A distance between maps via interleavings of relative Sullivan algebras

TL;DR

This paper introduces a pseudodistance between maps in the homotopy category using interleavings of persistence CDGAs, linking algebraic models with topological map comparisons.

Contribution

It develops a novel pseudodistance on the homotopy set of maps via interleavings of persistence CDGAs, extending the algebraic tools for topological map analysis.

Findings

Interleaving distance in the homotopy category defines a pseudodistance between maps.

Persistence CDGAs differ from persistence cochain complexes in their interleaving properties.

Computational examples demonstrate the pseudodistance between maps.

Abstract

In this article, we consider extended tame persistence commutative differential graded algebras (CDGAs) associated with relative Sullivan algebras. In particular, if the relative Sullivan algebra is a model for a map between spaces, then the persistence CDGA is isomorphic to the persistence object obtained by a Postnikov tower for the map with the polynomial de Rham functor in the homotopy category of extended tame persistence CDGAs. Moreover, the interleaving distance in the homotopy category (IHC) in the sense of Lanari and Scoccola enables us to introduce a pseudodistance on the homotopy set of maps via the persistence CDGA models for maps. In contrast to persistence cochain complexes, the IHC of persistence CDGAs does not coincide with the cohomology interleaving distance in general. Due to the reason, we also discuss formalities of a persistence CDGA with interleavings. Computational examples of the pseudodistances between maps are showcased.