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We introduce lifespan functors, which are endofunctors on the category ofpersistence modules that filter out intervals from barcodes according to theirboundedness properties. They can be used to classify injective and projectiveobjects in the category of barcodes and the category of pointwisefinite-dimensional persistence modules. They also naturally appear in dualityresults 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 inducedby morphisms between filtrations. This lays the groundwork for the efficientcomputation of barcodes for images, kernels, and cokernels of such morphisms.

Lifespan Functors and Natural Dualities in Persistent Homology