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Hopf monoidal comonads
Abstract
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.
... For a monad t on a monoidal category A, in [20] the additional structure was described which is equivalent to a monoidal structure on the Eilenberg-Moore category A t such that the forgetful functor A t → A is strict monoidal. The explicit computations of [20] in Cat (considered with the Cartesian product of categories as the monoidal structure) were replaced in [19], [8] by abstract arguments about more general monoidal bicategories. Beyond a wide generalization, thereby also a conceptually different proof was obtained. ...
... For any strict symmetric monoidal 2-category M, there is a strict symmetric monoidal 2-category M 10 whose 0-cells are the pseudomonoids (also called monoidal objects e.g. in [25] or monoidales e.g. in [8]), the 1-cells are the monoidal 1-cells, and the 2-cells are the monoidal 2-cells in M. Symmetrically, there is a strict symmetric monoidal 2-category M 01 whose 0-cells are again the pseudomonoids, but the 1-cells are the opmonoidal 1-cells, and the 2-cells are the opmonoidal 2-cells in M. Moreover, these constructions commute with each other (see Section 1). ...
... Because of this mixed nature of lifting; that is, since the different ingredients are lifted along different functors u t and f t , we do not expect it to be described by some 2-functor (as in the situations of [23] and [8]). Instead, in this paper we deal with strict symmetric monoidal double categories and define their (p, q)-oidal objects (see Section 6). ...
Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in strict symmetric monoidal 2-categories. Namely, any strict symmetric monoidal 2-category admits a strict symmetric monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in . Dually, there is a strict symmetric monoidal 2-category of pseudomonoids, opmonoidal 1-cells and opmonoidal 2-cells in . Extending a construction due to Aguiar and Mahajan for , we may apply the first construction p-times and the second one q-times (in any order). It yields a 2-category . A 0-cell therein is an object A of together with p+q compatible pseudomonoid structures; it is termed a (p+q)-oidal object in . A monad in is called a (p,q)-oidal monad in ; it is a monad t on A in together with p monoidal, and q opmonoidal structures in a compatible way. If has monoidal Eilenberg-Moore construction, and certain (Linton type) stable coequalizers exist, then a (p+q)-oidal structure on the Eilenberg-Moore object of a (p,q)-oidal monad (A,t) is shown to arise via a strict symmetric monoidal double functor to Ehresmann's double category of squares in , from the double category of monads in in the sense of Fiore, Gambino and Kock. While q ones of the pseudomonoid structures of are lifted along the `forgetful' 1-cell , the other p ones are lifted along its left adjoint. In the particular example when is an appropriate 2-subcategory of , this yields a conceptually different proof of some recent results due to Aguiar, Haim and L\'opez Franco.
... Weak Hopf algebras were one of the inspiring examples to define Hopf algebroids [BS04], which in turn led to Hopf monads [BLV11]. Just as Hopf monads live in the monoidal 2-category of categories, functors and natural transformations, Hopf-type objects can be defined in any monoidal bicategory, and they have been studied in this way, see eg. [DS97;CLS10;Str12;BL16]. In the latter, all previous mentioned Hopf-type objects (as well as a few more) are interpreted as particular opmonoidal monads in suitable monoidal bicategories, and different classical characterizations of Hopf algebras are recovered at an extremely high level of generality. ...
... In fact, for any monoidal bicategory K, PsMon opl (K) is also a bicategory: if (f, φ, φ 0 ) and (g, ψ, ψ 0 ) are two oplax morphisms between pseudomonoids A and B, a 2-cell between them is some α : f ⇒ g which is compatible with the oplax structure 2-cells, see Appendix A.1. The similarly defined bicategory PsMon lax (K) was denoted by Mon(K) in [CLS10]. Furthermore, when K is a braided monoidal bicategory with braiding σ, the bicategory PsMon opl (K) obtains a monoidal structure itself: the multiplication and unit on the tensor product of two pseudomonoids A and B is ...
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\"oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
... Weak Hopf algebras were one of the inspiring examples to define Hopf algebroids [BS04], which in turn led to Hopf monads [BLV11]. Just as Hopf monads live in the monoidal 2-category of categories, functors and natural transformations, Hopf-type objects can be defined in any monoidal bicategory, and they have been studied in this way, see eg. [DS97;CLS10;Str12;BL16]. In the latter, all previous mentioned Hopf-type objects (as well as a few more) are interpreted as particular opmonoidal monads in suitable monoidal bicategories, and different classical characterizations of Hopf algebras are recovered at an extremely high level of generality. ...
