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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2013
   • (edited Feb 11th 2021)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI've mention a couple of times, e.g., in this [thread](https://nforum.ncatlab.org/discussion/5540/kleisli-category-for-a-monadoid/?Focus=43926#Comment_43926), 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeDec 19th 2013
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   Author: Mike Shulman
   Format: MarkdownItexOne 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](http://www.math.mcgill.ca/rags/difftl/difftl.ps.gz) -- 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexJust to check, his tangent categories are the same as our [[tangent (infinity,1)-categories]]? At least they agree for $Top$ and $E_{\infty}-Ring$.

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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, they are the same

   Yes, they are the same

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBut 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 $Top$?

   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 $Top$?

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2013
   • (edited Dec 19th 2013)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOh no, that's only for $\mathbf{H}$ a topos. But still, we shouldn't be just using $Top_{\ast}$?

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

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's a typo!

   Yes, that’s a typo!

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 19th 2013
   • (edited Dec 19th 2013)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOK, so should we be interested in $Exc^1(\mathbf{H},\mathbf{H}^{\ast/})$? Ought we to have an entry for cotangent $(\infty, 1)$-category?

   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?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013
    • (edited Dec 19th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes 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

    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

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, 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]]_.

    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.

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
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    Author: David_Corfield
    Format: MarkdownItexSo 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 calculus|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 $\mathcal{C}$.

    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 $\mathcal{C}$.

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexIn 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexEdit: 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 $Hom_{Ab}(\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}$?]

    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 $Hom_{Ab}(\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}$?]

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2013
    • (edited Dec 20th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, 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.

    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.

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRegarding 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.

    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.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 20th 2013
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    Author: David_Corfield
    Format: MarkdownItexSo 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?

    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?

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeDec 20th 2013
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    Author: Mike Shulman
    Format: MarkdownItexCould 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 $Sp$?

    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 $Sp$?

19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2013
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    Author: David_Corfield
    Format: MarkdownItexPerhaps 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](http://golem.ph.utexas.edu/category/2009/06/algebraic_geometry_for_categor.html)?

    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?