Categorification of Persistent Homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a natural construction of a category of e-interleavings of $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category and show that if the target category is abelian, so is this category of interleavings.