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

On a notion of entropy in coarse geometry

AbstractThe notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids

Quasi-uniform and syntopogenous structures on categories

Philosophiae Doctor - PhDIn a category C with a proper (E; M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a uni ed approach to the theory of closure, interior and neighbourhood operators. We then introduce and study an abstract notion of C asz ar's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is a base for a quasi-uniformity. This leads us to prove that for any idempotent closure operator c (interior i) on C there is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of completeness of objects and precompactness with respect to the quasi-uniformity de ned in a natural way are studied. The great relationship between quasi-uniformities and closure operators in a category inspires the investigation of categorical quasi-uniform structures induced by functors. We introduce the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities induced by pointed and copointed endofunctors. Amongst other things, it is shown that every quasi-uniformity on a re ective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every re ection morphism is continuous. The notion of continuity of functors between categories endowed with xed quasi-uniform structures is also introduced and used to describe the quasi-uniform structures induced by an M- bration and a functor having a right adjoint

Interior operators and their applications

Philosophiae Doctor - PhDCategorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting

On factorization structures, denseness, separation and relatively compact objects

We define morphism (E, M)-structures in an abstract category, develop their basic properties and present some examples. We also consider the existence of such factorization structures, and find conditions under which they can be extended to factorization structures for certain classes of sources. There is a Galois correspondence between the collection of all subclasses of X-morphisms and the collection of all subclasses of X-objects. A-epimorphisms diagonalize over A-regular morphisms. Given an (E, M)-factorization structure on a finitely complete category, E-separated objects are those for which diagonal morphisms lie in M. Other characterizations of E-separated objects are given. We give a bijective correspondence between the class of all (E, M)factorization structures with M contained in the class of all X-embeddings and the class of all strong limit operators. We study M-preserving morphisms, M-perfect morphisms and M-compact objects in a morphism (E, M)-hereditary construct, and prove some of their properties which are analogous to the topological ones.Mathematical SciencesM. Sc. (Mathematics