Coarse geometry: a foundational and categorical approach with applications to groups and hyperspaces

The topic of the manuscript is coarse geometry, also known as large-scale geometry, which is the study of large-scale properties of spaces. It found applications in geometric group theory after the work of Gromov, and in Novikov and coarse Baum-Connes conjectures. The thesis is divided into three parts. In the first one, we provide a foundational and categorical approach to coarse geometry. Large-scale geometry was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. However, coarse spaces are innerly symmetric objects, and thus are not suitable to parametrise asymmetric objects such as monoids and quasi-metric spaces. In order to fill the gap, we introduce quasi-coarse spaces. Moreover, we consider also semi-coarse spaces and entourage spaces. These objects induce para-bornologies, quasi-bornologies, semi-bornologies, pre-bornologies (also known as bounded structures) and bornologies, and this process is similar to the definition of uniform topology from a (quasi-)uniform space. We study all the notions introduced and recalled to find extensions of classical results proved for metric or coarse spaces, and similarities with notions and properties for general topology. Furthermore, we study the categories of those objects and the relations among them. In particular, since all of them are topological categories, we have a complete understanding of their epimorphisms and monomorphisms, and the description of many categorical constructions. Among them, of particular interest are quotients. We then focus our attention on Coarse, the category of coarse spaces and bornologous maps, discussing its closure operators and the cowellpoweredness of its epireflective subcategories, and its quotient category Coarse/~, which turns out to be balanced and cowellpowered. The second part is dedicated to study the large-scale geometry of algebraic objects, such as unitary magmas, monoids, loops and groups. In particular, we focus on coarse groups (groups endowed with suitable coarse structures) and we investigate their category. We study different choices, underlining advantages and drawbacks. With some restrictions on the coarse groups that we are considering, if we enlarge the class of morphisms to contain bornologous quasi-homomorphisms (and not just bornologous homomorphisms), every coarse inverse of a homomorphism which is a coarse equivalence is a quasi-homomorphism. This observation is connected to the notion of localisation of a category and could provide a categorical justification to the notion of quasi-homomorphism. Once the categories of coarse groups are fixed, inspired by the notion of functorial topologies, we can introduce functorial coarse structures on Grp, the category of groups and homomorphisms, and on TopGrp, the category of topological groups and continuous homomorphisms. Among them, we pay attention to the ones induced by cardinal invariants, and to those associated to the family of relatively compact subsets. As for the latter functorial coarse structure, we study the transformation of large-scale properties along Pontryagin and Bohr functors. The third part is devoted to coarse hyperspaces, which are suitable coarse structures on power sets of coarse spaces. This construction was introduced following the work of Protasov and Protasova and miming the classical notion of uniform hyperspace. We see how properties of the initial coarse space are reflected on the hyperspace. Since the coarse hyperspace is highly disconnected, it is convenient to consider some special subspaces of it. For example, if the base space is a coarse group, it is natural to consider the subspace structure induced on the lattice of subgroups, called subgroup exponential hyperballean. We show that both the subgroup exponential hyperballean and the subgroup logarithmic hyperballean, another coarse structure on the subgroup lattice, capture many properties of the group.