So, I was just reading the page on twisted cohomology, and I got really excited about the paragraph (sorry, I don’t know how to make it look “quotey”):

Higher-order approximations should involve a notion of higher-order forms of the tangent $(\infty,1)$-topos, in parallel with the relationship between the jet bundles and tangent bundle of a manifold. It is clear that whatever we may say in detail about the $k$th-jet $(\infty,1)$-topos $J^k\mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.

Is this paragraph explained at greater length anywhere?

I seem to recall some recent work of Ching and Arone on classifying second order approximations to the identity functor on spaces? Does this perhaps clarify what the next level of approximation might be? Or, in other words, what might be considered second order Eilenberg-Steenrod axioms?

I feel like this also goes back to a question of perennial interest of mine: if we think of the category of spectra as the tangent space to the category of spaces, then what’s the “quadratic space”? And what does it mean to complete with respect to this tower?

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