Adrom, Pouya
(2015)
Internal categories as models of homotopy types.
PhD thesis, University of Glasgow.
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Abstract
A homotopy ntype 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 0type is, up to weak homotopy equivalence (isomorphism of
homotopy groups), a discrete set. It is wellknown that a connected 1type 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 1types 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 2types can
be modeled, in the sense described above, by crossed modules of groups. A crossed module
is equivalently what in [Loday] is called a 1catgroup, but now often referred to as a
cat1
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