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The page in the nLab on tangent \infty-category is really awesome. Does anyone here know if this can be extended to higher degree “approximations” or is all the relevant information contained in the tangent category?
Thanks!
The page in the nLab on tangent (infinity,1)-category is really awesome.
Thanks!
if this can be extended to higher degree “approximations” or is all the relevant information contained in the tangent category?
You are probably asking if there is something like a ” jet bundle $(\infty,1)$-category”, is that right?
Hm, I don’t know. Interesting question.
Yeah, that’s precisely what I’m asking!
I’ve been reading some of Lazard’s old stuff on formal groups and looking at some deformation theory stuff, and it all seems to fit into this much more general framework. If we consider S-modules (where by S I mean the sphere spectrum), to be the tangent category (at S?) to the category of S-algebras, and we recall that at least in algebra this is the same thing as square-zero extensions of S, which is controlled by Hochschild homology, one starts to wonder if there are categories corresponding to “cube-zero” extensions, etc. Or perhaps non-unital S-algebras with higher degrees of nilpotence. Somehow these categories would lose the quality of being non-linear, but perhaps this is related to the Goodwillie calculus stuff as well.
Perhaps you already knew about this Urs, but this seems to be related to the page:
http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+–+infinitesimal+cohesion#JetBundleObjects
though I’d have to find a translator. :)
Perhaps you already knew about this Urs, but this seems to be related to the page:
Yes I know about that: I came up with the idea and wrote that page, in particular the section jet bundle objects.
But this is a formalization of $\infty$-jet bundles which is a priori different from that via tangent (infinity,1)-categories, which you asked about above.
Which is not to say that one could not maybe connect the two. Given a cohesive $\infty$-site $\mathcal{C}$, one could try to use its tangent $\infty$-category $T \mathcal{C}^{op}$ as an infinitesimal neighbourhood site and if that is made to work, then inside the resulting differential cohesive oo-topos one has a notion of $\infty$-jet bundles induced from the original tangent $\infty$-category.
That would morally be the $\infty$-jet version of the original tangency construction. But it is a construction quite a bit remote from a would-be explicit generalization of the construction of the tangent $\infty$-category itself.
At the very least there is a Goodwillie Calculus in various $\infty$-topoi, which is about higher derivatives. However, the notion of infinitessimal there isn’t a map, but an arbitrarily-connected map (which notion must make sense to have a Goodwillie Calculus…). I’m still trying to learn it myself, so I can’t say much more.
Hm, true, but on the other hand this sort of is subsumed by the original tangent (infinity,1)-catgeory-construction, in the following sense:
As Jacob Lurie nicely explains at then end of his article, Goodwillie calculus is about approximating the missing functoriality of the stabilization construction on suitable $\infty$-categories. Now the tangent $\infty$-category of some suitable $\infty$-category is the fiberwise stabilization of its codomain fibration. So in some sense at least the tangent $\infty$-category does accomodate for all Goodwillie derivatives.
Of course that’s a bit vague. Can you maybe say a bit more precisely what it is that you are actually after?
So unfortunately, I no nothing about the Goodwillie calculus story. I mean, I know the barest bones of the Goodwillie calculus, but have yet to see anything concrete.
A lot of this for me came from working with formal varieties, for instance in Lazard’s old stuff on formal groups. He defines a formal variety over $A$ to be a functor from the category of nilpotent (necessarily non-unital) $A$-algebras to pointed sets. There are a lot of different ways to talk about this story I think. My way might be a little outdated. The point is, algebras $X$ such that $X^2=0$ can be identified with $A$-modules. This is the tangent category (to the category of rings at the “point” $A$). Functors from the category of square-zero $A$-algebras to pointed sets can be thought of as something that is to formal varieties as 2-buds are to formal group laws. Then, algebras $X$ such that $X^3=0$ are sort of, the next level up. Somehow, when we get our hands dirty and start thinking about these things in terms of formal power series over $A$, we basically see that we’re building Taylor series, hence my question’s title.
Okay, so Lurie, in DAG IV, has this category $Ring^{+}$ of pairs $(A,M)$ where $A$ is a ring and $M$ is an $A$-module. It maps into $Fun(\Delta^1,Ring)$, where $(A,M)$ maps to the inclusion morphism, I believe, $A\to A\oplus M$. And the left adjoint to the map $Ring^+\to Fun(\Delta^1,Ring)\to Ring$ is the construction of the cotangent complex. Anyway, I feel like there should be some kind of simplicial category $\ldots Fun(\Delta^n,C)\to\ldots\to Fun(\Delta^2,C)\to Fun(\Delta^1,C)\to C$, and we’re currently only looking at the first level of this thing.
Have you looked back at Quillen’s cohomology of comm. algebra papers, and Luc Illusie’s thesis (in SLN about 1973). Going back to that will show some of the basics behind Lurie’s construction. (I think that the old papers a re often a good place to go, although the more recent stuff then needs to be tackled.)
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