Internal categories as models of homotopy types

Adrom, Pouya (2015) Internal categories as models of homotopy types. PhD thesis, University of Glasgow.

Abstract

A homotopy n-type is a topological space which has trivial homotopy groups above
degree n. Every space can be constructed from a sequence of such homotopy types, in a
sense made precise by the theory of Postnikov towers, yielding improving `approximations'
to the space by encoding information about the first n homotopy groups for increasing n.
Thus the study of homotopy types, and the search for models of such spaces that can be
fruitfully investigated, has been a central problem in homotopy theory.
Of course, a homotopy 0-type is, up to weak homotopy equivalence (isomorphism of
homotopy groups), a discrete set. It is well-known that a connected 1-type can be represented,
again up to weak homotopy equivalence, as the classifying space of its fundamental
group: this is the geometric realization of the simplicial set that is the nerve of the group
regarded as a category with one object. Another way to phrase this is that the homotopy
category of 1-types obtained by localizing at maps which are weak homotopy equivalences
| formally adding inverses for these | is equivalent to the skeleton of the category of
groups.
In [Mac Lane and Whitehead] it was proved that connected homotopy 2-types can
be modeled, in the sense described above, by crossed modules of groups. A crossed module
is equivalently what in [Loday] is called a 1-cat-group, but now often referred to as a
cat1

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Abstract homotopy theory, model categories, simplicial objects
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Steiner, Dr. Richard and Stevenson, Dr. Daniel
• Date of Award: 2015
• Depositing User: Pouya Adrom
• Unique ID: glathesis:2015-6228
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 31 Mar 2015 09:08
• Last Modified: 04 May 2015 14:52
• URI: https://theses.gla.ac.uk/id/eprint/6228