Abstract Abstract. The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product. As first generalisations, these notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A, employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For τ -bimonads H, properties of braided (Hopf) bialgebras were captured by requiring a Yang-Baxter operator τ : HH → HH. The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang-Baxter operators) as considered by Alonsó Alvarez and others. Introduction There are various generalisations of the notions of (weak) bialgebras and Hopf algebras in the literature, mainly for (braided) monoidal categories, and we refer to Böhm [7] , the introductions to AlonsoÁlvarez e.a. [2] , Böhm e.a. [10] , and [14, Remarks 36.18] for more information about these. Introduction Bimonads and Hopf monads on arbitrary categories were introduced in [24] and the purpose of the present paper is to develop a weak version of these notion, that is, the initial conditions on the behaviour of the involved distributive laws towards unit and counit are replaced by weaker conditions. Introduction where M is the category of k-modules, M H the category of right H-modules, and M H H denotes the category of mixed bimodules; the latter can also be considered as (M H ) H , that is, the category of H-comodules over M H where H is the lifting of the comonad − ⊗ H to M H . H is a Hopf algebra provided the functor − ⊗ H is an equivalence of categories (Fundamental Theorem of Hopf modules). Introduction Concentrating on the essential parts of this setting, we consider, for any category A, the diagram Introduction where T is some monad on A, G is some comonad on the category A T of T-modules, φ T and U G denote the respective free and forgetful functors, and K is any functor making the diagram commutative. Introduction Having such a diagram, one may ask when the functor K allows for a right adjoint K. This creates a monad P on A, a monad morphism ι : P → T , the free functor φ P : A → A P , and a functor K P : (A T ) G → A P , the Eilenberg-Moore comparison functor for the monad P . Introduction If A is Cauchy complete and P is a separable Frobenius monad, then the change-of-basefunctor ι ! : A P → A T exists (see Proposition 3.2). As a consequence, K P has a left adjoint L P (Proposition 4.6) leading to the commutative diagram Introduction So asking for a fundamental theorem leads to the question for properties of φ P and K P . Similar constructions apply when the functor K has a left adjoint. After assembling preliminaries in Section 2 and properties of separable Frobenius monads in Section 3, the theory just sketched is outlined in Section 4. Introduction Section 5 deals with the application of this to endofunctors H on a category A endowed with a monad H as well as a comonad structure H, and a weak monad-comonad entwining ω : HH → HH. Exploiting ideas from [34] and Böhm [6] , these data allow for the definition of a comonad G on A H as well as a monad T on A H (Propositions 5.3, 5.5). Hereby, the category A H H (ω) of mixed H-bimodules is isomorphic to the categories (A H ) G and (A H ) T (see Theorems 5.4, 5.6) . If ω is a compatible entwining (i.e. δ · m = Hm · ωH · Hδ, Section 5.1), there is a functor Introduction leading to commutativity of the diagram corresponding to (1.1). Conditions for the existence of a right and a left adjoint functor for K ω are investigated (Propositions 5.10, 5.11). These problems were considered in [24] for proper compatible entwinings ω. Introduction In Section 6, we define weak τ -bimonads, also called weak braided bimonads (Definition 6.2), based on a weak Yang-Baxter operator τ : HH → HH (Section 6.1). This type of operator was introduced by AlonsoÁlvarez e.a. in [1] for monoidal categories and is here adapted to the more general setting. The conditions on weak braided bimonads induce a weak monad-comonad entwining ω : HH → HH as well as a weak comonad-monad entwining ω : HH → HH and allow to refine the results in Section 5: the natural transformation ξ := Hε · ω · He : H → H is idempotent and its splitting yields a separable Frobenius monad H ξ (see Proposition 6.11). Then, if K ω has a right adjoint functor K, the induced monad P is just H ξ and the diagram corresponding to (1.2) can be completed. Introduction The weak bialgebras over a commutative ring k as considered by Böhm e.a. in [9] are weak braided bimonads in our sense (τ the ordinary twist, ξ = ε s ) and for this case some of our results are shown there, including the Frobenius