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Title: Higher-dimensional category theory : opetopic foundations
Author: Cheng, E. L. G.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2002
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The problem of defining a weak n-category has been approached in various different ways, but so far the relationship between these approaches has not been fully understood. The subject of this thesis is the 'opetopic' theory of n-categories, embracing a group of definitions based on the theory of 'opetopes'. This approach was first proposed by Baez and Dolan, and further approaches to the theory have been proposed by Hermida, Makkai and Power, and Leinster. The opetopic definition of n-category has two stages. First, the language for describing k-cells is set up; this, in the language of Baez and Dolan, is the theory of opetopes. Then, a concept of universality is introduced, to deal with composition and coherence. We first exhibit an equivalence between the three theories of opetopes as far as they have been proposed. We then give an explicit description of the category Opetope of opetopes. We also give an alternative presentation of the construction of opetopes using the 'allowable graphs' of Kelly and MacLane. The underlying data for an opetopic n-category is given by an opetopic set. The category of opetopic sets is described explicitly by Baez and Dolan; we prove that this category is in fact equivalent to the category of presheaves on Opetope. We then turn our attention to the fully definition of (weak) n-categories. We define for each n a category Opic-n-Cat of opetopic n-categories and 'lax n-functors'. We then examine low-dimensional cases, and exhibit an equivalence between the opetopic and classical theories for the cases n £ 2, giving in particular an equivalence between the opetopic and classical approaches to bicategories.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available