Zastosowania teorii kategorii w topologii ogólnej

Abstract

This dissertation is concerned with some notions of category theory in application to general topology and investigates to what extent the arguments making use of those notions can replace a more traditional approach. One of the generic examples is the notion of Fraisse sequence. Quite a number of categories consist of algebraic objects: groups, Abelian groups and monoids with group, Abelian group and monoid homomorphisms, respectively. An example of a monoid consisting of topological operations is the Kuratowski monoid. The dissertation is organized as follows. In subsection 1.2.1 we give some comments on ways of making use of concepts such as a tree, the space of branches, an ultra-metric and so forth. In section 2 we provide axioms for categories, basic definitions, notations and examples. We draw attention to a particular covariant functor, namely a diagram. At the beginning of section 3 we remind basic ideas concerning the limit of inverse sequence. Then we analyze a proof of Knaster-Reichbach theorem about extendibility of a mapping between two closed, nowhere dense subsets of the Cantor set and its version for scattered compact metric spaces. In section 4 we focus on a symmetric Cantorval. Beside the Cantor set and the sum of a finite family of closed intervals it is one of the possible forms the set of subsums of a convergent series with positive terms can take. In section 5 we remind the notions of the amalgamation and the inverse amalgamation property as well as their special cases - pushout and pullback, respectively. We discuss 1 these notions in two particular categories: S e t category whose objects are sets and whose arrows are functions and T op category whose objects are topological spaces and whose arrows are continuous mappings. In section 6 we present the main idea of the dissertation, namely Fraisse sequences: definition, criterion of existence and some properties. In subsection 6.2 we focus on the category F in s e top/A, i.e. a comma-category related to the opposite category of the category F in s e t . In section 7 we discuss the structure and basic facts concerning Fraisse limits in the context of inverse sequences. We show a way in which the properties of projective universality and projective homogeneity combine with the concept of generic object (for a given subcategory). Section 8 comprises comments th a t will be included in joint papers with the supervisors concerning, among others, a category of finite non-empty sets and its linear version. We claim here th a t the Cantor set admits a unique strictly positive probability measure th a t takes rationa values on clopen sets and satisfies certain homogeneity condition.