| Summary: | 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 cofibration category due to Baues [[1], [2]]. The aim in this thesis is to look at notions of cofibration category within the world of coarse geometry. In particular, there are several sensible notions of the structure of a coarse version of a cofibration category that we define here. Later we compare these notions and apply them to computations. To be precise, there are notions of homotopy groups in a Baues cofibration category. So we compare these groups as well for the different structures we have defined, and to the more concrete notion of coarse homotopy groups defined also in [10]. Going further, there is an abstract notion of a cell complex defined in the context of a cofibration category. In the coarse setting, we prove such cell complexes have a more geometric definition, 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 cofibration 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 define 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.
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