文档介绍：Higher-Dimensional Category Theory:
Opetopic Foundations
Eugenia L.-G. Cheng
Gonville and Caius College, Cambridge
A dissertation submitted for the degree of Doctor of Philosophy at the
University of Cambridge
March 2002
This dissertation is the result of my own work and includes nothing which
is the e of work done in collaboration. The statements made in
the ‘Related Work’ section of the Introduction, concerning which ideas are
original or novel, are to the best of my knowledge correct.
This dissertation is not substantially the same as any that I have submitted
for a degree or diploma or any other qualification at any other university.
Eugenia L.-G. Cheng
10th March 2002
Summary
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 ap-
proaches 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 position 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
Mac Lane.
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