COARSE VERSION OF HOMOTOPY THEORY (AXIOMATIC STRUCTURE)

In topology, homotopy theory can be put into an algebraic framework.
The most complete such framework is that of a Quillen model
Category [[15], [5]].
The usual class of coarse spaces appears to be too small to be a
Quillen model category. For example, it lacks a good notion of
products. However, there is a weaker notion of a co�bration category
due to Baues [[1], [2]].
The aim in this thesis is to look at notions of co�bration category
within the world of coarse geometry. In particular, there are several
sensible notions of the structure of a coarse version of a co�bration
category that we de�ne here.
Later we compare these notions and apply them to computations.
To be precise, there are notions of homotopy groups in a Baues
co�bration category. So we compare these groups as well for the
di�erent structures we have de�ned, and to the more concrete notion
of coarse homotopy groups de�ned also in [10].
Going further, there is an abstract notion of a cell complex de�ned
in the context of a co�bration category. In the coarse setting, we
prove such cell complexes have a more geometric de�nition, and
precisely we prove that a coarse CW-complex is a cell complex.
The ultimate goal of such computations is a version of the Whitehead
theorem relating coarse homotopy groups and coarse homotopy
equivalences for cell complexes. Abstract versions of the Whitehead
theorem are known for co�bration categories [1], so we relate these
abstract results to something more geometric.
Another direction of the thesis involves Quillen model categories.
As already mentioned, there are obstructions to the class of coarse
spaces being a Quillen model category; there is no apparent way
to de�ne category-theoretic products of coarse spaces. However,
such obvious objections vanish if we add extra spaces to the coarse
category. These extra spaces are termed non-unital coarse spaces
in [9]. We have proved most of Quillen axioms but the existence of
limits in one of our categories.