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Symmetries of data sets and functoriality of persistent homology

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Wojciech Chachólski, Alessandro De Gregorio, Nicola Quercioli and Francesca Tombari

The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked.
A data set for us is a finite collection of functions, called measurements, with a finite domain.
Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms.
Different choices of the set of endomorphisms encode different symmetries of the data set.
We describe various category structures on such enriched data sets and prove some of their properties such as decompositions and morphism formations.
We also describe a data structure, based on coloured directed graphs, which is convenient to encode the mentioned enrichment.
We show that persistent homology preserves only some aspects of these collection of enriched data sets however not all.
In other words persistent homology is not a functor on the entire category of enriched data sets.
Nevertheless we show that persistent homology is functorial locally.
We use the concept of set equivariant operator (SEO) to capture some of the information missed by persistent homology.
Moreover, we provide examples and give ways to construct such SEOs.

Keywords:
persistent homology, equivariant operators

2020 MSC:
55N31, 62R40, 68T09, 18D25

*Theory and Applications of Categories,*
Vol. 39, 2023,
No. 23, pp 667-686.

Published 2023-08-04.

http://www.tac.mta.ca/tac/volumes/39/23/39-23.pdf

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