and separability property of H ξ (= H s ) ( [9, Proposition 4.4] ). Introduction Eventually, in Section 7, weak braided Hopf monads are defined as weak braided bimonads H with a weak antipode (Definition 7.1). In monoidal categories, these correspond to the weak braided Hopf algebras considered in [2] , [3] . Introduction We show that for a braided bimonad in Cauchy complete categories the existence of an antipode is equivalent to the functor K ω having a left adjoint and a monadic right adjoint and this leads to an equivalence between the categories of H ξ -modules and A olH H (ω) (Fundamental Theorem 7.6). Introduction Examples for our weak braided bimonads and weak braided Hopf monads are the weak braided Hopf algebras in strict monoidal Cauchy complete categories considered by Alonsó Alvarez et al. in [1, 2, 3] , the weak bimonoids and weak Hopf monoids in braided monoidal categories as defined by Pastro et. al. in [26] and also showing up in [10, Sections 3, 4] . These all subsume the braided Hopf algebras considered, e.g., by Takeuchi [32] and Schauenburg [27] and, of course, the weak Hopf algebras in vector space categories introduced by Böhm et al. in [11] . Moreover, we generalise the bimonads and τ -Hopf monads defined on arbitrary categories in [24] and these include, for example, bimonoids in duoidal categories (e.g. [25] ). Introduction Opmonoidal monads T = (T, m, e) on strict monoidal categories (V, ⊗, I) were also called bimonads by Bruguières et al. in [13] and these were generalised to weak bimonads in monoidal categories by Böhm et al. in [10] . As pointed out in [23, Section 5] , the bimonads from [13] yield a special case of an entwining of the monad T with the comonad − ⊗ T (I), where T (I) has a coalgebra structure derived from the opmonoidality of T. To transfer the structures from [10] to arbitrary categories one has to consider weak entwinings between monads and comonads. It is planned to elaborate details for this in a subsequent article. Preliminaries Monads and comonads. Recall that a monad T on a category A (or shortly an A-monad T) is a triple (T, m, e) where T : A → A is a functor with natural transformations m : T T → T , e : 1 → T satisfying the usual associativity and unitality conditions. A T-module is an object a ∈ A with a morphism h : T (a) → a subject to associativity and unitality conditions. The (Eilenberg-Moore) category of T-modules is denoted by A T and there is an adjunction e T , ε T : φ T ⊣ U T : A T → A, with φ T : A → A T and U T : A T → A given by the respective assignments a → (T (a), m a ) and (a, h) → a, while e T = e and (ε T ) (a, h) = h for each (a, h) ∈ A T . Monads and comonads. If T = (RF, RεF, η) is the monad generated on A by an adjoint pair η, ε : F ⊣ R : B → A, then there is the comparison functor K T : B → A T which assigns to each object b ∈ B the Talgebra (R(b), R(ε b )), and to each morphism f : Monads and comonads. The functor R is called monadic (resp. premonadic) if the comparison functor K T is an equivalence of categories (resp. full and faithful). 2.2. Theorem. (Beck, [5] ) Let η, ǫ : F ⊣ R : B → A be an adjunction, and T = (RF, RεF, η) the corresponding monad on A. 2.2. (1) The comparison functor 2.2. ) has a coequaliser in B. (2) R is monadic if and only if it is conservative and for (a, h) ∈ A T , the pair of morphisms (F (h), ε F (a) ) has a coequaliser and this coequaliser is preserved by R. 2.2. Dually, a comonad G on A (or shortly an A-comonad G) is a triple (G, δ, ε) where G : A → A is a functor with natural transformations δ : G → GG, ε : G → 1, and G-comodules are objects a ∈ A with morphisms θ : a → G(a). Both notions are subject to coassociativity and counitality conditions. The (Eilenberg-Moore) category of G-comodules is denoted by A G and there is a cofree functor 2.2. which is right adjoint to the forgetful functor 2.2. If η, ε : F ⊣ R : A → B is an adjoint pair and G = (F R, F ηR, ε) is the comonad on A associated to (R, F ), then one has the comparison functor 2.2. One says that the functor F is precomonadic if K G is full and faithful, and it is comonadic if K G is an equivalence of categories. Cauchy completeness. A morphism e : A → A in A is idempotent if e 2 = e and A is said to be Cauchy complete if for every idempotent e : a → a, there exists an object a and morphisms p : a → a and i : a → a such that ip = e and pi = 1 a . In this case, (a, i) is the equaliser of e and 1 a and (a, p) is the coequaliser of e and 1 a . Hence any category admitting either equalisers or