Lifespan functors and natural dualities in persistent homology

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 co-kernels of such morphisms.

Keywords

barcode, persistent homology, duality, injectivity, projectivity

2010 Mathematics Subject Classification

16G20, 18G05, 55U10, 57N15

Full Text (PDF format)

This research has been supported by the German Research Foundation (DFG) through the Collaborative Research Center SFB/TRR 109 Discretization in Geometry and Dynamics; the Collaborative Research Center SFB/TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics; the Cluster of Excellence EXC-2181/1 STRUCTURES; and the Research Training Group RTG 2229 Asymptotic Invariants and Limits of Groups and Spaces.

Received 15 October 2021

Received revised 25 July 2022

Accepted 21 September 2022

Published 22 November 2023