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2020

Persistent homotopy theory

Jardine, J. F.

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Understand

Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms.

These homotopy types are viewed as systems (or functors), which are defined on a parameter space.
The category of systems of spaces admits a partial homotopy theory that is based on controlled equivalences, suitably defined, that are the output of homotopy stability results.

Persistent homotopy theory · Around

Built on

Homotopy limits, completions and localizations
A. K. Bousfield and D. M. Kan · 1972
Earlier work this paper cites.
Higher algebraic K K -theory. I
Daniel Quillen · 1973
Earlier work this paper cites.
Homotopy properties of the poset of nontrivial p p -subgroups of a group
Daniel Quillen · 1978
Earlier work this paper cites.

Similar

Simplicial Homotopy Theory
P. G. Goerss and J. F. Jardine · 1999
Cited alongside, same era.
Simplicial approximation
J. F. Jardine · 2004
Cited alongside, same era.
Path categories and resolutions
J. F. Jardine · 2010
Cited alongside, same era.

Then

Universality of the homotopy interleaving distance
Original
Andrew J. Blumberg and Michael Lesnick · 2017
Later among the works it cites.
A Distance Between Filtered Spaces Via Tripods
F. Memoli · 2017
Later among the works it cites.

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