coequalizers is Cauchy complete. Proof. The result follows from the fact that the forgetful functor U G : A G → A preserves and creates coequalisers, while the functor U T : A T → A preserves and creates equalisers. ⊔ ⊓ 2.5. Split (co)equalisers. Recall (e.g. from [20] ) that a diagram Cauchy completeness. with p∂ 0 = p∂ 1 is said to be split by a pair of morphisms i : x → a ′ and s : Cauchy completeness. A pair of morphisms (∂ 0 , ∂ 1 : a ⇒ a ′ ) in A is called split if there exists a morphism s : a ′ → a with ∂ 0 s = 1 and ∂ 1 s∂ 0 = ∂ 1 s∂ 1 . In this case, ∂ 1 s : a ′ → a ′ is an idempotent, and if we assume A to be Cauchy complete and if Cauchy completeness. is a split coequaliser. Conversely, if the above diagram is a split coequaliser, then s makes the pair (∂ 0 , ∂ 1 : a ⇒ a ′ ) split. Thus, when A is Cauchy complete, a pair (∂ 0 , ∂ 1 : a ⇒ a ′ ) is part of a split coequaliser diagram if and only if it is split. Note that split (co)equalisers are absolute, i.e., they are preserved by any functor. Cauchy completeness. If Cauchy completeness. The dual notions are those of cosplit pairs and F -cosplit pairs. Cauchy completeness. Given a monad T (resp. comonad G) on A and a category X, one may consider the functor category [X, A] and the induced monad [X, T] (resp. comonad [X, G]) thereon, whose functor part sends a functor F : X → A to the composite T F : X → A (resp. GF : X → A). Symmetrically, one has the induced monad [T, X] (resp. comonad [G, X]) on [A, X], whose functor-part sends F : A → X to F T : A → X (resp. F G : A → X). Cauchy completeness. A morphism of (T, S)-bimodules is a morphism of left T-modules which is simultaneously a morphism of right S-modules. We write [X, A] [S,T] for the corresponding category. Cauchy completeness. 2.7. Canonical modules. Let T = (T, m, e) be an arbitrary monad on A. Using the associativity and unitality of the multiplication m, we find that for any functor M : X → A, the pair (T M, mM ) is a left T-module. Moreover, if ν : M → M ′ is a natural transformation, then T ν : T M → T M ′ is a morphism of left T-modules. Cauchy completeness. Symmetrically, for any functor N : A → Y, the pair (N T, N m) is a right T-module, and if ν : N → N ′ is a natural transformation, then νT : N T → N T ′ is a morphism of right T-modules. In particular, (T, m) can be regarded as a right as well as a left T-module; again by the associativity of m, (T, m, m) is a (T, T)-bimodule. Moreover, if S is another monad on A and ι : S → T is a monad morphism, then Cauchy completeness. 2.8. Tensor product of module functors. Let T be a monad on A and X and Y arbitrary categories. If (N, ρ) is a left T-module and (M, ̺) a right T-module, then their tensor product (over T) is defined as the object part of the coequaliser Cauchy completeness. The action ζ is uniquely determined by the property Cauchy completeness. Proof. A left G-comodule functor is a pair (F, θ), where F : X → A is a functor and θ : F → GF is a natural transformation inducing commutativity of the diagrams Cauchy completeness. Symmetrically, right G-comodule functors A → Y are defined. 2.11. Cotensor product of comodule functors. Let G be a comonad on A and X, Y arbitrary categories. If (C, θ) is a right G-comodule functor A → X and (D, ϑ) is a left G-comodule functor Y → A, then their cotensor product (over G) is defined as the object part of the equaliser 2.12. Monads and adjunctions. For categories A, B and C, consider the adjunctions η, ε : F ⊣ R : A → C and η ′ , ε ′ : F ′ ⊣ R ′ : B → C, with corresponding monads T and T ′ , and let [16] 2.11. When this is the case, the unit η of the adjunction K ⊣ K is the unique natural transformation 1 → KK yielding commutativity of the diagram 2.11. while the counit ε is the unique natural transformation KK → 1 making the diagram 2.11. commute. Precomposing the image of the last square under R ′ with ηR ′ and using the fact that R ′ γ K is a monad morphism T ′ → T, we get the commutative diagram 2.11. Since R ′ ε ′ · η ′ R ′ = 1 by one of the triangular identities for F ′ ⊣ R ′ , one gets 2.11. 