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Pseudodistributive laws
, 2004
"... We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and ..."
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We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and we show how the definition and the main theorems about it may be used to model several such structures simultaneously. Specifically, we address the relationship between pseudodistributive laws and the lifting of one pseudomonad to the 2category of algebras and to the Kleisli bicategory of another. This, for instance, sheds light on the preservation of some structures but not others along the Yoneda embedding. Our leading examples are given by the use of open maps to model bisimulation and by the logic of bunched implications.
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.
Balanced Coalgebroids
, 2000
"... A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of ..."
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A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasibialgebras, quasitriangular dual quasibialgebras, and balanced quasitriangular dual quasibialgebras; the latter of which is a quantum group when equipped with a compatible antipode. As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.
The simplicial interpretation of bigroupoid 2torsors
, 902
"... Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we intr ..."
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Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2torsor. In that case, the Duskin nerve of the canonical projection is precisely the DuskinGlenn simplicial 2torsor, introduced in [25]. 1
The
, 708
"... periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories ..."
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periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories
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.
BIEQUIVALENCES IN TRICATEGORIES
"... Abstract. We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses. ..."
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Abstract. We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.
THE CORE OF ADJOINT FUNCTORS
"... Abstract. There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the homenriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categ ..."
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Abstract. There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the homenriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting. 1.
ON THE COHERENCE CONDITIONS FOR
, 907
"... Abstract. We survey the development of the formal theory of pseudomonads, the analogue for pseudomonads of the formal theory of monads. One of the main achievements of the theory is a satisfactory axiomatisation of the notion of a pseudodistributive law between pseudomonads. ..."
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Abstract. We survey the development of the formal theory of pseudomonads, the analogue for pseudomonads of the formal theory of monads. One of the main achievements of the theory is a satisfactory axiomatisation of the notion of a pseudodistributive law between pseudomonads.