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