2.14. Comonads and adjunctions. Again let η, ε : 2.11. A → C be adjunctions and now consider the corresponding comonads G and G ′ . Let 2.11. A be a commutative diagram and define Theorem. In the situation described in 2.14, suppose that F ′ is precomonadic. Then K has a right adjoint K if and only if the following equaliser exists in [C, B], When this equaliser exists, the unit η of the adjunction K ⊣ K is the unique natural transformation 1 → KK yielding commutativity of the diagram Theorem. In the situation described in 2.14, suppose that F ′ is precomonadic. Then K has a right adjoint K if and only if the following equaliser exists in [C, B], while the counit ε of the adjunction K ⊣ K is the unique natural transformation KK → 1 making the diagram 2.16. The restriction-and change-of-base functors. Any morphism 2.16. of monads on A (that is, a natural transformation ι : S → T such that ι · e S = e T and ι · m S = m T · (ιι)) gives rise (see [5] ) to the functor 2.16. called the restriction-of-base functor. It is clear that ι * makes the diagram 2.16. Since the forgetful functor U S : A S → A is clearly monadic, it follows from the Theorem 2.13 that ι * has a left adjoint ι ! : A S → A T if and only if the pair of natural transformations when it exists, is called the change-of-base functor. Recalling that for any (2) It is easy to see that when it exists, is called the change-of-base functor. Recalling that for any is just the pair Thus, if the coequaliser diagram (2.6) exists and if its image under the functor [ A S , U T ] : [A S , A T ] → [A S , A] is again a coequaliser, we get a coequaliser T ST T m T ·T ιT G G m T T ·T ιT G G T T U T qι * φ T G G U T ι ! ι * φ T . But since (T, m T · T ι) is a right S-module, while (T, m T · ιT ) is a left S-module (see Section 2.7), one concludes that U T ι ! ι * φ T = T ⊗ S T and that U T qι * φ T = can T,T S . 2.18. Proposition. If ι : S → T is a morphism of A-monads such that the change-of-base functor ι ! : A S → A T exists, then the diagram A φ T 2 2 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ φ S G G A S ι ! A T commutes (up to isomorphism). Proof. Since U S · ι * = U T (see Section 2.16) and φ S (resp. ι ! ) is left adjoint to U S (resp. ι * ), the result follows by uniqueness of left adjoints. commutes (up to isomorphism). ⊔ ⊓ Separable Frobenius monads The crucial role of separable Frobenius functors (e.g. [29] ) in the theory of weak bimonads was pointed out by Szlachányi in [31] and such functors are used by Böhm et al. in [10] as an integral part of their definition of weak bimonads on monoidal categories. In this section we show that in our approach separable Frobenius monads S are of interest since they imply the existence of the change-of-base functor for monad morphisms S → T . 3.1. Definition. An A-monad S = (S, m S , e S ) is called Frobenius if S has an A-comonad structure G = (S, δ S , ε S ) with commutative diagram 3.1. Dually, a comonad G = (S, δ S , ε S ) is called Frobenius if there exists a monad structure S = (S, m S , e S ) such that Diagram (3.1) commutes. S (resp. G) is called Frobenius separable if, moreover, m S · δ S = 1. Proposition. Let S be a Frobenius separable monad on a Cauchy complete category A. Then for any morphism ι : S → T of monads, the change-of-base functor ι ! : A S → A T exists. Proposition. Let S be a Frobenius separable monad on a Cauchy complete category Proof. We claim that, under our assumptions, (2.6) is a split pair, a splitting morphism being the composite Proposition. Let S be a Frobenius separable monad on a Cauchy complete category Indeed, that ̺ · π = 1 follows from the commutativity of the diagram Proposition. Let S be a Frobenius separable monad on a Cauchy complete category ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Proposition. Let S be a Frobenius separable monad on a Cauchy complete category Here the square and the curved region commute since (φ T , ̺) is a right S-module by Section 2.7(vii), while the triangle commutes by separability of the monad S. Proposition. Let S be a Frobenius separable monad on a Cauchy complete category in which the curved region commutes since S is assumed to be Frobenius, the right-hand parallelogram commutes since ̺ is a morphism of right S-modules, while the other regions commute by naturality of composition. Thus the whole diagram is commutative, implying -since m S · Se S = 1 -that Proposition. Let S be a Frobenius separable monad on a Cauchy complete category In a similar manner one proves that There exist bijective correspondences (see [17] ) between: Comparison functors • comonad morphisms from the comonad generated by F ⊣ R to the comonad G; Comparison functors These bijections are constructed as follows. If Comparison functors , and the collection {θ b , b ∈ B} constitutes a natural transformation θ : Comparison functors Next, for any left G-comodule structure θ : F → GF , the composite Comparison functors is a morphism from the comonad generated by F ⊣ R to the comonad G. Then the corre- Comparison functors while the corresponding left G-comodule structure θ : F → GF on F is the composite Comparison functors Proof. Comparison functors Then γ K is the composite KR Comparison functors − − → φ G and using the fact from Section 2.14 that U G γ K is just the comonad morphism t K : F R → G induced by the triangle, an easy calculation shows that Comparison functors where β : R → RG is the right G-module structure on R corresponding to the triangle (4.1). Thus, when the right adjoint K of K exists, it is determined by the equaliser diagram Comparison functors It is easy to see that for any (a, θ) ∈ A G , the (a, θ)-component of (4.2) is the equaliser diagram Comparison functors It is easy to verify, using (2.5) that if σ is the counit of the adjunction K ⊣ K, then for any (a, θ) ∈ A G , one has , θ) ) . Comparison functors Suppose now that K exists, write P for the monad on B generated by the adjunction K ⊣ K, and consider the corresponding comparison functor K P : , θ), K(σ (a, θ) ) for any (a, θ) ∈ A G . Moreover, K P K = φ P and U P K P = K. The situation may be pictured as Comparison functors In order to proceed, we need the following (see [28, Lemma 21.2.7] ). Comparison functors with corresponding monads T and T ′ , respectively, and let Comparison functors be a commutative diagram of categories and functors.Then the composite Comparison functors Suppose again that K exists and consider the natural transformation ι : P → RF , where Comparison functors 4.5. Proposition. ι : P → RF is a monad morphism from the monad P to the monad generated by the adjunction F ⊣ R. Proof. Applying Proposition 4.4 to the diagram in which U G K = F , gives that the natural transformation Proof. Applying Proposition 4.4 to the diagram is a monad morphism from the monad P to the monad generated by the adjunction F ⊣ R. , θ) ) by (4.4), it follows that, for each b ∈ B, the b-component of the above natural transformation is the composite Proof. Applying Proposition 4.4 to the diagram which is easily verified to be just the morphism ι b : P (b) → U F (b). This completes the proof. ⊔ ⊓ We are mainly interested in the case where the functor F is monadic. So, our standard situation of interest, and our standard notation, will henceforth be as follows. We consider a monad T = (T, m T , e T ) on A, a comonad G on A T and an adjunction η , σ : ⊔ ⊓ G is a functor with U G K = φ T . Write P = (P, m P , e P ) for the monad on A generated by the adjunction K ⊣ K and write ι : P → T for the induced morphism of monads. This is pictured in the diagram ⊔ ⊓ in which K P : (A T ) G → A P is the Eilenberg-Moore comparison functor for the monad P, and thus K P K = φ P and U P K P = K. Proposition. In the situation above, the functor K Proof. According to Section 2.16, ι ! : A P → A T exists if and only if for each (a, g) ∈ A P , the pair of morphisms (φ T (g), m T a · φ T (ι a )) has a coequaliser in A T , while by Proposition 2.2(1), L P : A P → (A T ) G exists if and only if the pair of morphisms (K(g), σ K(a) ) has a coequaliser in (A T ) G . Proposition. In the situation above, the functor K Since the functor U G : (A T ) G → A T preserves and creates coequalisers, it suffices to show that the image of the pair ( Proposition. In the situation above, the functor K T a · φ T (ι a )) and thus the result follows. Proposition. In the situation above, the functor K ⊔ ⊓ Now assume that the change-of-base functor ι ! : A P → A T exists, that is, K P : (A T ) G → A P admits a left adjoint L P : A P → (A T ) G . Thus, for any (a, g) ∈ A P , ι ! (a, g) is given by be the coequaliser Proposition. In the situation above, the functor K Since ι ! = U G L P by Proposition 4.6 and ι ! · φ P = φ T by Proposition 2.18, both triangles in the diagram Proposition. In the situation above, the functor K commute. Write G 1 (respectively G 2 ) for the A T -comonad generated by the adjunction φ T ⊣ U T (respectively ι ! ⊣ ι * ), and consider the related comonad morphism t φ P : Proposition. In the situation above, the functor K corresponding to the left (respectively right) triangle in the above diagram (see Sections 2.14 and 4.1). Since U P K P = K and φ P (respectively L P ) has a right adjoint U P (respectively K P ), it follows -by uniqueness of right adjoints -that L P · φ P = K. Thus we may apply [23, Proposition 1.21] to obtain the equality Proposition. In the situation above, the functor K Recall from Section 2.16 that ι ! can be obtained as the coequaliser of Diagram (2.6). 