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I’ve mention a couple of times, e.g., in this thread, some ideas of Tom Goodwillie on developing analogues of differential geometry for certain $\infty$-categories. For example, he speaks of two connections, a tangent and a cotangent one, deriving from base change and cobase change, and the difference between them amounting to smash product of spectra.
Tom has kindly shown me a grant proposal he wrote sketching these ideas. Perhaps we could have a discussion here about it. I can pass on the proposal to those interested.
One thing I would like to understand is whether these ideas can be fit into one of the existing categorical frameworks for differential geometry, such as synthetic differential geometry or differential categories – or perhaps a categorification thereof.
For instance, Goodwillie defines the tangent category of a pointed category $P$ to be the category $E(S_*,P)$ of excisive functors from pointed spaces to $P$, and the tangent category at $x$ of an unpointed category $C$ to be the tangent category of $C_x$ (the category of sectioned objects over $x$), serving as a sort of “infinitesimal neighborhood” of $x$ in $C$.
This seems to me reminiscent of the SDG definition of the tangent space at $x$ to a microlinear object $M$ as the set of maps from the “nilsquare line” $D$ into $M$ which take $0$ to $x$. Any such map factors through the “infinitesimal neighborhood” of $x$, and the condition of taking $0$ to $x$ seems perhaps analogous to excisivity? Moreover, $D$ is exactly the nilsquare neighborhood of $0$ in the standard line $R$, just as $S_*$ is the category of sectioned objects over the one-point space in the standard category $S$ of spaces. Is there anything here?
The condition that 0 be taken to x is explicit in the goodwillie setup. Apart from this excisiveness somehow encodes linearity. And this seems to be a crucial difference to sdg, where it is instead all maps out of D that give tangents.
Just to check, his tangent categories are the same as our tangent (infinity,1)-categories? At least they agree for $Top$ and $E_{\infty}-Ring$.
Yes, they are the same
But in the case of $E_{\infty}-Ring$, we want $E(E_{\infty}-Ring_{\ast}, E_{\infty}-Ring_x)$ don’t we to match our
$Exc^1(\mathbf{H}^{\ast/},\mathbf{H}) \simeq T\mathbf{H}$ is the collection of parameterized spectra in $\mathbf{H}$, hence the tangent (∞,1)-topos of $\mathbf{H}$?
So it’s only $E(Top_{\ast}, C_x)$, when $C$ is $Top$?
Oh no, that’s only for $\mathbf{H}$ a topos. But still, we shouldn’t be just using $Top_{\ast}$?
Yes, that’s a typo!
OK, so should we be interested in $Exc^1(\mathbf{H},\mathbf{H}^{\ast/})$? Ought we to have an entry for cotangent $(\infty, 1)$-category?
Yes but pointed H must be pointed InftyGrpd throughout. This is a bad typo in the entry. I just tried to fix it but difficult on the phone. Also the lab seems to be down again
Okay, back on my usual machine for a moment. I have restarted the nLab and then fixed that typo in exmaple 1 of n-excisive (∞,1)-functor.
So we also need to change at jet (infinity,1)-category?
Given a differentiable (∞,1)-category $\mathcal{C}$, then the (∞,1)-category of n-excisive functors from the pointed objects in $\mathcal{C}$ itself to $\mathcal{C}$ behaves like the bundles of order-$n$ Goodwillie derivatives over all objects of $\mathcal{C}$. Hence this is the analog of the $n$th order jet bundle in Goodwillie calculus.
In particular for $n = 1$ this is the tangent (∞,1)-category of $\mathcal{C}$.
In Goodwillie’s proposal we have
$T_{x}C = E(Top_{*}, C_x)$ and $CoT_{x}C = E(C_x, Top_{*})$, for the cotangent category.
Then since $E(C_{x}, D_{y}) \simeq Hom(T_x C, T_y D)$, and $T_{*} Top_{*} \simeq Sp$, he can write
$CoT_{x}C = E(C_x, Top_{*}) \simeq Hom(T_x C, Sp).$So he could also write
$T_{x}C = E(Top_{*}, C_x) \simeq Hom(Sp, T_x C).$So nothing surprising.
Edit: changed the previous comment, so this is now wrong.
[Hmm, rambling on, wouldn’t this make $Sp$ play the role of $D$ in SDG, and then also $R$? But what of $R^D \simeq R \times R$? Restricted to preservation of 0, that’s just $R$ on the right.
What is $Hom(Sp, Sp)$? Should one restrict to those preserving the zero object? Is this just $T_{*} Sp$, which is $Sp$ itself?
Is that just like $Hom_{Ab}(\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}$?]
Yes, right, have fixed it.
Concerning how to think of this as SDG: there is “2-algebraic geometry”, or rather “$(\infty,2)$-algebraic geometry” where commutative rings or $E_\infty$-rings are replaced by symmetric monoidal $\infty$-categories. Aspects of this appear for instance in Tannaka duality for geometric stack.
In such a context it would seem to be fairly straightforward to talk about higher analogs of SDG. In particular it seems that regarding $(Sp, \wedge)$ as a 2-algebra this way, there are useful statements obtained. For instance its prime ideals, hence the points of the “spectrum of spectra” apparently correspond to the Morava E-theory spectra. At least roughly, I may have to recall the details.
Now however in Goodwillie calculus the tensor product structure does not play a role. Instead it’s just the colimits and limits that somehow encode the structure of “varieties”. Somehow I suppose one needs to say this correctly and then the higher SDG analog would become apparent.
Regarding my #14, I’ve added the characterization to Spec of its universal property as the free stable locally presentable (infinity,1)-category on one generator.
So just as the universal property of $\mathbb{Z}$ as a free abelian group on one generator induces a product
$Hom_{Ab} (F(*), F(*)) \simeq Hom_{Set} (*, U F(*)),$did $Spec$ have to have a product?
Could we please not use $Spec$ for the category of spectra? To me $Spec$ is the functor of taking the spectrum of something, e.g. the Zariski spectrum of a ring. How about $Sp$?
Perhaps context would be enough to reduce to $Sp$, despite other uses, as in symplectic groups. But maybe something longer would be better. $Spect$ or $Spectra$.
Anyway, I’d like to hear more about Urs’ remarks in #15. Didn’t we have Cafe discussions about a 2-algebraic geometry in that tempestuous thread on Algebraic Geometry for Category Theorists?
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