On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces

Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology

A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, permuting the encoding of topological features by so called barcodes or equivalently persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between persistent bars through the algebraic properties of its underlying lattice structure. In this paper, we investigate the topos of sheaves over such algebra, as well as discuss its construction and potential for a generalised simplicial homology over it. In particular we are interested in establishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio Mathematica. The new version has restructured arguments, clearer intuition is provided, and several typos correcte

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres