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Quantum categories, star autonomy, and quantum groupoids
 in "Galois Theory, Hopf Algebras, and Semiabelian Categories", Fields Institute Communications 43 (American Math. Soc
, 2004
"... Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers distributed ..."
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Cited by 19 (9 self)
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Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers distributed over by tensoring with an object. The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's starautonomous categories is more suitable than autonomy ( = compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects". 1.
Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics
 Lecture Notes in Pure and
"... Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given b ..."
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Cited by 13 (2 self)
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Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg [15] together with their bimonad description given by the author in [18] here we characterize the ”long ” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1.
MONADS AND COMONADS ON MODULE CATEGORIES
"... known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodu ..."
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Cited by 12 (10 self)
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known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B, −) and HomA(C, −) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find
The monoidal EilenbergMoore construction and bialgebroids
"... Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, app ..."
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Cited by 12 (3 self)
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Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, appear as the special case when T has also a right adjoint. Street’s 2category of monads then leads to a natural definition of the 2category of bialgebroids. Contents
From left modules to algebras over an operad: application to combinatorial Hopf algebras
, 2006
"... The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness resu ..."
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Cited by 7 (2 self)
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The purpose of this paper is two fold: we study the behaviour of the forgetful functor from Smodules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from Smodules to graded vector spaces. Left modules over an operad P are treated as Palgebras in the category of Smodules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends Palgebras to Palgebras. If P is a Hopf operad the functor O sends Hopf Palgebras to Hopf Palgebras. If the operad P is regular one gets two different structures of Hopf Palgebras in the category of graded vector spaces. We develop the notion of unital infinitesimal Pbialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra.
Lawvere completeness in Topology
, 2008
"... It is known since 1973 that Lawvere’s notion of (Cauchy)complete enriched category is meaningful for metric spaces: it captures exactly Cauchycomplete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)categories and show that it has an interestin ..."
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Cited by 5 (3 self)
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It is known since 1973 that Lawvere’s notion of (Cauchy)complete enriched category is meaningful for metric spaces: it captures exactly Cauchycomplete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)categories and show that it has an interesting meaning for topological spaces and quasiuniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)category structure which plays a key role: it is Lawverecomplete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)categories.
GALOIS FUNCTORS AND ENTWINING STRUCTURES
, 909
"... Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of 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 injecti ..."
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Cited by 2 (2 self)
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Abstract. Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of 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 grouplike natural transformation g: I → 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. Bruguières and A. Virelizier. As wellknow, for any set G the product G × − 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 Galois objects in the sense of Chase and Sweedler.
On Hopf algebras and their generalizations
 Comm. Algebra
, 2007
"... Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduce ..."
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Cited by 2 (0 self)
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Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.
On the derived category of an algebra over an operad, in preparation
"... Dedicated to Mamuka Jibladze on the occasion of his 50th birthday Abstract. We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad. ..."
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Cited by 2 (1 self)
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Dedicated to Mamuka Jibladze on the occasion of his 50th birthday Abstract. We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.