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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?
The idea first came up here, I think.
That’s righ, that’s where I first wrote this. But one should say that I wrote this following persistent prodding by you (David C.).
I still think that it seems clear that this is something that should eventually be important. But I haven’t thought about it any further yet.
Hey Urs,
So, these other things, you say, are in between abelian and non-abelian cohomology right? I mean, in my mind, there’s non-abelian cohomology, then 1-stable cohomology (e.g. with coeffs in a sheaf of spectrum objects) with coefficients in an element of the tangent category, then the next levels should somehow be “higher” than stable cohomology. For instance, quadratic, etc. Then, these should complete to some kind of… completed infinitesimal category on the category of spaces.
Does that make sense? Is that a decent way to think of it?
Yes, that’s the idea:
in a precise sense,
twisted nonabelian cohomology is the cohomology of the codomain fibration $\mathbf{H}^I \to \mathbf{H}$
by Goodwillie calculus the tangent space to this is the tangen (infinity,1)-topos $T \mathbf{H}\to \mathbf{H}$ whose cohomology is twisted stable cohomology.
So whatever you come up with in terms of “higher order Goodwillie jets” will be “in between” these two.
As David kept emphasizing, “Goodwillie jets” seems the most obvious thing to wonder about, in view of Goodwillie calculus. Do you know if anyone did have any thoughts on this?
Unfortunately, I’m not aware of any available work on this specific idea. I know that Gijs Heuts, a student of Jacob Lurie’s, has some interest in this topic. An excerpt from an e-mail from Jacob on the topic is:
“The tangent bundle to a presentable infty-category C can be identified with the infty-category of 1-excisive functors from pointed spaces (or spectra) into C. One version of n-jets would be to think about n-excisive functors instead (for n > 1, the distinction between spaces and spectra will matter). I think that Tom Goodwillie has put some thought into this sort of thing. My student Gijs Heuts has also been thinking about something along similar lines.”
I have a feeling also that it’s related to work on the completion tower of an operad done by Kathryn Hess and John Harper here .
If we have
$\array{ \Stab(\mathbf{H}) \\ \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.$and
$\array{ \mathbf{H} \\ \downarrow^{\mathrlap{incl}} \\ \mathbf{H}^I \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,,$what should we have as inclusion here
$\array{ ???(\mathbf{H}) \\ \downarrow^{\mathrlap{incl}} \\ J^n\mathbf{H} \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,?$I guess we call it $J^n_{\ast}(\mathbf{H})$.
Then, is there a construction for $J^n(\mathbf{H})$ parallel to the one giving the cotangent complex functor for $\mathbf{H}$ in the case of $T(\mathbf{H})$?
Well, could this possibly be related to the idea of “higher extensions”that are classified by higher hochschild cohomology groups? I forget quite how this story goes, but it’s something like the fact that the tangent space looks like square zero extensions, and then we want to take square zero extensions of square zero extensions, etc. Sorry this is really imprecise, it’s been a while since I’ve thought about this. I’m thinking in particular of the case of formal group laws, since that’s the sort of thing I worked on explicitly.
From this paper, it seems that $J^2_{\ast}(\infty Grpd)$ has as objects pairs $\{A_1, A_2\}$, where $A_1$ is a spectrum and $A_2$ is a spectrum with action of the symmetric group $\Sigma_2$, and some map relating $A_1$ and $A_2$ via the Tate construction.
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