The power of pullbacks: meets and joins in persistence
Primoz Skraba, Mikael Vejdemo-Johansson, and João Pita Costa
Abstract. Persistent homology and its variants is an important tool in computational topology and topological data analysis. In this paper, we introduce a novel algebraic characterization of per- sistence inspired by concepts from lattice theory. By defining the notion of a meet and a join of vector spaces in a diagram, persistence is defined as the image between these two spaces. After reviewing the relevant concepts, we introduce algorithms for computing the relevant structures and give two applications: computing the largest injective object of an arbitrary commuta- tive diagram of vector spaces and a parallel algorithm for computing zigzag persistence. The constructions have nice theoretical properties but are constructive, resulting in concrete algo- rithms. For each case, we give an analysis of the algorithms and conclude with a discussion of open questions and further directions.
JOÃO PITA COSTA 2014