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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 32 (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.
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 6 (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
Equivalence of Borcherds Gvertex algebras and axiomatic vertex algebras
, 1999
"... In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds ’ notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations ..."
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Cited by 6 (1 self)
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In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds ’ notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations of singularities to their inputs. In particular we show that this description of a vertex algebra allows us to present generalised notions of rationality, commutativity and associativity as natural consequences of the definition. Finally, we show that for a certain choice of vertex group, axiomatic vertex algebras correspond bijectively to algebras in the relaxed multilinear category of representations of a vertex group.
Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
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We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.
Abstract
, 1999
"... In this paper we describe how to give a particular global category of rings and modules the structure of a relaxed multi category, and we describe an algebra in this relaxed multi category such that vertex algebras appear as such algebras. Key words: Multicategory, relaxed multicategory, vertex alge ..."
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In this paper we describe how to give a particular global category of rings and modules the structure of a relaxed multi category, and we describe an algebra in this relaxed multi category such that vertex algebras appear as such algebras. Key words: Multicategory, relaxed multicategory, vertex algebra, ring and module. Our intention for this paper is to describe a method for giving the category of modules for a cocommutative, coassociative Hopf algebra, the structure of a
Contents
, 2008
"... In this paper, firstly, we introduce a higherdimensional analogue of hypergraphs, namely ωhypergraphs. This notion is thoroughly flexible because unlike ordinary ωgraphs, an ndimensional edge called an ncell has many sources and targets. Moreover, cells have polarity, with which pasting of cell ..."
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In this paper, firstly, we introduce a higherdimensional analogue of hypergraphs, namely ωhypergraphs. This notion is thoroughly flexible because unlike ordinary ωgraphs, an ndimensional edge called an ncell has many sources and targets. Moreover, cells have polarity, with which pasting of cells is implicitly defined. As examples, we also give some known structures in terms of ωhypergraphs. Then we specify a special type of ωhypergraph, namely directed ωhypergraphs, which are made of cells with direction. Finally, besed on them, we construct our weak ωcategories. It is an ωdimensional variant of the weak ncategoreis given by Baez and Dolan [2]. We introduce ωidentical, ωinvertible and ωuniversal cells instead of universality and balancedness in [2]. The whole process of our definition is in parallel with the way of regarding categories as graphs with composition and identities.
Contents
, 2002
"... We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad cons ..."
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We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad construction on suitable monads. We use this to give an explicit description of the relationship between BaezDolan and Leinster opetopes.
fcmulticategories
, 1999
"... What fcmulticategories are, and two uses for them. fcmulticategories are a very general kind of twodimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. ..."
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What fcmulticategories are, and two uses for them. fcmulticategories are a very general kind of twodimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories.