Lifespan Functors and Natural Dualities in Persistent Homology

TL;DR

This paper introduces lifespan functors that classify and filter persistence modules based on boundedness, aiding in duality analysis and efficient barcode computations for morphisms in persistent homology.

Contribution

It presents lifespan functors as a new tool for classifying and analyzing persistence modules, extending duality results and enabling efficient barcode computations.

Findings

Lifespan functors classify injective and projective objects.

They facilitate duality results in persistent (co)homology.

They enable efficient computation of barcodes for images, kernels, and cokernels.

Abstract

We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for the efficient computation of barcodes for images, kernels, and cokernels of such morphisms.