ACA 2015: Persistence over a topos of variable sets
The basic technique of standard persistent homology identifies a global structure by inferring high-dimensional structure from low-dimensional representations and studying properties of a continuous space by the analysis of a discrete sample of it, assembling points into global structure. It is formalised with several different approaches to the underlying algebraic structures: persistence modules have been defined as graded modules over k[t], or as graded modules over a quiver of type A_n. Each of these formalisations have brought extensions of both algorithmics and the scope of what TDA methods can be envisioned and studied. An alternative to set theory for the foundation of Mathematics based on topoi (i.e., cartesian closed categories with enough structure to produce a classifier of subobjects for each object) providing effective means for transferring results and techniques between distinct fields.
The idea of applying sheaves to encode the shape of persistent homology is not itself new. The novelty of our approach is to consider a topos theoretic approach to set theory permitting ideas like time variable sets and provides tools to consider such a common framework, an appropriate approach to generalise a time-driven theory such as persistence. In this topos-based approach, persistent homology is computed from the internal homology of simplicial complexes over a set theory in which elements have encoded lifetimes. In such a setting, a filtered topological space corresponds to a topological space where parts of the space come in at later times; the construction of the homology functor immediately provides homology groups where elements come in and go away as time flows. This leads to a formulation in the topos setting for the various flavours of persistence that have emerged so far: standard, multi-dimensional and zigzag persistence. For each of these cases, the recognition of an underlying algebraic structure has contributed both to the identification of new problems and to the development of new algorithms.
Persistence permutes the encoding of topological features by multisets of pairs of real numbers called persistence diagrams, where the birth and death time of every connected component of the sample of a given topological space is recorded. Each element of one persistence diagram corresponds to a basis element of the correspondent module. For a given time point t, we can determine the local Betti number at t by counting the number of points (b,d) in the multiset such that b < t < d. This can be visualized either as counting points in a quadrant or as counting bars intersecting a vertical line. There is nothing that keeps us from doing this for longer spans of query time intervals -- we can ask for points (b,d) such that b < x < y < d for some interval (x,y). This produces Betti numbers that persist for at least the time period (x,y).
The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars through the algebraic properties of its underlying lattice structure. The category of sheaves over this Heyting algebra constitutes a topos, behaving as a category of sheaves of sets over a topological space. We consider the topos of sheaves over the algebra of lifetimes P, denoted by [P^{op},Set]. We explore a topos-based approach to foundations for persistent homology. We will show that the considered topos exhibits the same features we are expecting from a persistent topology. Within the set theory of this topos, we can develop semi-simplicial sets, chain complexes, and a combinatorial homology theory that reimplements classical persistent homology by introducing the persistence aspects already at the level of set theory. We shall also consider the representation of this category in Vect, its relation with the generalised persistence modules, and discuss theorems that permit us a step forward towards stability results that can be reached at the underlying algebra level. Finally, we describe some ideas on how a choice of a different base Heyting algebra can generate other shapes of a persistence theory, thus potentially unifying multidimensional, tree-based, DAG-based and zig-zag persistence under a common foundation.