Structures in higher-dimensional category theory

Abstract

This paper, written in 1998, aims to clarify various higher categorical structures, mostly through the theory of generalized operads and multicategories. Chapters I and II, which cover this theory and its application to give a definition of weak n-category, are largely superseded by my thesis (math.CT/0011106), but Chapters III and IV have not appeared elsewhere. The main result of Chapter III is that small Gray-categories can be characterized as the sub-tricategories of the tricategory of 2-categories, homomorphisms, strong transformations and modifications; there is also a conjecture on coherence in higher dimensions. Chapter IV defines opetopes and a category of n-pasting diagrams for each n, which in the case n=2 is a definition of the category of trees.

81 pages, written 1998

References in corpus (4)

• Quantum Cohomology of a Product
• A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen
• Limits in $n$-categories
• Meromorphic tensor categories

Cited by in corpus (9)

• A Prehistory of n-Categorical Physics
• Operads in Higher-Dimensional Category Theory
• Homotopy Algebras for Operads
• Opetopic bicategories: comparison with the classical theory
• Weak n-categories: opetopic and multitopic foundations
• Weak n-categories: comparing opetopic foundations
• Topology and Higher-Dimensional Category Theory: the Rough Idea
• An alternative characterisation of universal cells in opetopic n-categories
• Comparing algebraic and non-algebraic foundations of $n$-category theory