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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

http://arXiv.org/abs/math.CT/0011106

The summary follows.

Tom



Operads in Higher-Dimensional Category Theory

Tom Leinster

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
strong sense.

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
first paragraph.

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.