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Atrium > Mathematics, Physics & Philosophy: open problems in axiomatic cohesion

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2018
    • (edited Jul 18th 2018)

    Since Dan and Felix and others asked me to list good problems (i.e. potential theorems of interest whose proof should be within reach) in cohesive/elastic/solid modal homotopy theory, I started making a list here. Just a start, for the moment.

    In addition to the problem of formalizing the fundamental theorem of calculus (i.e. Stokes theorem, an issue we had discussed at some length here a while back) there is so far only one new item: To show that the bosonic body of a supermanifold is an ordinary manifold (here).

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 19th 2018

    How about something concerning PDEs? There was discussion of the h-principle.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2018
    • (edited Jul 19th 2018)

    I am not sure which proposition one would try to show regarding the h-principle. But since just stating it seems to be a nice example of combining coreduction with shape, I have added a remark to the list, under “further directions” (here)

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 20th 2018

    Could one generate some problems by specifying a cohesive/elastic/solid entity through a HIT and then looking to establish some property?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2018

    David, not quite sure what you mean to suggest here. Are you thinking of automatically generating propositions? It seems unlikely to me that this will be fruitful. Instead, it takes the usual creativity of research mathematics here to come up with the right statements, and then with their proofs.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2018
    • (edited Jul 20th 2018)

    Prodded by a request on the HoTT discussion group (here) I added an item Knot theoy to the list of “further directions” (i.e. the list of topics which should have a formalization in cohesive HoTT, but where I currently don’t know what a useful proposition to prove would be):

    The very idea of equivalence classes of knots (embeddings of smooth manifolds up to smooth isotopy) is one in differential cohesion (“elastic homotopy theory”), combining concepts of differential geometry (embeddings of smooth manifolds) with homotopy (a smooth isotopy between embeddings being a smooth homotopy/path in the smooth space of embeddings).

    This should lend itself to formalization in the differential cohesion (elasticity) fragment of (eq:TheProgressionOfModalities)

    ℜ ⊣ ℑ ⊣ & ∨ ∨ ʃ ⊣ ♭ ⊣ ♯ \array{ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && ʃ &\dashv& \flat &\dashv& \sharp }

    Using the infinitesimal shape modality ℑ\Im we may speak about smooth manifolds, as in Wellen 17, and form a homotopy type [Σ,X] emb↪[Σ,X][\Sigma,X]_{emb} \hookrightarrow [\Sigma,X] of embeddings of smooth manifolds. Then using the actual shape modality ʃʃ we obtain ʃ[Σ,X] embʃ [\Sigma,X]_{emb}. A path in this type is a smooth isotopy. Hence the 0-truncation

    Knots Σ(X)≔|ʃ[Σ,X] emb| 0 Knots_\Sigma(X) \;\coloneqq\; \big\vert ʃ [\Sigma,X]_{emb} \big\vert_0

    is the type of knots of form the Σ\Sigma in XX.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 20th 2018

    Re #5, no I just meant is there a chance to work out properties of specific entities rather than general results. Some of the first things achieved in synthetic homotopy theory were some homotopy groups of spheres. So how about proving something about a specific pde?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 20th 2018
    • (edited Jul 21st 2018)

    Or to tie in your #6 with my suggestion of specificity, how about establishing π 2V 2(ℝ 3)=0\pi_2 V_2(\mathbb{R}^3) = 0 showing all immersions of S 2S^2 into ℝ 3\mathbb{R}^3 are isotopic, mentioned in John Francis’s notes.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 20th 2018

    Well of course that’s the easy part. Establishing the Smale-Hirsch immersion theorem has to happen too.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2018

    Regarding “specific entities”, I am wondering if we can get around the issue of not having a constructive definition of the smooth ℝ 1\mathbb{R}^1 by “cancelling two ambiguities” against each other:

    If we first declare that shape is homotopy localization at unspecified type 𝔸 1\mathbb{A}^1 and then focus on the shape of 𝔸 n\mathbb{A}^n-manifolds (with 𝔸 n≔(𝔸 1) × n\mathbb{A}^n \coloneqq (\mathbb{A}^1)^{\times_n}) we might still get somewhere.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 21st 2018

    You said somewhere that in the analytic supercohesive case that 𝔸 1\mathbb{A}^1 would be ℂ\mathbb{C}, i.e., shape is localization there. What does this mean for what can be expected to be provable about 𝔸 n\mathbb{A}^n-manifolds?

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2018
    • (edited Jul 21st 2018)

    I just realized that long ago I had already written out a proof that Aufhebung ℑX⇝≃ℑX\overset{\rightsquigarrow}{\Im X} \simeq \Im X implies that

    η X ℑ⇝≃η X⇝ ℑ \overset{\rightsquigarrow}{\eta^{\Im}_X} \;\simeq\; \eta^{\Im}_{\overset{\rightsquigarrow}{X}}

    Now reproduced here.

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