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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.
This is a collation of the operative parts of a proposal submitted to the NSF concerning
"... cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themse ..."
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cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themselves, should eventually become part of the standard mathematical culture. PROJECT DESCRIPTION EXPLORATIONS OF HIGHER CATEGORICAL STRUCTURES AND THEIR APPLICATIONS PROPOSED RESEARCH 1. Historical background and higher homotopies Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper [47]. The language they introduced transformed modern mathematics. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Higher category theory concerns higher level notions of naturality, which can be viewed as maps between natural transformations, and maps between such maps, and so for
Project Description:
"... d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surf ..."
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d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through nmanifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an ncategory with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb
On comparing definitions of "weak n–category"
, 2001
"... 1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand and this is more relevant here, I start "from scratch", and thus what I say can be understood with little technical ..."
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1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand and this is more relevant here, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally about technical matters that are written down formally elsewhere, where they can be studied further. [The text in square brackets [] is either some technical explanation, or a digression.] 2. Terminology First, some terminological conventions. I will use the word "category " in its most general sense: weak ωcategory. This is completely inclusive: all sorts of "categories " are categories ow. here are two extensions of the original meaning: "weak", and "omegadimensional". Weak " signifies an indeterminate notion; there are several different specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would