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I'm reading these notes

Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M \to N$.)

I've gathered from context that there is a connection between formal immersions and the sheaf of sections $\mathcal{F}(U):=\Gamma(V_n(TU) \times_{GL_n} \mathrm{Imm}(\mathbb R^n,N))$ (defined for an arbitrary open $\mathbb R^n \to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don't know why, although I assume it agrees in the sense of functor calculus.

My questions are the following: is the topological sheaf of formal immersions isomorphic to $\mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the ``obvious" maps $\mathrm{Imm}(U,N) \to \mathrm{Imm}^f(U,N),\,\,\, f \mapsto (df,f)$?

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  • $\begingroup$ The last obvious map is 1-jet prolongation. The basic motivation here is to realize Hirsch-Smale as an instance of the H-principle in the sense of the posted notes (via the scanning map) $\endgroup$
    – Andres Mejia
    Commented Jun 11, 2020 at 0:12
  • $\begingroup$ I think the idea is that we can view a formal immersion as immersing a neighborhood of p into the tangent space of $g(p)$. This coincides with how we view the linearization of a diff invariant sheaf because the fiber over a point of the linearization is a colimit of the sheaf on values of trivial neighborhoods, and these trivial neighborhoods are exactly what the tangent space approximates via the exponential map (which we use to define the scanning map). $\endgroup$
    – Connor Malin
    Commented Jun 11, 2020 at 15:45
  • $\begingroup$ @ConnorMalin thank you for your comment, it was very helpful! What remains for me is writing down an explicit map (and hopefully it’s inverse.) I will try to work it out later today using your insights. $\endgroup$
    – Andres Mejia
    Commented Jun 11, 2020 at 18:31
  • $\begingroup$ You probably need to choose a type of exponential map to define the isomorphism. Probably a good idea to choose this so it coincides with the exponential used to define the scanning map if you want the square to commute. $\endgroup$
    – Connor Malin
    Commented Jun 11, 2020 at 22:05
  • $\begingroup$ Check out section 37 of these notes: people.math.harvard.edu/~kupers/teaching/272x/book.pdf . To be linear (at a manifold) is to have the map to the first taylor approximation a weak equivalence. It seems to me after reading this that satisfying the h-principle is equivalent to being linear. $\endgroup$
    – Connor Malin
    Commented Jun 19, 2020 at 16:30

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