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Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 28 (11 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Algebras of higher operads as enriched categories II
 In preparation
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the ..."
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Cited by 13 (8 self)
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of nglobular sets from any normalised (n + 1)operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak ncategories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
Symmetric monoidal completions and the exponential principle among labeled combinatorial structures
 THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... We generalize Dress and Müller's main result in [5]. We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponential principle in a symmetric monoidal category. We show th ..."
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Cited by 4 (2 self)
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We generalize Dress and Müller's main result in [5]. We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponential principle in a symmetric monoidal category. We show that for any groupoid G, the !G of presheaves on the symmetric monoidal completion !G of G satisfies the exponential principle. The main result in [5] reduces to the case G = 1. We discuss two notions of functor between categories satisfying the exponential principle and express some well known combinatorial identities as instances of the preservation properties of these functors. Finally, we give a characterization of G as a subcategory of !G.
Free Products of Higher Operad Algebras
, 909
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, ..."
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Cited by 4 (4 self)
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an noperad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure
SYMMETRIC MONOIDAL COMPLETIONS AND THEEXPONENTIAL PRINCIPLE AMONG LABELED COMBINATORIAL STRUCTURES.
"... ABSTRACT. We generalize Dress and M"uller's main result in [5]. We observe thattheir result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponentialprinciple in a symmetric monoidal categ ..."
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ABSTRACT. We generalize Dress and M&quot;uller's main result in [5]. We observe thattheir result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponentialprinciple in a symmetric monoidal category. We show that for any groupoid G, the category c!G of presheaves on the symmetric monoidal completion!G of G satisfies theexponential principle. The main result in [5] reduces to the case G = 1. We discuss twonotions of functor between categories satisfying the exponential principle and express
DENSE MORPHISMS OF MONADS
"... Abstract. Given an arbitrary locally finitely presentable category K and finitary monads T and S on K, we characterize monad morphisms α: S − → T with the property that the induced functor α ∗ : KT − → KS between the categories of EilenbergMoore algebras is fully faithful. We call such monad morphi ..."
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Abstract. Given an arbitrary locally finitely presentable category K and finitary monads T and S on K, we characterize monad morphisms α: S − → T with the property that the induced functor α ∗ : KT − → KS between the categories of EilenbergMoore algebras is fully faithful. We call such monad morphisms dense and give a characterization of them in the spirit of Beth’s definability theorem: α is a dense monad morphism if and only if every Toperation is explicitly defined using Soperations. We also give a characterization in terms of epimorphic property of α and clarify the connection between various notions of epimorphisms between monads.