29 pages; version 2 minor corrections and added references, also added remark 4.3; title changed from "Weak bimonads" to "Weak bimonads and weak Hopf monads"; to appear in Journal of Algebra
We define a weak bimonad as a monad T on a monoidal category M with the property that the Eilenberg-Moore category M^T is monoidal and the forgetful functor from M^T to M is separable Frobenius. Whenever M is also Cauchy complete, a simple set of axioms is provided, that characterizes the monoidal structure of M^T as a weak lifting of the monoidal structure of M . The relation to bimonads, and the relation to weak bimonoids in a braided monoidal category are revealed. We also discuss antipodes, obtaining the notion of weak Hopf monad.
For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières 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 s…
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 inter…
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 $τ$-bimonads $H$, properties of braided (Hopf) bialgebras were captured by requiring a Yang-Baxter operator $τ:…
Monoidal functors U:C --> M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on M. Such monads will be called bimonads. Treating bimonads as abstract "quantum groupoids" we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi's x_R-bialgebras, appear as the special case when T has also a right adjoint. Street's 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.
Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler's Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodule-monad and a algebra-comonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler's result (to the setting of Hopf Galois extensions).
We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a cat…
Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad $T$ on a magma category $\mathcal{C}$ suc…
We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. Indeed, any monoidal adjunction between autonomous categories gives rise to a Hopf monad. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke's criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again defined in a non-braided setting).
We survey the theory of Hopf monads on monoidal categories, and present new examples and applications. As applications, we utilise this machinery to present a new theory of cross products, as well as analogues of the Fundamental Theorem of Hopf algebras and Radford's biproduct Theorem for Hopf algebroids. Additionally, we describe new examples of Hopf monads which arise from Galois and Ore extensions of bialgebras. We also classify Lawvere theories whose corresponding monads on the category of sets and functions become Hopf, as well as Hopf monads on the poset of natural numbers.
The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z-C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad Z_T on C, the centralizer of T, and a canonical distributive law of T over Z_T. By Beck's theory, this has two consequences. On one hand, D_T=Z_T T is a quasitriangular Hopf monad…
Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals then Bruguieres and Virelizier showed when the category of modules inherits this structure of being monoidal with duals, and this gave rise to what they called a Hopf monad. In this paper it is shown that there are good diagrammatic descriptions of dinatural transformations which allows the three-dimensional, object-free nature of their constructions to become…
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive adjunction, if and only if $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further,…
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 cate…
An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its left dual and right dual objects are isomorphic. Given such an object and a choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects which intertwine with $P$ and $Q$ in a compatible manner. We show that this category lifts the monoidal structure of $\mathcal{C}$ and the closed structure of $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitable colimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct a family of Hopf monads on arbitrary closed monoidal categories…
We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due to Bruguières) can be seen as Hopf monads in $\mathsf{Span}| \mathsf{Cat}$ while Hopf group monoids in a braided monoidal category $V$ (in the spirit of Turaev and Zunino), and Hopf categories over $V$ (by Batista, Caenepeel and Vercruysse) both turn out to be Hopf monads in $\mathsf{Span}| V$. Hopf group monoids and Hopf categories are Hopf monads on a distinguished type of monoidales fitti…
We introduce Hopf polyads in order to unify Hopf monads and group actions on monoidal categories. A polyad is a lax functor from a small category (its source) to the bicategory of categories, and a Hopf polyad is a comonoidal polyad whose fusion operators are invertible. The main result states that the lift of a Hopf polyad is a strong (co)monoidal action-type polyad (or strong monoidal pseudofunctor). The lift of a polyad is a new polyad having simpler structure but the same category of modules. We show that, under certain assumptions, a Hopf polyad can be `wrapped up' into a Hopf monad. This…
Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg-Moore category of the induced monad. Making appropriate ass…
We study integrals of Hopf monoids in duoidal endohom categories of naturally Frobenius map monoidales in monoidal bicategories. We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals. It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories.
We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the relative (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.
In this article Hopf parametric adjunctions are defined and analysed within the context of the 2-adjunction of the type $\mathbf{Adj}$-$\mathbf{Mnd}$. In order to do so, the definition of adjoint objects in the 2-category of adjunctions and in the 2-category of monads for $Cat$ are revised and characterized. This article finalises with the application of the obtained results on current categorical characterization of Hopf Monads.
In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidal bicategory \M, the analogue of the category of Hopf modules for $A$ is an Eilenberg-Moore construction for a certain monad in $\mathbf{Hom}(\M^{\mathrm{op}},\mathbf{Cat})$. We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under Eilenberg-Moore objects of a 2-category to a endo-homomorphism o…
{\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 groupli…
Alain Bruguieres, in his talk [1], announced his work [2] with Alexis Virelizier and the second author which dealt with lifting closed structure on a monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal monad. Our purpose here is to generalize that work to the context internal to an autonomous monoidal bicategory. The result then applies to quantum categories and bialgebroids.
We study monoidal comonads on a naturally Frobenius map-monoidale $M$ in a monoidal bicategory $\mathcal M$. We regard them as bimonoids in the duoidal hom-category $\mathcal M(M,M)$, and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in $\mathcal M$, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf…
If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}\mbox{-mod}$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}\mbox{-mod}$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra the…
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…
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to Z and collapsing H_0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.
We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative descr…
We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adj…
Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street's weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl^{(n)}(K), of (n+1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl^{(0)}(K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl^{(n)}(K) contains Mnd^{n+1}(K) as a full 2-subcategory. For the local idempotent closure K^ of K, extending the multiplicati…
Distributive laws between two monads in a 2-category $\CK$, as defined by Jon Beck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\CK$ of monads with different 2-cells from $\mathrm{Mnd}\CK$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others, they are comonads in $\mathrm{Mnd}\CK$. Actually, as pointed out by John Power and Hiroshi Watanabe, there are a number of dual possibilities for mixed dis…
We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvabl…
We study Kan extensions in three weakenings of the Eilenberg-Moore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad $T$ on a double category $\mathcal K$, we consider the double categories consisting of pseudo $T$-algebras, `weak' vertical $T$-morphisms, horizontal $T$-morphisms and $T$-cells, where `weak' means either `lax', `colax' or `pseudo'. Denoting these double categories by $\mathsf{Alg}_w T$, where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (p…
