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15
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 38 (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.
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
Paths in double categories
 Theory Appl. Categ
"... Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, w ..."
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Cited by 16 (2 self)
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Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fcmulticategories, with representable identities in the second case.
Generic morphisms, parametric representations and weakly cartesian monads
 THEORY APPL. CATEG
, 2004
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A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 12 (1 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
Homotopy algebras for operads
"... We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically ..."
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Cited by 10 (1 self)
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We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy Palgebra in M is, provided only that some of the morphisms in M have been marked out as ‘homotopy equivalences’. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any nfold loop space is an nfold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A∞spaces, A∞algebras and nonstrict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on ‘change of base’, e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we
THE SPAN CONSTRUCTION
"... Abstract. We present two generalizations of the Span construction. The first generalization ..."
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Abstract. We present two generalizations of the Span construction. The first generalization
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.
1 The pseudomonoidality quartet
, 2004
"... A roadmap to the unification of weak categorical ..."
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Summary
, 2006
"... Two distinct pools of Src family tyrosine kinases regulate PDGFinduced DNA synthesis and actin dorsal ruffles ..."
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Two distinct pools of Src family tyrosine kinases regulate PDGFinduced DNA synthesis and actin dorsal ruffles