On the stability of the persistence topos.
Toposys Final Meeting, Krakow University, 2015
Recently, several (co)sheaf theoretic approaches were proposed to explore existing generalizations of persistence, and persistence related topics such as Reeb graphs. Our approach is based on the generalization of set theory by a topos of sheaves over a Heyting algebra of lifetimes of topological features, encoded in a persistence diagram or a barcode. 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 the algebra of lifetimes and discuss the construction of 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. Furthermore, we will discuss stability results that can be reached at the underlying algebra level.