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replaced various occurences of “map” by more specific terms (bundle morphism, smooth function, continuous function)
Added plural redirect.
There is a constructive eversion via the Roseman moves of 2-knot theory. Scott Carter has more than one book discussing it, it is a beautiful topic. But I would be astonished if HoTT can express it.
I would love a proof assistant for knot theory, and would even try to write one, but I have no idea even in theory how to formulate a good syntactics that captures both local moves and less local considerations. Somehow the fact that diagrammatic knot theory lives in the plane, and the geometric manipulations that are possible in the plane, seem essential, and I have no idea how to capture this 2D aspect syntactically.
Now is there any possibility of proving this with differential cohesive HoTT?
Sounds like a major research program. For comparison, the problems that I started to make a list of (here) are meant to be essentially “formalization ready”: We have a formal classical proof, or at least a detailed sketch of one. It’ still a bit of work from there to a computer checked proof, but at least there is a clear strategy.
Here you seem to be asking pretty generic questions, in comparison. Any questions relating to the actual smooth $n$-spheres will first need to solve the problem of narrowing in the synthetic language on the “standard model”, modeled on standard smooth manifolds.
As I had tried to say elsewhere, maybe one way to approach this is to impose the following axioms to differential cohesion:
ask shape to be homotopy localization at a homogeneous type $\mathbb{A}^1$ (as in Def. 4.8 in Felix’s arXiv:1806.05966)
ask for an identification $\flat(\mathbb{A}^1) \simeq \mathbb{R}^1_{Cauchy}$
require that composed with the counit $\mathbb{R}^{1}_{Cauchy} \simeq \flat(\mathbb{A}) \to A$ this respects homogeneous type structure.
Intuitively it seems that this should make $\mathbb{A}^n$-manifolds (Def. 7.1), hence in particular $n$-spheres, behave much like actual smooth $n$-manifolds, maybe enough so to prove some classical theorem.
Do you happen have a more precise source for this than Francis’ note? It remains a little unclear what kind of “fibering” he is referring to, and what exactly the space $Map(M,N)$ is.
In Lecture 3 he specifies that the fibering is the forgetful map to smooth functions of manifolds. But he notes also that
The base space $Map^{sm}(M,N)$ is a comparatively easy space to study since it is homotopy equivalent to all continuous maps $Map(M,N)$
Meanwhile, this paper sets out defining formal immersions in terms of underlying continuous maps. Hmm, is there consensus?
Anyway the story needs to continue to see how Smale’s theory is an example of the h-principle, for some open $Diff(M)$-invariant differential relation on $M$, a section of which is a formal immersion.
the space of all formal immersions fibers over the space of smooth maps $M \to N$, with fiber over some $f:M \to N$ being the space of injective maps $T M \to f^{\ast}T N$ over $M$. The space of smooth maps is homotopy equivalent to the space of all continuous maps (due to integrating against a smoothing kernel), whose homotopy type might be more accessible (for example vanish). (Homotopy Sheaves and h-Principles).
So let’s keep with smooth maps, and point out the homotopy equivalence to all continuous maps.
Okay, thanks! That would be good to make explicit in the entry. What kind of fibration is it?
I’m not sure exactly.
I added that Smale-Hirsch immersion theory is an h-principle situation.
Okay, let’s maybe make a note to check later. Also, this will depend on the actual choice of spaces, not just their homotopy types. (I mean, if there is to be a chance to formalize the argument, these details will matter.)
Sure.
To my amateur eye it seems that there might be parts of this domain which are amenable to differential cohesive formalization. I’m not sure at what degree of generality it would be best to attack. Some statements at least seem expressible:
Consider a smooth $n$-manifold $M$ and a smooth fibration over it, $E \to M$, and $\mathcal{R}$ a subspace of the $k$-th jet space of $E$, i.e., a differential relation. If $M$ is open as a manifold, and $\mathcal{R}$ is an open subspace and is $Diff(M)$-invariant, then the image (under formation of $k$-jets) of smooth sections of $E$ in smooth sections of $\mathcal{R}$ is weakly homotopy equivalent to smooth sections of $\mathcal{R}$.
Is that expressible with your additional axioms from #5?
The proposition itself (not its proof) will have a fairly straightforward expression in HoTT equipped with an infinitesimal shape modality, not needing any further axioms.
Now the proof, I don’t know. Very likely this will require further axioms. To see that, it would be useful to have a very clean version of the classical proof of this fact laid out, and its key ingredients isolated.
A proof is given in Holonomic approximation and Gromov’s h-principle, but I doubt that would be easy to formalize, and also quite concisely here.
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