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So the idea, it seems, is that we can have persistent forms of any kind of generalized cohomology theory.
What’s going on at the abstract level of cohomology as hom-space in an $(\infty, 1)$-topos? Something like the hom-space of a filtered object in an (∞,1)-category to some coefficient space? I guess one could filter the latter too.
Maybe up to a slight distinction:
One could declare that for any filtered diagram $D$, the corresponding persistent (co)homology is that internal to the $\infty$-topos over $D$. This would include filtered coefficients, yes, which is a generality that is not currently being considered.
But for such a “generalized (co)homology theory of filtered spaces” to deserve being related to “persistency”, there ought to be – I suppose – some analog of the “first fundamental theorem of persistence” (here, we recently talked about it) which guarantees that some barcode of persistent invariants may always be extracted, and then some analog of the “second fundamental theorem” (namely the stability theorem, which we had talked about here), guaranteeing that these invariants don’t vary erratically with the underlying filtered data.
Indeed, in the generalization to persistent homotopy, this stability result is the first thing people looked into establishing (cf. Blumberg-Lesnick 17, Jardine 19 – both remaining unpublished, though?!)
I have currently no idea how much this result would generalize away from the base $\infty$-topos.
That make sense. I wonder what’s the abstract general of why these fundamental theorems work.
I have further expanded (and slightly re-arranged) the Idea-section:
one new sub-section (here) highlights how persistent Cohomotopy provides an effective answer to the question for data points meeting fixed but noisy targets indicators. (this is the same material that I just added at TDA, too)
Another one (here) briefly comments on the relation to framed cobordism theory, via Pontryagin’s theorem.
have now added a pointer to the slides that I have been preparing: New Foundations for TDA – Cohomotopy
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