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Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
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 (8 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.
Weak Hopf monoids in braided monoidal categories
 Algebra Number Theory
"... Abstract. We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R. Contents ..."
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Cited by 5 (0 self)
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Abstract. We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R. Contents
ON ∗AUTONOMOUS CATEGORIES OF TOPOLOGICAL MODULES
"... Abstract. Let R be a commutative ring whose complete ring of quotients is Rinjective. We show that the category of topological Rmodules contains a full subcategory that is ∗autonomous using R itself as dualizing object. In order to do this, we ..."
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Abstract. Let R be a commutative ring whose complete ring of quotients is Rinjective. We show that the category of topological Rmodules contains a full subcategory that is ∗autonomous using R itself as dualizing object. In order to do this, we
Example
, 2004
"... Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras ..."
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Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e:A æÆ æ k such that: e ( ab) = 0 for all a ŒA implies b = 0.