Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2022

    starting someting – not done yet but need to save

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2022

    now some substance in place:

    an Idea-section (here),

    the definition due to FK17 (here)

    with a graphics illustrating its simplest instructive example (here)

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 22nd 2022

    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 (,1)(\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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2022

    Maybe up to a slight distinction:

    One could declare that for any filtered diagram DD, the corresponding persistent (co)homology is that internal to the \infty-topos over DD. 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.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 23rd 2022

    That make sense. I wonder what’s the abstract general of why these fundamental theorems work.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2022
    • (edited May 23rd 2022)

    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.

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2022

    where in the Examples-section it had “it is intuitively clear that…” I have now added (here) an animated graphics illustrating this

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2022
    • (edited May 26th 2022)

    have now added a pointer to the slides that I have been preparing: New Foundations for TDA – Cohomotopy

    diff, v9, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)