On some coarse geometric notions inspired by topology and category theory

Coarse geometry is the study of the large scale properties of spaces. The interest in large scale properties is mainly motivated by applications to geometric group theory and index theory, as well as to important open problems such as the Novikov Conjecture. In this thesis, we introduce and study coarse versions of the following classical topological notions: connectedness, monotone-light factorizations, extension theorems, and quotients by properly discontinuous group actions. We will draw on the analogy between large scale geometry and topology as well as on the perspective of category theory using Roe's coarse category. In the first of four research chapters, we look at a large scale connectedness condition arising from the coarse category and show that it coincides with the topological connectedness of the Higson corona. In the second, we introduce coarse versions of monotone and light maps (calling them coarsely monotone and coarsely light maps respectively) and show that these maps constitute a factorization system on the coarse category. We also show that coarsely light maps preserve some important large scale properties. In the third research chapter, we unify the proof of three extension theorems: the classical Tietze Extension Theorem from topology, Katetov's extension theorem for uniform spaces, and an extension theorem for slowly oscillating functions (an important class of functions in coarse geometry). The unification is achieved via a general extension theorem for neighbourhood operators. In the final research chapter, we study warped spaces associated to group actions on metric spaces, focussing in particular on coarsely discontinuous actions which we introduce as large scale analogues of properly discontinuous actions in topology. For such actions, we relate the (maximal) Roe algebra of the warped space with the crossed product of the (maximal) Roe algebra of the original space and the group, and prove a deck transformation result.