Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 14th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhere did we get to with those discussions on what kinds of thickened points there are, or alternatively, what kinds of cohesive $(\infty, 1)$-categories there are? So, there are plain, smooth and synthetic differential versions of $\infty-Grpd$ and $super \infty-Grpd$. Was the idea that these are just low truncations of versions of some 'generalized' super $\infty-Grpd$? There was then the question of when the tangent to a cohesive $(\infty, 1)$-topos is also a cohesive $(\infty, 1)$-topos? We had $T(\infty Grpd) = parameterized spectra$, but what are $T(super \infty Grpd)$ and $T^2(\infty Grpd) = T(parameterized spectra)$? And, does $T(smooth \infty-Grpd)$ equal smooth parameterized spectra?

   Where did we get to with those discussions on what kinds of thickened points there are, or alternatively, what kinds of cohesive $(\infty, 1)$-categories there are?

   So, there are plain, smooth and synthetic differential versions of $\infty-Grpd$ and $super \infty-Grpd$. Was the idea that these are just low truncations of versions of some ’generalized’ super $\infty-Grpd$?

   There was then the question of when the tangent to a cohesive $(\infty, 1)$-topos is also a cohesive $(\infty, 1)$-topos?

   We had $T(\infty Grpd) = parameterized spectra$, but what are $T(super \infty Grpd)$ and $T^2(\infty Grpd) = T(parameterized spectra)$?

   And, does $T(smooth \infty-Grpd)$ equal smooth parameterized spectra?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 14th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi David, I still don't have much news on these questions, good as they are...

   Hi David,

   I still don’t have much news on these questions, good as they are…

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 14th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI was thinking that would make a fun story for philosophers - 'The history of the point', from Euclid's 'That which has no part' to the thickened point underlying a cohesive $(\infty, 1)$-topos.

   I was thinking that would make a fun story for philosophers - ’The history of the point’, from Euclid’s ’That which has no part’ to the thickened point underlying a cohesive $(\infty, 1)$-topos.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 15th 2013
   • (edited Aug 15th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi David, yes certainly. I would be more than happy to further expand on this. It's just somehow right now it feels more urgent to me to work out [more](http://ncatlab.org/nlab/show/motivic%20quantization) of the synthetic description of QFT in cohesive infinity-toposes, and that and its examples such as the "brane bouquet" is mostly absorbing me at the moment. We should find a student to look more into the topic of mapping the space of cohesion! :-)

   Hi David,

   yes certainly. I would be more than happy to further expand on this. It’s just somehow right now it feels more urgent to me to work out more of the synthetic description of QFT in cohesive infinity-toposes, and that and its examples such as the “brane bouquet” is mostly absorbing me at the moment.

   We should find a student to look more into the topic of mapping the space of cohesion! :-)

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 16th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSure there are golden nuggets everywhere, and who better than you to choose where to prospect. I wonder though, in a spirit of speculation, if you had an army of postdocs, where they might be pointed. You'd think it would be profitable to have some head off to homotopy theory in search of stable cohesion. And there must be a way to link up with number theory. I wonder if >[[Arithmetic Chow groups]] are refinements of ordinary Chow groups analogous to how ordinary differential cohomology refines ordinary cohomology, might be a clue.

   Sure there are golden nuggets everywhere, and who better than you to choose where to prospect.

   I wonder though, in a spirit of speculation, if you had an army of postdocs, where they might be pointed. You’d think it would be profitable to have some head off to homotopy theory in search of stable cohesion. And there must be a way to link up with number theory.

   I wonder if

   Arithmetic Chow groups are refinements of ordinary Chow groups analogous to how ordinary differential cohomology refines ordinary cohomology,

   might be a clue.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 16th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHmm, if the analogy above is roughly right and from [here](http://andreasholmstrom.org/thesis.pdf‎) >...Arakelov motivic cohomology...relates to motivic cohomology in the sense of Voevodsky roughly in the same way that the arithmetic Chow groups of Gillet and Soule relate to ordinary Chow groups, and if analogies compose, then is Holmstrom onto some kind of cohesion? Any experts on Arakelov motivic cohomology on hand to explain [I](http://arxiv.org/abs/1012.2523) and [II](http://arxiv.org/abs/1205.3890)?

   Hmm, if the analogy above is roughly right and from here

   …Arakelov motivic cohomology…relates to motivic cohomology in the sense of Voevodsky roughly in the same way that the arithmetic Chow groups of Gillet and Soule relate to ordinary Chow groups,

   and if analogies compose, then is Holmstrom onto some kind of cohesion?

   Any experts on Arakelov motivic cohomology on hand to explain I and II?

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 30th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexCould the refinement of algebraic K-theory to differential algebraic K-theory of [Bunke/Tamme](http://arxiv.org/abs/1209.6451) fit with those refinements in #5 and #6 above? This was [mentioned](http://nforum.mathforge.org/discussion/5313/global-equivariant-stable-homotopy-theory/?Focus=42287#Comment_42287) as a possible case of cohesive thickening. And Bunke and Tamme write > differential algebraic K-theory may also be seen as a variant of arithmetic algebraic K-theory, so somehow there's a relation between the differential and the arithmetic which brings us back to #5.

   Could the refinement of algebraic K-theory to differential algebraic K-theory of Bunke/Tamme fit with those refinements in #5 and #6 above? This was mentioned as a possible case of cohesive thickening. And Bunke and Tamme write

   differential algebraic K-theory may also be seen as a variant of arithmetic algebraic K-theory,

   so somehow there’s a relation between the differential and the arithmetic which brings us back to #5.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 30th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe. I need to study this.

   Maybe. I need to study this.

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 30th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHmm, I [see](http://arxiv.org/abs/1205.3890) >Arakelov motivic cohomology is a generalization of arithmetic K-theory and arithmetic Chow groups.

   Hmm, I see

   Arakelov motivic cohomology is a generalization of arithmetic K-theory and arithmetic Chow groups.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2013
    • (edited Oct 30th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThis is closely related to the [Bunke-Tamme article](http://arxiv.org/abs/1209.6451) which we discussed recently. Let me tell you what we are after next in this context: given that we have smooth (cohesive) versions of ordinary cohomology and of K-theory spectra, next we want a smooth version of the elliptic cohomoloy spectra. This should be not just probed by smooth manifolds, but also be parameterized over elliptic curves, clearly. Therefore we expect to build smooth elliptic cohomology in the $\infty$-topos $$ Sh_\infty(SmoothMfd, Sh_\infty(Schemes)) \longrightarrow Sh_\infty(Schemes) $$ which is cohesive over the $\infty$-topos over a suitable big site of suitable schemes.

    This is closely related to the Bunke-Tamme article which we discussed recently.

    Let me tell you what we are after next in this context:

    given that we have smooth (cohesive) versions of ordinary cohomology and of K-theory spectra, next we want a smooth version of the elliptic cohomoloy spectra. This should be not just probed by smooth manifolds, but also be parameterized over elliptic curves, clearly. Therefore we expect to build smooth elliptic cohomology in the $\infty$-topos

    $$Sh_\infty(SmoothMfd, Sh_\infty(Schemes)) \longrightarrow Sh_\infty(Schemes)$$

    which is cohesive over the $\infty$-topos over a suitable big site of suitable schemes.