20150615

Bedlewo 2015: The Topos Foundation of Persistence.A topos theoretic approach to set theory permits ideas like time variable sets and provides tools for unification of techniques for mathematics. An important application targets topological data analysis, and in particular persistent homology. A topos theoretic foundation for persistence allows for a common framework on the study of several aspects of persistence, illuminating the nature of concepts, establishing new theorems, suggesting more general examples, and promoting new lines of investigation. In this talk we shall look at the topos of sheaves over a Heyting algebra of lifetimes, discuss its construction and potential for a generalized simplicial homology over it. Moreover, we also describe the sheafification process through the etale space construction that permits us to compute persistent homology in the most concrete cases. We shall also consider the representation of this category in Vect, its relation with the generalised persistence modules, and describe an algorithm for the computation of the indecomposables in this framework. Furthermore, we will discuss theorems that permit us a step forward towards stability results that can be reached at the underlying algebra level.

Bellow you can find the slides of this talk, its long abstract and the paper on what it is based on, under revision at Demonstratio Mathematica.