4.7. Proposition. If K P : (A T ) G → A P admits a left adjoint, then t φ P = qι * . 4.7. Proof. Applying the results of Section 2.14 to the left triangle in Diagram (4.7) gives that t φ P = ι ! γ, where γ is the composite 4.7. Here η is the unit of the adjunction ι ! ⊣ ι * , which (as a result of Theorem 2.15 applied to the commutative diagram U P · ι * = U T ) is a unique natural transformation making the diagram 4.7. Here γ ′ is the composite 4.7. Since U P γ ′ = ι and ι · e P = e T , the diagram 4.7. commute. Since U P ε P · e P U P = 1 by one of the triangular identities for the adjunction φ P ⊣ U P , it follows that U P η is the composite 4.7. In particular, U P ηφ P is the composite 4.7. Since, by Remark 2.17(1), qφ P is the composite 4.7. Since, by functoriality of composition, T ι·e T P = e T T ·ι and since m T ·e T = 1, one concludes that U P ηφ P = ι. Now since for any (a, h) ∈ A T , (ε T ) (a, h) = h, one concludes that γ (a,h) is the composite 4.7. h − → a. Now, since by Remark 2.17, 4.7. is the coequaliser defining ι ! (P (a), m P a ) = ι ! (φ P (a)), it follows that ι ! (γ (a, h) ) = (t φ P ) (a, h) is the unique morphism making the diagram 4.7. commute. Since ι a ·e P a = e T a and m T a ·T (e T a ) = 1 = h·T (e T a ), it follows from the commutativity of the above diagram that (t φ P ) (a, h) = ι ! (γ (a,h) ) = q ι * (a, h) , as claimed. ⊔ ⊓ Weak entwinings Let H be an endofunctor on any category A, admitting both a monad H = (H, m, e) and a comonad H = (H, δ, ε) structure, and define The class Nat(H, H) of all natural transformations from H to itself allows for the structure of a monoid by defining the (convolution) product of any two ϕ, ϕ ′ ∈ Nat(H, H) as the composite ϕ * ϕ ′ = m · ϕϕ ′ · δ. The identity for this product is e · ε : H → H. Weak entwinings Recall that weak entwinings of tensor functors were defined by Caenepeel and De Groot in [15] and a more general theory was formulated by Böhm (e.g. Weak entwinings and is said to be compatible if Weak entwinings It is easily checked that Mixed bimodules. We write A H H (ω) for the category of mixed H-bimodules, whose objects are triples (a, h, θ), where (a, h) ∈ A H , (a, θ) ∈ A H with commutative diagram Mixed bimodules. and whose morphisms are those in A which are H-module as well as H-comodule morphisms. Mixed bimodules. The following is a particular case of [6, Proposition 5.7]. Proposition. Let H = (H, H, ω) be a weak entwining on A. Then the composite is an idempotent, and if Proposition. Let H = (H, H, ω) be a weak entwining on A. Then the composite is its splitting, then there is a comonad G = ( G, δ, ε) on A H , whose functor part takes an arbitrary (a, h) ∈ A H to Proposition. Let H = (H, H, ω) be a weak entwining on A. Then the composite and whose comultiplication δ and counit ε evaluated at (a, h) are the composites, respectively, Proposition. Let H = (H, H, ω) be a weak entwining on A. Then the composite We call G the comonad induced by H = (H, H, ω). Obviously, U H G = G. Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories with the inverse given by Φ −1 ((a, h), ζ) = (a, h, i (a,h) · ζ). Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories Proof. Since p i = 1, it is clear that ΦΦ −1 = 1. To show that Φ −1 Φ = 1, consider an arbitrary object (a, h, θ) ∈ A H H (ω). In the diagram Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories ; Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories the square commutes by naturality of e, while the trapezium commutes since (a, h, θ) ∈ A H H (ω). Since h · e = 1, this means Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories is its splitting, then there is a comonad T = ( T , m, e) on A H , whose functor part takes an arbitrary (a, θ) ∈ A H to Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories , and whose multiplication m and unit e, evaluated at an H-comodule (a, θ), are the composites, respectively, Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and G the induced comonad on A H . Then there is an isomorphism of categories We call T the monad induced by H = (H, H, ω) . Obviously, U H T = T . Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories (a,h) , θ). Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Precomposing K ω with Φ and Φ ′ gives functors Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories leading to commutative diagrams Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories We will use that the splitting of Γ (from 5.3) leads to a splitting of κ, Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Symmetrically, the splitting of Γ ′ (from 5.5) leads to a splitting of κ ′ , Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories where κ ′ is the composite HH Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories (2) For any a ∈ A, the φ H (a)-component for t, Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories is the unique morphism leading to commutativity of the diagram Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories , the left G-comodule structure α : φ H → Gφ H on φ H corresponding to the left triangle in (5.5), has for its a-component α a = p (H(a), ma ) · δ a . It then follows from Section 4.1 that for any (a, h) ∈ A H , the (a, h)-component t (a,h) is the composite Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories which, by naturality of composition, is the same as Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories is a unique morphism j : Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Our general results from Section 2 now yield: Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories 5.10. Proposition. Let H = (H, H, ω) be a compatible weak entwining on A. Then the functor K : A → (A H ) G (and hence also K ω : A → A H H (ω)) has a right adjoint if and only if, for any (a, h, θ) ∈ A H H (ω), the pair of morphisms Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories has an equaliser in A. Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Proof. Since the functor U G : (A H ) G → A is clearly (pre)comonadic, it follows from Theorem 2.15 that the functor K : A → (A H ) G admits a right adjoint if and only if for any ((a, h), ν) ∈ (A H ) G , the pair Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories where β : U H → U H G is the right G-comodule structure on U H : A H → A induced by the triangle (5.5) (see Section 4.1), has an equaliser in A, which -since i : G → HU H is a (split) monomorphism -is the case if and only if the pair Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories has one. According to Propositions 5.8 and 4.1, β (a, h) is the composite Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Since κ · δ = δ by (5.4) and Γ = i · p, it follows by naturality of i that the diagram (H(a),ma) G G G(H(a), m a ) G(h) Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories is commutative. So we have Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Thus, the functor K : A → (A H ) G has a right adjoint if and only if for any ((a, h), ν) ∈ (A H ) G , the pair of morphisms Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories has an equaliser. Recalling that Φ : Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories has a coequaliser in A. Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Symmetric to 5.1 one may consider 5.12. Weak comonad monad entwinings. For a natural transformation ω : HH → HH, define the natural transformation Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories and is said to be compatible if Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Here we get Theorem. Let H = (H, H, ω) be a weak entwining on a Cauchy complete category A and T the induced monad on A H . Then there is an isomorphism of categories Certainly, the theory for this notion will be similar to that for monad comonad entwinings. However, the mixed bimodules (as in 5.2) do not play the same role here but are to be replaced by liftings to Kleisli categories. Nevertheless, comonad monad entwinings will enter the picture in the next section. Weak braided bimonads In the theory of Hopf algebras H over a field k, the twist map for k-vector spaces M, N , tw M,N : M ⊗ k N → N ⊗ k M , plays a crucial part. In particular it helps to commute H ⊗ k − with itself by tw H,H : H ⊗ k H → H ⊗ k H. Generalising this to monoidal categories, often a braiding is required, that is, a condition on the whole category. It was observed (e.g. in [24] ) that it can be enough to have such a twist only for the functor H under consideration, that is, a natural isomorphism τ : HH → HH satisfying the Yang-Baxter equation. For the study of weak braided Hopf algebras, AlonsoÁlvarez e.a. suggested in [1, Definition 1.2] to consider, for any object D in a monoidal category, a weak Yang-Baxter operator Weak braided bimonads which is not necessarily invertible but only regular. Here we take up this notion and formulate it for any functor on an arbitrary category. Weak braided bimonads 6.1. Weak Yang Baxter operator. Given an endofunctor H : A → A, a pair of natural transformations