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13
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.
The periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories
, 2007
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ON THE COHERENCE CONDITIONS FOR Pseudodistributive Laws
, 2009
"... 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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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.
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.
Hopf modules for autonomous pseudomonoids and the monoidal centre
, 2007
"... Abstract. In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel’d double. If A is a map pseudomonoid in a monoidal bicategory M, the analogue of the category of Hopf modules for ..."
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Abstract. In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel’d double. If A is a map pseudomonoid in a monoidal bicategory M, the analogue of the category of Hopf modules for A is an EilenbergMoore construction for a certain monad in Hom(M op,Cat). We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under EilenbergMoore objects of a 2category to a endohomomorphism of tricategories on Bicat. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case the Hopf module construction for A always exists. We use these results to study the lax centre of a left autonomous map pseudomonoid. We show that the lax centre is the EilenbergMoore construction for a certain monad on A (one existing if the other does). If A is also right autonomous, then the lax centre equals the centre. We look at the examples of the bicategories of Vmodules and of comodules in V, and obtain the Drinfel’d double of a coquasiHopf algebra H as the centre of H. 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.
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.
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
Descent on 2fibrations and 2regular 2categories
"... Abstract. We consider pseudodescent in the context of 2fibrations. A 2category of descent data is associated to a 3truncated simplicial object in the base 2category. A morphism q in the base induces (via commaobjects and pullbacks) an internal category whose truncated nerve allows the definiti ..."
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Abstract. We consider pseudodescent in the context of 2fibrations. A 2category of descent data is associated to a 3truncated simplicial object in the base 2category. A morphism q in the base induces (via commaobjects and pullbacks) an internal category whose truncated nerve allows the definition of the 2category of descent data for q. When the 2fibration admits direct images, we provide the analogous of the BeckBénabouRoubaud theorem, identifying the 2category of descent data with that of pseudoalgebras for the pseudomonad q ∗ Σq. We introduce a notion of 2regularity for a 2category R, so that its basic 2fibration of internal fibrations cod: Fib(R) → R admits direct images. In this context, we show that essentiallysurjectiveonobjects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beckstyle pseudomonadicity theorem.