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categories: Thesis: `Operads in higher-dimensional category theory'
My PhD thesis, `Operads in higher-dimensional category theory', is now
(approved and) available electronically as math.CT/0011106 - that is, at
The summary follows.
Operads in Higher-Dimensional Category Theory
The purpose of this dissertation 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 omega-category. This theory of operads and
multicategories has various other applications too: for instance, to the
opetopic approach to n-categories expounded by Baez, Dolan and others, and to
the theory of enrichment of higher-dimensional categorical structures. We
sketch some of these further developments, without exploring them in full.
We start with a look at bicategories (Chapter 1). Having reviewed the basics
of the classical definition, we define `unbiased bicategories', in which
n-fold composites of 1-cells are specified for all natural n (rather than the
usual nullary and binary presentation). We go on to show that the theories
of (classical) bicategories and of unbiased bicategories are equivalent, in a
The heart of this work is the theory of generalized operads and
multicategories. More exactly, given a monad T on a category E, satisfying
simple conditions, there is a theory of T-operads and T-multicategories. In
Chapter 2 we set up the basic concepts of the theory, including the important
definition of an algebra for a T-multicategory. In Chapter 3 we cover an
assortment of further operadic topics, some of which are used in later parts
of the thesis, and some of which pertain to the applications mentioned in the
Chapter 4 is a (proposed) definition of weak omega-category, a modification
of that given by Batanin (Adv Math 136 (1998), 39-103). Having given the
definition formally, we take a long look at why it is a *reasonable*
definition. We then explore weak n-categories (for finite n), and show that
weak 2-categories are exactly unbiased bicategories.
The four appendices take care of various details which would have been
distracting in the main text. Appendix A contains the proof that unbiased
bicategories are essentially the same as classical bicategories. Appendix B
describes how to form the free T-multicategory on a given T-graph. In
Appendix C we discuss various facts about strict omega-categories, including
a proof that the category they form is monadic over an appropriate category
of graphs. Finally, Appendix D is a proof of the existence of an initial
object in a certain category, as required in Chapter 4.