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12
ENRICHED INDEXED CATEGORIES
"... Abstract. We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an Sindexed monoidal category V. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. ..."
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Abstract. We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an Sindexed monoidal category V. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. We then describe the appropriate notion of “limit ” for such enriched indexed categories, and show that they admit “free cocompletions” constructed as usual with a Yoneda embedding.
Multivariable adjunctions and mates
, 2012
"... We present the notion of “cyclic double multicategory”, as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n + 1 functors of n variables. Furthermore, we ..."
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We present the notion of “cyclic double multicategory”, as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n + 1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The workis motivated byand appliedtoRiehl’s approach
Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings
"... We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, thes ..."
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We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to nvariables, and prove that, in a manner analogous to that of butterflies, these multiextensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal category or stack, treating the multiplicative structure as a bimonoidal functor with respect to the additive one. In the context of the multilinear functor calculus, we view the bimonoidal structure as an instance of the general notion of pseudomonoid. We show that when the structure is ringlike, i.e. the pseudomonoid is a stack whose fibers are categorical rings, we can recover the classification by the third Mac Lane cohomology of a ring with values in a bimodule.
COMPOSITION OF MODULES FOR LAX FUNCTORS
"... Abstract. We study the composition of modules between lax functors of weak double categories. We adapt the bicategorical notion of local cocompleteness to weak double categories, which the codomain of our lax functors will be assumed to satisfy. We introduce a notion of factorization of cells, which ..."
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Abstract. We study the composition of modules between lax functors of weak double categories. We adapt the bicategorical notion of local cocompleteness to weak double categories, which the codomain of our lax functors will be assumed to satisfy. We introduce a notion of factorization of cells, which most weak double categories of interest possess, and which is sufficient to guarantee the strong representability of composites of modules between lax functors whose domain satisfies it.
Kleisli enriched
"... For a monad S on a category whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules ( = profunctors, distributors), for some quantale V. The assignment V is shown to belong to a global adjunc ..."
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For a monad S on a category whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules ( = profunctors, distributors), for some quantale V. The assignment V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative, unital) quantale V the prototypical example of a modular monad, namely the presheaf monad on V Cat, the category of (small) Vcategories. We discuss in particular the question whether the Hausdorff monad on V Cat is modular.
ALGEBRAIC KAN EXTENSIONS IN DOUBLE CATEGORIES
"... Abstract. We study Kan extensions in three weakenings of the EilenbergMoore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad T on a double category K, we consider the double categories consisting of pseudo Tal ..."
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Abstract. We study Kan extensions in three weakenings of the EilenbergMoore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad T on a double category K, we consider the double categories consisting of pseudo Talgebras, ‘weak ’ vertical Tmorphisms, horizontal Tmorphisms and Tcells, where ‘weak ’ means either ‘lax’, ‘colax ’ or ‘pseudo’. Denoting these double categories by Algw(T), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise) Kan extensions can be lifted along the forgetful double functor Algw(T) → K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler’s result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids. 1.
unknown title
"... Cyclic multicategories, multivariable adjunctions and mates by EUGENIA CHENG, NICK GURSKI AND EMILY RIEHL A multivariable adjunction is the generalisation of the notion of a 2variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n C 1 functors of n variables. ..."
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Cyclic multicategories, multivariable adjunctions and mates by EUGENIA CHENG, NICK GURSKI AND EMILY RIEHL A multivariable adjunction is the generalisation of the notion of a 2variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n C 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual ” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities soencoded.
MaxPlanckInstitut für Mathematik Preprint Series 2010 (114) Kleisli enriched
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LAX FORMAL THEORY OF MONADS, MONOIDAL APPROACH TO BICATEGORICAL STRUCTURES AND GENERALIZED OPERADS
"... Abstract. Generalized operads, also called generalized multicategories and Tmonoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including sym ..."
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Abstract. Generalized operads, also called generalized multicategories and Tmonoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to bicategorical structures. 1.