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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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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 &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
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 38 (12 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.
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n < ..."
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Cited by 37 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Deriving Bisimulation Congruences using 2categories
, 2003
"... We introduce Grelativepushouts (GRPO) which are a 2categorical generalisation of relativepushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisim ..."
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Cited by 34 (10 self)
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We introduce Grelativepushouts (GRPO) which are a 2categorical generalisation of relativepushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that su#ciently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell.
Categorical structures enriched in a quantaloid: Categories, distributions and functors
 Theory Appl. Categ
"... We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is th ..."
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Cited by 26 (4 self)
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We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is the supremum of the presheaf of objects “totally below ” it); and also are they the Qcategories of regular presheaves on a regular Qsemicategory. As a particular case, the Qcategories of presheaves on a Qcategory are precisely the “totally algebraic” cocomplete Qcategories (every object is the supremum of the “totally compact” objects below it). We think that these results should be part of a yettobeunderstood “quantaloidenriched domain theory”. 1
Categorical and combinatorial aspects of descent theory, [arXiv:math/0303175
"... There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern ..."
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There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves nonabelian cohomology, stacks, torsors, homotopy, and higherdimensional categories. Many of the ideas are scattered through the literature or are folklore; a few are new. Section Headings
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Mackaay: Categorical representations of categorical groups
, 2004
"... The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1 ..."
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Cited by 25 (0 self)
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The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1