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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2013
    • (edited Feb 11th 2021)

    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.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 19th 2013

    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 PP to be the category E(S *,P)E(S_*,P) of excisive functors from pointed spaces to PP, and the tangent category at xx of an unpointed category CC to be the tangent category of C xC_x (the category of sectioned objects over xx), serving as a sort of “infinitesimal neighborhood” of xx in CC.

    This seems to me reminiscent of the SDG definition of the tangent space at xx to a microlinear object MM as the set of maps from the “nilsquare line” DD into MM which take 00 to xx. Any such map factors through the “infinitesimal neighborhood” of xx, and the condition of taking 00 to xx seems perhaps analogous to excisivity? Moreover, DD is exactly the nilsquare neighborhood of 00 in the standard line RR, just as S *S_* is the category of sectioned objects over the one-point space in the standard category SS of spaces. Is there anything here?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013

    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.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2013

    Just to check, his tangent categories are the same as our tangent (infinity,1)-categories? At least they agree for TopTop and E RingE_{\infty}-Ring.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013

    Yes, they are the same

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2013

    But in the case of E RingE_{\infty}-Ring, we want E(E Ring *,E Ring x)E(E_{\infty}-Ring_{\ast}, E_{\infty}-Ring_x) don’t we to match our

    Exc 1(H */,H)THExc^1(\mathbf{H}^{\ast/},\mathbf{H}) \simeq T\mathbf{H} is the collection of parameterized spectra in H\mathbf{H}, hence the tangent (∞,1)-topos of H\mathbf{H}?

    So it’s only E(Top *,C x)E(Top_{\ast}, C_x), when CC is TopTop?

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2013
    • (edited Dec 19th 2013)

    Oh no, that’s only for H\mathbf{H} a topos. But still, we shouldn’t be just using Top *Top_{\ast}?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013

    Yes, that’s a typo!

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 19th 2013
    • (edited Dec 19th 2013)

    OK, so should we be interested in Exc 1(H,H */)Exc^1(\mathbf{H},\mathbf{H}^{\ast/})? Ought we to have an entry for cotangent (,1)(\infty, 1)-category?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013
    • (edited Dec 19th 2013)

    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

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013

    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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013

    So we also need to change at jet (infinity,1)-category?

    Given a differentiable (∞,1)-category 𝒞\mathcal{C}, then the (∞,1)-category of n-excisive functors from the pointed objects in 𝒞\mathcal{C} itself to 𝒞\mathcal{C} behaves like the bundles of order-nn Goodwillie derivatives over all objects of 𝒞\mathcal{C}. Hence this is the analog of the nnth order jet bundle in Goodwillie calculus.

    In particular for n=1n = 1 this is the tangent (∞,1)-category of 𝒞\mathcal{C}.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)

    In Goodwillie’s proposal we have

    T xC=E(Top *,C x)T_{x}C = E(Top_{*}, C_x) and CoT xC=E(C x,Top *)CoT_{x}C = E(C_x, Top_{*}), for the cotangent category.

    Then since E(C x,D y)Hom(T xC,T yD)E(C_{x}, D_{y}) \simeq Hom(T_x C, T_y D), and T *Top *SpT_{*} Top_{*} \simeq Sp, he can write

    CoT xC=E(C x,Top *)Hom(T xC,Sp). CoT_{x}C = E(C_x, Top_{*}) \simeq Hom(T_x C, Sp).

    So he could also write

    T xC=E(Top *,C x)Hom(Sp,T xC). T_{x}C = E(Top_{*}, C_x) \simeq Hom(Sp, T_x C).

    So nothing surprising.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)

    Edit: changed the previous comment, so this is now wrong.

    [Hmm, rambling on, wouldn’t this make SpSp play the role of DD in SDG, and then also RR? But what of R DR×RR^D \simeq R \times R? Restricted to preservation of 0, that’s just RR on the right.

    What is Hom(Sp,Sp)Hom(Sp, Sp)? Should one restrict to those preserving the zero object? Is this just T *SpT_{*} Sp, which is SpSp itself?

    Is that just like Hom Ab(,)=Hom_{Ab}(\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}?]

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)

    Yes, right, have fixed it.

    Concerning how to think of this as SDG: there is “2-algebraic geometry”, or rather “(,2)(\infty,2)-algebraic geometry” where commutative rings or E 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,)(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.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013

    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.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013

    So just as the universal property of \mathbb{Z} as a free abelian group on one generator induces a product

    Hom Ab(F(*),F(*))Hom Set(*,UF(*)), Hom_{Ab} (F(*), F(*)) \simeq Hom_{Set} (*, U F(*)),

    did SpecSpec have to have a product?

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2013

    Could we please not use SpecSpec for the category of spectra? To me SpecSpec is the functor of taking the spectrum of something, e.g. the Zariski spectrum of a ring. How about SpSp?

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2013

    Perhaps context would be enough to reduce to SpSp, despite other uses, as in symplectic groups. But maybe something longer would be better. SpectSpect or SpectraSpectra.

    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?