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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 $\mathbb{A}$, employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For $\tau$-bimonads $H$, properties of braided (Hopf) bialgebras were captured by requiring a Yang-Baxter operator $\tau:HH\to HH$. 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 Alonso \'Alvarez and others.

Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincar\'e-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework., Comment: 17 pages

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base category by a comodule category for some comonoid and considering a comparison functor to generalised Hopf modules. For duoidal categories, Bohm, Chen and Zhang investigated a comparison functor to the Hopf modules over a bimonoid induced by the two monoidal structures given in such categories. In both approaches fundamental theorems are proved and the purpose of this paper is to show that these can be derived from the theory of Galois functors.

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^{\alpha} & \Mor_\A (A,R(B))\ar@<0.5ex>[l]^{\beta}}.$$ $(L,R)$ is an {\em adjoint pair} provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\A$, $LR$ defines a comonad on the category $\B$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of {\em weak monads} and {\em weak comonads} and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors $(L,R,\alpha,\beta)$ with $\alpha = \alpha\dcirc \beta\dcirc \alpha$ and $\beta = \beta \dcirc\alpha\dcirc\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\A$ gives rise to a regular pairing between $\A$ and the category of {\em compatible (co)modules}., Comment: 19 pages; shortened revised and corrected version with some change of terminology

For a generalisation of the classical theory of Hopf algebra over fields, A. Brugui\`eres and A. Virelizier study opmonoidal monads on monoidal categories (which they called {\em bimonads}). In a recent joint paper with S. Lack the same authors define the notion of a {\em pre-Hopf monad} by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case %Hopf monads may be considered as a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered.

In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$ with the coalgebra $C$ provided by the evaluation map $\ev:C^*\ot_R C\to R$. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$, $$\beta_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad and $\bG=(G,\delta_G,\ve_G)$ is a comonad on $\A$, additional compatibility conditions are imposed on a pairing between $\bT$ and $\bG$. If such a pairing is given and is rational, and $\bT$ has a right adjoint monad $\bT^\di$, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the $\bT$-modules $\A_{\rT}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.

{\em Galois comodules} over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of {\em Galois functors} over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a {\em grouplike natural transformation} $g:I\to G$ generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Brugui\`{e}res and A. Virelizier. As well-know, for any set $G$ the product $G\times-$ defines an endofunctor on the category of sets and this is a Hopf monad if and only if $G$ allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to {\em Galois objects} in the sense of Chase and Sweedler.

For an associative ring $R$, let $P$ be an $R$-module with $S=\End_R(P)$. C.\ Menini and A. Orsatti posed the question of when the related functor $\Hom_R(P,-)$ (with left adjoint $P\ot_S-$) induces an equivalence between a subcategory of $_R\M$ closed under factor modules and a subcategory of $_S\M$ closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphism and the counit is a monomorphism. A module $P$ inducing these properties is called a $\star$-module. The purpose of this paper is to consider the corresponding question for a functor $G:\B\to \A$ between arbitrary categories. We call $G$ a {\em $\star$-functor} if it has a left adjoint $F:\A\to \B$ such that the unit of the adjunction is an {\em extremal epimorphism} and the counit is an {\em extremal monomorphism}. In this case $(F,G)$ is an idempotent pair of functors and induces an equivalence between the category $\A_{GF}$ of modules for the monad $GF$ and the category $\B^{FG}$ of comodules for the comonad $FG$. Moreover, $\B^{FG}=\Fix(FG)$ is closed under factor objects in $\B$, $\A_{GF}=\Fix(GF)$ is closed under subobjects in $\A$.

Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\Hom_A(\C,-)$-modules (called $\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\C$-comodules and $\Hom_A(\C,-)$-modules are equivalent provided $\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom_R(H-)$ is a Hopf bimonad on $\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\Hom_R(H,-)$-bimodules are both equivalent to $\M_R$., Comment: 35 pages, LaTeX

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to {\em monoidal} categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category $\A$ and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on $\A$: we define a {\em bimonad} on $\A$ as an endofunctor $B$ which is a monad and a comonad with an entwining $\lambda:BB\to BB$ satisfying certain conditions. This $\lambda$ is also employed to define the category $\A^B_B$ of (mixed) $B$-bimodules. In the classical situation, an entwining $\lambda$ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws $\tau:BB\to BB$ satisfying the Yang-Baxter equation ({\em local prebraidings}) which induce an entwining $\lambda$ and lead to an extension of the theory of {\em braided Hopf algebras}. An antipode is defined as a natural transformation $S:B\to B$ with special properties and for categories $\A$ with limits or colimits and bimonads $B$ preserving them, the existence of an antipode is equivalent to $B$ inducing an equivalence between $\A$ and the category $\A^B_B$ of $B$-bimodules. This is a general form of the {\em Fundamental Theorem} of Hopf algebras.

We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined., Comment: Final version

Generalising the notion of Galois corings, Galois comodules were introduced as comodules $P$ over an $A$-coring $\cC$ for which $P_A$ is finitely generated and projective and the evaluation map $\mu_\cC:\Hom^\cC(P,\cC)\ot_SP\to \cC$ is an isomorphism (of corings) where $S=\End^\cC(P)$. It was observed that for such comodules the functors $\Hom_A(P,-)\ot_SP$ and $-\ot_A\cC$ from the category of right $A$-modules to the category of right $\cC$-comodules are isomorphic. In this note we call modules $P$ with this property {\em Galois comodules} without requiring $P_A$ to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. These comodules are close to being generators and have some common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions., Comment: 24 pages

A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element. Biseparable corings are defined, and a conjecture relating them to Frobenius corings is proposed., Comment: 17 pages, LaTeX