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Nori diagrams and persistent homology. (English) Zbl 1464.18002

Inspired by similarities perceived by the authors between the persistent homology in topological data analysis and the theory of Nori motives in algebraic geometry, the present paper builds a bridge between the two fields by giving a unified treatment of both formalisms.
In broad strokes, the main developments are as follows. §1 introduces a version of Nori’s construction in the context of persistent homology, up to and including the identification of the resulting category as a category of comodules over a suitable bialgebra (Proposition 1.8.1). This can be interpreted as providing a notion of motive for topological data analysis, which can perhaps play a role there similar to the one of motives in algebraic geometry. §2 and §3 introduce a general version of the persistence construction for objects fibred over a partially ordered base object. §5 instantiates this idea in the case of Nori motives over a partially ordered base scheme.
Beyond this core theme of the paper, §6 contains some material on model category structures in persistent homology, amounting to working with the universal model category in D. Dugger’s sense [Adv. Math. 164, No. 1, 144–176 (2001; Zbl 1009.55011)] generated by a certain category of data sets. §7 considers a probabilistic variant [M. Marcolli, J. Geom. Phys. 140, 26–55 (2019; Zbl 1420.81007)] of G. Segal’s \(\Gamma\)-space construction for categories with sums [Topology 13, 293–312 (1974; Zbl 0284.55016)]. §8 ends with an interesting list of further directions concerning the use of persistent homology ideas in other subfields of pure mathematics.
Some of the arguments in the paper contain some minor inaccuracies. In particular, the equivalence of the category of functors \((\mathbb{R},\le) \to \mathrm{Vec}\) with the category of \(\mathbb{R}[x]\)-modules must be understood as involving \(\mathbb{Z}\)-graded \(\mathbb{R}[x]\)-modules, implying that the Nori diagram construction considered there also needs to be formulated in a \(\mathbb{Z}\)-graded version. Lemma 1.5.1(2) is incorrect as stated; in fact it can be shown that the assignment on objects \(F \mapsto F_{\mathbb{Z}}\) cannot be extended to a functor at all. (Here’s how: let \(F\) be a barcode diagram supported on the interval \([2,3]\), and \(F'\) the direct sum of \(F\) with a barcode diagram supported on \([0,1]\). Then \(\mathrm{id}_F\) factors through \(F'\), but \(\mathrm{id}_{F_\mathbb{Z}}\) does not factor through \(F'_{\mathbb{Z}}\).)

MSC:

18-02 Research exposition (monographs, survey articles) pertaining to category theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
55N31 Persistent homology and applications, topological data analysis
14C15 (Equivariant) Chow groups and rings; motives
18M25 Tannakian categories
18N40 Homotopical algebra, Quillen model categories, derivators
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