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have expanded a bit more, but it remains preliminary.
This might deserve to be called “stability theorem”, because that’s how the result is referred to in the TDA literature, but unfortunately it’s too unspecific a term for use outside of that community.
I am hoping to say more on the relation to well groups, but I don’t understand that well enough yet.
Is this interest of yours related to its cohomotopic aspect?
Yes, I should finally write that up on the nLab.
But also, I wanted to list and explain the actual main theorems in persistent homology theory.
The first main theorem, in hindsight, is Gabriel’s theorem. This serves to establish that persistence really is a concept, even for zigzag-modules.
Next the stability theorem here. This establishes that persistence is a useful concept.
Is there a further theorem of this rank?
Maybe not at this point. But the cohomotopy story might help show that the restriction to linear/homological data that must be assumed for these theorems is not actually truthful to the subject matter.
I did read a little on the subject a while ago, but never gained much sense of key results. I guess the theorems mentioned in surveys such as Ghrist’s should be important, such as Theorem 2.3.
There’s an isometry theorem mentioned on slide 21 of the talk Categories for persistent homology.
Interesting to see you say this, because it shows how intransparent the literature is, compared to the relative triviality of its subject matter: Because these two theorems are just the two we are talking about!
Namely theorem 2.3 in Ghrist is just Gabriel’s theorem generalized a little to infinite quivers, and the isometry theorem is just a little strengthening of the stability theorem.
The diamond principle would be another real theorem in persistence. I dont yet have a feeling for just how important this is.
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