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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 <= 2 ..."
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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.
THE CATEGORY OF OPETOPES AND THE CATEGORY OF OPETOPIC SETS
 THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... We give an explicit construction of the category Opetope of opetopes. We prove that the category of opetopic sets is equivalent to the category of presheaves over Opetope. ..."
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Cited by 4 (1 self)
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We give an explicit construction of the category Opetope of opetopes. We prove that the category of opetopic sets is equivalent to the category of presheaves over Opetope.
Polynomial functors and opetopes
 In preparation
"... We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and exam ..."
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We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the BaezDolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the BaezDolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the BaezDolan construction. The calculus of opetopes is also wellsuited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts.
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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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
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Comparing operadic theories of ncategory
, 2008
"... We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of ..."
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We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of ncategories of the former kind and produce a globular operad whose algebras are the ncategories we started with. We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by an operad P in the category V. We define weak ncategories by iterated weak enrichment using a series of parametrising operads Pi. We then show how to construct from such a theory an ndimensional globular operad for each n ≥ 0 whose algebras
HIGHER DIMENSIONAL ALGEBRAS VIA COLORED PROPS
, 809
"... Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs ..."
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Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs P, we obtain higher categorical versions of polycategories, 2fold monoidal categories, topological quantum field theories, and so on.