Primož Škraba, João Pita Costa
Abstract. The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence based on the study of the order structure of its correspondent lattice. Its algorithmic construction leads to two operations on homology groups which describe a diagram of spaces as a complete Heyting algebra, which is a generalization of a Boolean algebra. We investigate some of the properties of this lattice, the algorithmic implications of it, and some possible applications.
Persistent (co)homology is one of the central objects of study in applied and computational topology. Numerous extensions have been proposed to the original formulation including zig-zag persistence and multidimensional persistence, whereas the original persistence looks at a filtration (i.e., an increasing sequence of spaces). Zig-zag persistence extended the theory and showed that the direction of the maps does not matter, using tools from quiver theory. In multidimensional persistence, multifiltrations are considered. In this paper, we also look at the problem of persistence in more general diagrams of spaces using tools from lattice theory. There is another key difference in this work however. Rather than try to find a decomposition of the diagram of spaces into indecomposables, we concentrate on pairs of spaces within diagrams addressing the more difficult problem of indecomposables in the sequel paper.
Lattice theory is the study of order structures. The deep connections between topology and lattice theory has been known since the work of Stone, showing a duality between Boolean algebras and certain compact and Hausdorff topological spaces, called appropriately Stone spaces. In the first section of this paper we present the basic concepts of lattice theory. These preliminaries mostly refer to classical results on distributive lattices and Heyting algebras, and can be skipped by the reader that is familiar with the subject.
A description of the topological background follows in the second section, reviewing the main concepts and results of Persistent Homology and suggesting several examples that are a motivation to this study. In the following section we describe the order structure of our input diagram of spaces by a partial order induced by certain maps between vector spaces, and show that this order provides a lattice structure. We construct the meet and join operations using the natural concepts of limits and colimits of linear maps, and show that this construction stabilizes. We shall see that the constructed lattice is a complete Heyting algebra, one of the algebraic objects of biggest interest in topos theory.
From the latter results we discuss connections with persistent homology, and give a different perspective on several aspects of this theory. In particular, we look at diagrams of spaces and retrieve general laws both based on concrete examples (like standard or zig-zag persistence) and on the interpretation of laws derived from the lattice theoretic analysis. Finally we introduce a few algorithmic applications which we will develop further in a subsequent paper.
JOÃO PITA COSTA 2014