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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.
Modelling linear logic without units (preliminary results). Available at http://arxiv.org/abs/math/0504037
, 2005
"... Proof nets for MLL − , unitfree Multiplicative Linear Logic (Girard, 1987), provide elegant, abstract representations of proofs. Cutfree MLL − proof nets form a category, under path composition 1, in the manner of EilenbergKellyMac Lane graphs (Eilenberg and Kelly, 1966; Kelly and Mac Lane, ..."
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Cited by 13 (1 self)
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Proof nets for MLL − , unitfree Multiplicative Linear Logic (Girard, 1987), provide elegant, abstract representations of proofs. Cutfree MLL − proof nets form a category, under path composition 1, in the manner of EilenbergKellyMac Lane graphs (Eilenberg and Kelly, 1966; Kelly and Mac Lane,
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
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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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
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
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
"... Abstract. We show that the (co)endomorphism algebra of a sufficiently separable “fibre ” functor into Vectk, for k a field of characteristic 0, has the structure of what we call a “unital ” von Neumann core in Vectk. For Vectk, this particular notion of algebra is weaker than that of a Hopf algebra, ..."
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Abstract. We show that the (co)endomorphism algebra of a sufficiently separable “fibre ” functor into Vectk, for k a field of characteristic 0, has the structure of what we call a “unital ” von Neumann core in Vectk. For Vectk, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group. 1.
∗AUTONOMOUS CATEGORIES IN QUANTUM THEORY
, 2006
"... mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗autonomous partially ordered sets A = (A, p, j, S) ..."
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Cited by 1 (1 self)
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mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗autonomous partially ordered sets A = (A, p, j, S)
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.
A CATEGORY OF QUANTUM CATEGORIES
"... Abstract. Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebro ..."
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Abstract. Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures.
The 2category of quantum categories
, 910
"... We describe the 2category of quantum categories. ..."