Categorical and quantitive topology and its applications

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Much of the recent progress in mathematical research has been achieved by the import of ideas and methods of one area to another. From its beginnings in the 1940s, categorical language and theory have been developed to facilitate such interaction. Although being only one of many, this original overarching aspect of category theory continues to strive, and our research proposal aims to reinforce it in areas which, by comparison to other disciplines of mathematics, have so far only moderately been influenced or aided by category theory: functional analysis and general topology. In order to broaden its applicability, we want to extend a more quantitative approach to general topology beyond metrizability, by investigating spaces equipped with a distance function that may assume values other than numbers, such as probability functions. ***A particular type of such spaces should allow for a categorical unification of novel results in functional analysis that are based on "approximate" notions, defined to miss the standard one only by a preset margin. In Banach space theory, a prominent classical example is the so-called Gurarii space, which is separable and has an approximate extension property with respect to isometric embeddings of finite-dimensional spaces. We have recently established a categorical existence proof which, when applied to metric spaces, yields also the existence of the so-called Urysohn universal space. Our first goal is to crystallize the categorical principles that would allow us to give not only an existence but also a unified uniqueness proof for these two classical spaces, and then exploit them for other types of objects, especially for spaces with additional properties or structures of interest in topology or functional analysis.***In a metric space (infinite distances permitted) one has a numerically defined closest distance from a point x to any set A of points of the space. So-called approach spaces come with an axiomatized point-set-distance function. They faithfully encompass, as categories, both metric and topological spaces, i.e., degenerate approach spaces where the distance from x to A may only be 0 or infinite (x lies in or outside the closure of A). For novel applications, we study spaces where the "distances" may not only be binary or numerical, but take values in a well-structured lattice. Moreover, by considering distances from points to features of the space in question other than its subsets, such as ultrafilters, we employ algebraic methods to study special types of spaces, such as the ones permitting the satisfactory formation of their function spaces. As a thesis project we seek to find situations when the totality of such spaces behaves dually to another category, usually of an algebraic type, which will then allow for the transfer of information between both types. Another thesis project will study "connected" objects and transformations in these and even broader contexts.*****