Mohamad, Nadia (2013) COARSE VERSION OF HOMOTOPY THEORY (AXIOMATIC STRUCTURE). PhD thesis, University of Sheffield.
Abstract
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.
Metadata
Supervisors: | Mitchener, Paul |
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Publicly visible additional information: | nadiamg74@yahoo.com |
Awarding institution: | University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Identification Number/EthosID: | uk.bl.ethos.577441 |
Depositing User: | Mrs Nadia Mohamad |
Date Deposited: | 09 Aug 2013 10:32 |
Last Modified: | 03 Oct 2016 10:45 |
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