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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 ∞-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 Cx (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∞−Ring.
Yes, they are the same
But in the case of E∞−Ring, we want E(E∞−Ring*,E∞−Ringx) don’t we to match our
Exc1(H*/,H)≃TH is the collection of parameterized spectra in H, hence the tangent (∞,1)-topos of H?
So it’s only E(Top*,Cx), when C is Top?
Oh no, that’s only for H a topos. But still, we shouldn’t be just using Top*?
Yes, that’s a typo!
OK, so should we be interested in Exc1(H,H*/)? Ought we to have an entry for cotangent (∞,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 𝒞, then the (∞,1)-category of n-excisive functors from the pointed objects in 𝒞 itself to 𝒞 behaves like the bundles of order-n Goodwillie derivatives over all objects of 𝒞. Hence this is the analog of the nth order jet bundle in Goodwillie calculus.
In particular for n=1 this is the tangent (∞,1)-category of 𝒞.
In Goodwillie’s proposal we have
TxC=E(Top*,Cx) and CoTxC=E(Cx,Top*), for the cotangent category.
Then since E(Cx,Dy)≃Hom(TxC,TyD), and T*Top*≃Sp, he can write
CoTxC=E(Cx,Top*)≃Hom(TxC,Sp).So he could also write
TxC=E(Top*,Cx)≃Hom(Sp,TxC).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 RD≃R×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 HomAb(ℤ,ℤ)=ℤ?]
Yes, right, have fixed it.
Concerning how to think of this as SDG: there is “2-algebraic geometry”, or rather “(∞,2)-algebraic geometry” where commutative rings or E∞-rings are replaced by symmetric monoidal ∞-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,∧) 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 ℤ as a free abelian group on one generator induces a product
HomAb(F(*),F(*))≃HomSet(*,UF(*)),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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