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12
Quantum categories, star autonomy, and quantum groupoids
 in &quot;Galois Theory, Hopf Algebras, and Semiabelian Categories&quot;, 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 dist ..."
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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 &quot;quantum category&quot;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 &quot;Hopf algebra with several objects&quot;. 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.
Monoidal Morita equivalence
"... Let A be an algebra over the commutative ring k. It is well known that the category MA of right Amodules is cocomplete, Abelian and the right regular object AA is a small projective generator. The latter three properties means precisely that the functor Hom A(A, ) : MA → Mk preserves coproducts, pr ..."
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Cited by 3 (1 self)
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Let A be an algebra over the commutative ring k. It is well known that the category MA of right Amodules is cocomplete, Abelian and the right regular object AA is a small projective generator. The latter three properties means precisely that the functor Hom A(A, ) : MA → Mk preserves coproducts, preserves cokernels and it is faithful, respectively. In fact this functor is monadic and has a right adjoint. It is also well known [8] that the above properties characterize such categories: For a klinear category C to be equivalent to MA for some kalgebra A it is sufficient that C is cocomplete, Abelian and possesses a small projective generator. Of course the algebra A is determined by C only up to Morita equivalence. The analogue question for monoidal module categories has been studied by B. Pareigis in [10]. With the advent of quantum groupoids it is worth reconsidering the question. Therefore we are interested in monoidal structures on MA admitting a strong monoidal forgetful functor to the category RMR of bimodules over some other kalgebra R. In the special case of R = k one obtains that A is a bialgebra [11]. The general case leads to bialgebroids [12]. The importance of module categories
SCALAR EXTENSION OF BICOALGEBROIDS
, 707
"... Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided ju ..."
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Abstract. After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of [2] and [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule
GENERALIZED HOPF MODULES FOR BIMONADS
"... Abstract. 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 comodulemonad ..."
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Abstract. 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 comodulemonad and a algebracomonoid 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).
SKEW MONOIDALES, SKEW WARPINGS AND QUANTUM CATEGORIES
"... Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax un ..."
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Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudomonoids) in any monoidal bicategory M. These are skewmonoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures. 1.
FIBER FUNCTORS, MONOIDAL SITES AND TANNAKA DUALITY FOR BIALGEBROIDS
, 2009
"... What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phùng Hô Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstr ..."
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What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phùng Hô Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed
BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. 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 monoidal c ..."
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Abstract. 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 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