-)dualising the results o f the previous sections, functors b et w een the re- sp ectiv e categories with pairs of adjunctions and the resp ectiv e categories of con texts can b e constructed. Appropriate Eilen b erg- Mo ore catego ries for v arious con t exts can b e deﬁned thu s yielding a conv erse construction and leading to the deﬁnition of a comparison functor in eac h case, a nd to the solutio n of the corresp onding mor it abilit y problem. 6.2. Bicategories. Adjoint pairs, Morita a nd T ak euchi con texts hav e a natural for- m ulat io n within the fra mework of bicategories; see, for example, the bicategorical form ulation o f wide Morita con texts in [7]. W e b eliev e t ha t our w ork can, taking in t o accoun t the needed (computational) care but without a n y conceptual problems, b e transferred to this (mor e general) setting. How ev er, w e preferred to f orm ulate the results of this pap er in the pres en t w ay , as this pres en ta tion migh t b e clearer, A BECK-TYPE THEOREM FOR A MOR I T A CONTEXT 33 more accessible a nd w e b eliev e tha t ev en in this g eneralit y it co v ers a lready enough in t erestering examples a nd applications. A c kno wledgements The authors w ould lik e to thank Viola Bruni and George Janelidze for helpful commen ts. The researc h of A. V azque z Marquez is supp o rted b y the CONA CYT gran t no. 20 8351/303 031. Reference s [1] M. Ba rr and C. W ells, T o po ses, triples and theories, R epr. The ory Appl. Cate g. , No. 1 2 (2005) pp. 1–28 7 [2] J.M. Beck, T riples, alg ebras and cohomology , PhD Thesis , Columbia Univ ersity , 1967 ; R epr. The ory Appl. Cate g. , No. 2 (2003 ), pp. 1–59. [3] G. B¨ ohm and C. Menini, Pre-tor sors a nd Galois como dules over mixed dis tr ibutive laws, P reprint 2008, arXiv:08 06.12 1 2 . T o a ppea r in A ppl. Cate g. Str. [4] T. Brzezi ´ nski a nd J . V ercruy sse, Bimo dule herds, J . A lgebr a 321 (2009 ), 267 0–270 4. [5] F. Casta ˜ no-Iglesias, J. G´ omez - T or recillas, Wide Mo rita contexts and equiv alences o f como dule categorie s, J. Pur e Appl. A lgebr a , 131 (199 8), 21 3–225 . [6] S. Eilenberg a nd J. C. Mo ore, Adjoin t functors a nd triples, Il linois J. Math. , 9 (19 65), 3 81–39 8. [7] L. E l Kao utit, Wide Morita contexts in bicategor ies, Ar ab. J. Sci. Eng. , 33 No. 2C (2008 ), 153–1 73. [8] B. Mesa blishvili, Monads of eﬀective descent t ype a nd comonadicity , The ory Appl. Cate g. 16 (2006), 1–45. Dep ar tment of Ma thema tics, Sw ansea Un iversity, Singleton P ark, Sw ansea S A2 8PP, U. K. E-mail addr ess : T. Brzezi nski@s wansea.ac.uk Dep ar tment of Ma thema tics, Sw ansea Un iversity, Singleton P ark, Sw ansea S A2 8PP, U. K. E-mail addr ess : 39 7586@s wansea .ac.uk F a cul ty of Engineering, V rije Universiteit Brussel (VU B), B-1050 Brussels, Belgium E-mail addr ess : jv ercruy @vub.a c.be

The Eilenberg-Moore category and a Beck-type theorem for a Morita context

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