Towards a topos foundation of persistence

Topos theory is often presented with the potential for the unification of Mathematics, enriching is foundations and thus taking the place that set theory has occupied over the passed generations. The construction of a foundation by topoi has recently been established for quantum theory by A. Doring and C. J. Isham. Persistent homology is a vibrant area of research into applied and computational algebraic topology using multi-scale methods to study the topology of point clouds as a route for approximating topological features of an unobservable geometric object generating samples. In this work, we explore the idea of a topos-based approach as a potential unifying language for various generalizations of persistence in use today. Such an approach would encode the various flavors of persistent homology as a topos of sheaves over a certain Heyting algebra encoding the shape of the persistence theory. This is a joint work of Joao Pita Costa with Primož Škraba and Mikael Vejdemo-Johansson.