τ, τ ′ : HH → HH is said to be a weak YB-pair provided the following equalities hold: Weak braided bimonads and for ∇ := τ · τ ′ , Weak braided bimonads The conditions in (6.2) are the usual Yang-Baxter equations for τ and τ ′ , respectively. The equations in (6.3) are obviously satisfied if τ ′ = τ −1 and in this case τ is known as Yang-Baxter operator. Weak braided bimonads 6.2. Definition. Let H = (H, m, e) a monad, H = (H, δ, ε) a comonad on A, and τ, τ ′ : HH → HH a weak YB-pair with Weak braided bimonads For vector space categories and (finite dimensional) tensor functors H ⊗ − with τ the twist map, these conditions were introduced in [11, Definition 1] . For monoidal categories and monoidal functors the conditions are those for a weak braided bialgebra introduced and studied by AlonsoÁlvarez e.a. [1, 2] and we can -and will -freely use essential parts of their results in our situation. Note that if ∇ is the identity of H, the conditions (1)- (4) in the definition describe the invertible double entwinings considered in [23] . Weak braided bimonads The following observations provide the key to our previous results. • (H, H, ω) is a compatible weak (comonad monad) entwining. Weak braided bimonads Proof. As easily seen, condition (5) in 6.2 yield the equalities (5.3) and (5.11) and also implies (5.2(i)) and (5.10(i)) for ω and ω, respectively (e.g. [8] , [23] ). Now Propositions 2.3 and 2.5 in [2] show the equations in (5.2)(ii) and (5.10)(ii). Weak braided bimonads ⊔ ⊓ Direct inspection yields the technical observation: Weak braided bimonads 6.6. Lemma. Suppose that f, g : X → X are idempotents in an arbitrary category such Weak braided bimonads is a (split) equaliser diagram, while Weak braided bimonads Henceforth we work over a Cauchy complete category A and suppose that Weak braided bimonads is a splitting of ξ. Weak braided bimonads 6.7. Proposition. In the situation of Proposition 5.10, the diagram Weak braided bimonads Proof. Since Hm · δH · eH = χH · δ by Proposition 6.4 (4), we have to show that the diagram Weak braided bimonads is a split equaliser. Let us first show that the pair Weak braided bimonads is cosplit by the morphism Hε : HH → H. Indeed, since Hε · δ = 1 and Hε · χH · δ = χ · Hε · δ = χ, it remains to show that χH · δ · χ = δ · χ. For this, consider the diagram Weak braided bimonads in which the • regions (1), (2), (5) and (6) commute by naturality of composition; Weak braided bimonads • region (3) commutes by coassociativity of δ; Weak braided bimonads • region (4) commutes by Proposition 6.4 (5); Weak braided bimonads • the curved regions commute by 6.4 (4). Hence the whole diagram commutes, implying Weak braided bimonads So the pair (6.6) is cosplit by the morphism Hε and hence one finds its equaliser by splitting the idempotent χ = Hε · χH · δ. But since ξ · χ = χ and χ · ξ = ξ by Proposition 6.4 (6) and ι ξ · q ξ is the splitting of the idempotent ξ (see (6.4)), it follows from Lemma 6.6 that (6.5) is a (split) equaliser diagram. Weak braided bimonads ⊔ ⊓ Symmetrically, we have: Weak braided bimonads 6.8. Proposition. In the situation of Proposition 5.11, the diagram Weak braided bimonads Since Hm · δH · eH = χH · δ and εH · mH · Hδ = m · χH by Proposition 6.4(4) and (5), Propositions 6.7 and 6.8 immediately yield: Weak braided bimonads is a (split) equaliser yielding the monad H ξ = (H ξ , m ξ , e ξ ) with m ξ = q ξ · m · (ι ξ ι ξ ) and e ξ = q ξ · e, and Weak braided bimonads is a (split) coequaliser yielding the comonad Weak braided bimonads The next result provides the technical data to show Frobenius separability. 6.10. Lemma. Let H = (H, H, τ ) be a weak braided bimonad and consider the composite Weak braided bimonads For this, m ξ · υ = e ξ and one has commutativity of the diagrams Weak braided bimonads According to Theorem 2.15, the unit of the adjunction K ω ⊣ K is the (unique) natural transformation η : 1 → KK ω = H ξ such that the diagram Weak braided bimonads e e ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Weak braided bimonads is the unique morphism making the diagram Weak braided bimonads commute and we calculate (recall (6.4)) Weak braided bimonads This completes the proof. ⊔ ⊓ By symmetry, we also have: ⊔ ⊓ 6.13. Proposition. Let H = (H, H, τ ) be a weak braided bimonad on A and suppose the functor ⊔ ⊓ is the comonad generated by the adjunction
{'paper_id': '119179409', 'title': 'The Fundamental Theorem for weak braided bimonads', 's2_url': 'https://api.semanticscholar.org/CorpusID:119179409', 'source': 'S2ORC'}