... In fact, for any monoidal bicategory K, PsMon opl (K) is also a bicategory: if (f, φ, φ 0 ) and (g, ψ, ψ 0 ) are two oplax morphisms between pseudomonoids A and B, a 2-cell between them is some α : f ⇒ g which is compatible with the oplax structure 2-cells, see Appendix A.1. The similarly defined bicategory PsMon lax (K) was denoted by Mon(K) in [CLS10]. Furthermore, when K is a braided monoidal bicategory with braiding σ, the bicategory PsMon opl (K) obtains a monoidal structure itself: the multiplication and unit on the tensor product of two pseudomonoids A and B is ...
We introduce the notion of (op)lax Hopf monoids in a monoidal 2-category and show that Hopf V-categories are a particular type of oplax Hopf monoids in the 2-category Span|V. We also introduce Frobenius V-categories as the Frobenius objects in this category.
... Let (C, ⊗, I, a, l, r) be a monoidal category, (F, δ, ε) be a comonad on C, and (F, F 2 , F 0 ) : C → C be a monoidal functor. Then recall from [18] or [19] that F is called a monoidal comonad (or a bicomonad) on C if δ and ε are both monoidal natural transformations, i.e. the following compatibility conditions hold for any X, Y ∈ C: ...
In this paper, we define and study quasi-monoidal comonads on a monoidal category. It generalize the (Hom type) coquasi-bialgebras to a non-braided setting. We investigate their corepresentations and their coquasitriangular structures. We also discuss their gauge equivalence relations.
... We obtain a 2-category PsMon lax (K) for any monoidal 2-category K, which is sometimes denoted by Mon(K) [CLS10]. By changing the direction of the 2-cells in Equation (B.7) and the rest of the axioms appropriately, or asking for them to be invertible, we have 2-categories PsMon opl (K) and PsMon(K) of pseudomonoids with oplax or (strong) morphisms between them. ...
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a network model, a monoidal variant of Joyal's combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green's graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.
... We obtain a 2-category PsMon (K) for any monoidal 2-category K, which is sometimes denoted by Mon(K) [CLS10]. By changing the direction of the 2-cells in (19) and the rest of the axioms appropriately, or asking for them to be invertible, we have 2-categories PsMon op (K) and PsMon(K) of pseudomonoids with oplax or (strong) morphisms between them. ...
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namelylax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigatethe relation between this ‘global’ monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a ‘fibrewise’ one where the fibresare monoidal and the reindexing functors strongly preserve the structure, first hintedby Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
This is
a review of Monoidal Grothendieck construction by Moeller, Joe; Vasilakopoulou, Christina
This paper presents the definition of the cotwists of a bicomonad, which offers a generalization of cotwists of bimonoids in duoidal categories. We show that a new bicomonad can be deduced from the cotwist, and their corepresentations are monoidal isomorphic. As applications, the cotwists of bialgebroids and of BiHom-bialgebras are discussed.
We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which .
Because an exact pairing between an object and its dual is extraordinarily natural in the object, ideas of R. Street apply to yield a definition of dualization for a pseudomonoid in any autonomous monoidal bicategory as base; this is an improvement on Day and Street, Adv. in Math.
129 (1997), Definition 11, p. 114. We analyse the dualization notion in depth. An example is the concept of autonomous (which, usually in the presence of a symmetry, also has been called “rigid” or “compact”) monoidal category. The antipode of a quasi-Hopf algebra H in the sense of Drinfeld is another example obtained using a different base monoidal bicategory. We define right autonomous monoidal functors and their higher-dimensional analogue. Our explanation of why the category Comodf
(H) of finite-dimensional representations of H is autonomous is that the Comodf
operation is autonomous and so preserves dualization.
Hopf monads are identified with monads in the 2-category OpMon of mon- oidal categories, opmonoidal functors and transformations. Using Eilenberg-Moore ob- jects, it is shown that for a Hopf monad S, the categories Alg(Coalg(S)) and Coalg(Alg(S)) are canonically isomorphic. The monadic arrows OpMon are then characterized. Finally, the theory of multicategories and a generalization of structure and semantics are used to identify the categories of algebras of Hopf monads.
We give an explicit description of the free completion EM ( K ) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category Mnd ( K ) of monads in K . We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM ( K ) as the 2-category of monads rather than Mnd ( K ) . We also introduce the wreaths in K ; these are the objects of EM ( EM ( K )) , and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts.
We give an explicit description of the free completion EM(K) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category Mnd(K) of monads in K. We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM(K) as the 2-category of monads rather than Mnd(K). We also introduce the wreaths in K; these are the objects of EM(EM(K)), and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts.
We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] 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 category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid.Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras).We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedlerʼs Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).