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• CommentRowNumber1.
• CommentAuthorLuis_Scoccola
• CommentTimeJul 17th 2022

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• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 17th 2022
• (edited Jul 17th 2022)

I am taking the liberty of renaming from “persistent object” to “persistence object” – first to better rhyme on “persistence module”, but also since it seems more logical: whether or how much a persistence object is actually persistent will have to be seen.

I have tried to check the references you gave on the terminology, but I can’t find them use the term either way (I admit I am just on my phone here, which makes a comprehensive search more tedious).

One reference which agrees with “persistence object” is

Further on terminological quibbles: Despite its title, the first reference

does not seem to be about categorification in the established technical sence, but about category theoretic formulations, which is technically different. So I have changed the wording around this item.

In the Idea-section I made some additions to further bring out the Idea via its main examples:

This is a concept with an attitude: One calls such functors “persistence objects” when one is interested in determining their persistence diagrams or other measures of “persistence” as used in topological data analysis.

The main example in this context arises when $C$ is a category of vector spaces or more generally a category of modules, in which case one speaks of persistence modules as used in persistent homology. Alternatively, $C$ could be a category of groups, such as homotopy groups, or even of full homotopy types, which is the case of interest in persistent homotopy.

And then after the mentioning of interleaving distance:

The key property which one will typically demand of a good theory of persistence objects is a notion of persistence diagrams (measuring “how persistent” a given persistence object is) which is stable with respect to interleaving distance.

• CommentRowNumber3.
• CommentAuthorLuis_Scoccola
• CommentTimeJul 17th 2022

Thank you for the improvements (and sorry that I didn’t add comments on the original contribution; I understand how the system works now).

Regarding persistent vs persistence, I think different people use different conventions. The following paragraph is my opinion on the subject; I’m curious to know what you think.

In my mind, a persistence X is an X that describes the persistent features of a certain Y. Examples: A persistence module that was obtained by taking the homology of a functor Y : P -> Top is a module that describes the persistent topological features of the functor Y; the persistence diagram of a (tame) functor Y : R -> Vect is a diagram that describes the persistent features of the functor Y. Thus, the usual terms persistence module and persistence diagram are consistent with this interpretation. Calling a persistence module persistence vector space would not be consistent, since a persistence module is not a vector space, but a collection of vector spaces with extra structure. On the other hand, a persistent X is a parametrized X (typically parametrized by a poset). The term persistent homology is consistent, as it is the parametrized homology of a parametrized, e.g, topological space. See also Remark 2.6 in The Fiber of the Persistence Map for Functions on the Interval by Justin Curry, where, it seems, the terms persistent vector space and persistent set were introduced.

Here are some references that use different conventions for persistent/persistence (object/topological space/vector space/set):

Persistent X:

• Persistent cup-length. M Contessoto, F Mémoli, A Stefanou, L Zhou
• The Fiber of the Persistence Map for Functions on the Interval. Justin Curry
• Decorated Merge Trees for Persistent Topology. Justin Curry, Haibin Hang, Washington Mio, Tom Needham, and Osman Berat Okutan
• Topology and Data. Gunnar Carlsson (uses both conventions)
• Topological pattern recognition for point cloud data. Gunnar Carlsson (uses both conventions)
• Magnitude meets persistence. Homology theories for filtered simplicial sets. Nina Otter
• Multiplicative persistent distances. Gregory Ginot and Johan Leray
• Locally Persistent Categories And Metric Properties Of Interleaving Distances. Luis Scoccola ;-)
• Rectification of interleavings and a persistent Whitehead theorem. E Lanari, L Scoccola ;-)

Persistence X:

• 1-Dimensional intrinsic persistence of geodesic spaces. Žiga Virk
• Topology and Data. Gunnar Carlsson (uses both conventions)
• Topological pattern recognition for point cloud data. Gunnar Carlsson (uses both conventions)
• A brief history of persistence. Jose A. Perea
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 18th 2022

Thanks for all these details!, and in particular for the pointer to that remark 2.6. Looking it up, I see that it reads:

To the author’s knowledge, the terms “persistent set” and “persistent vector space” are being used here for the first time. The reasons for bringing this new terminology into circulation are twofold: Firstly, the pronounciation of “persistent set” sounds better than “persistence set,” which would be the logical shortening of “persistence module valued in Set.” Secondly, an attempt to maintain grammatical consistency suggests that we use “persistent vector spaces” following “persistent homology,” rather than “persistence modules.”

All this would be great to add to the entry! (Including this quote) to orient the reader.

I admit that I am not fully convinced that we should insist that “a persistence module is a persistent vector space”, but I gather that a globally convincing convention may not be possible and I don’t feel strongly about it.

So if you feel like undoing my renaming of the entry, please feel free to do so (or let me know and I do it.)