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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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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
Frobenius Algebras and ambidextrous adjunctions
, 2006
"... In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fu ..."
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In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2category and utilizing the free completion under EilenbergMoore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2category M, and replacing the 2categoryD into which it embeds by a semistrict 3category. To state this more powerful result, we must first define the notion of a ‘Frobenius pseudomonoid’, which categorifies that of a Frobenius object. We then define the notion of a ‘pseudo ambijunction’, categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2category arises from a pseudo ambijunction in some semistrict 3category.
Polycategories via pseudodistributive laws
"... In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomo ..."
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In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘twosided Kleisli bicategory’ of this pseudodistributive law are precisely symmetric polycategories. 1
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
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.
∗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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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)
ON FROBENIUS ALGEBRAS IN RIGID MONOIDAL CATEGORIES
, 2008
"... We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal catego ..."
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We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
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.
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