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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.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am about to create [[D-scheme]], but currently the Lab is down and the server does not react to my login attempts...

   I am about to create D-scheme, but currently the Lab is down and the server does not react to my login attempts…

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I have created an entry [[D-scheme]]

   okay, I have created an entry D-scheme

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 17th 2011
   • (edited Jun 17th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere is such an entry already! In a variant, [[diffiety]]. See also [[D-geometry]], and especially [[crystal]].

   There is such an entry already! In a variant, diffiety. See also D-geometry, and especially crystal.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay. You need to remember including the relevant redirects when you create an entry! I'll inlcude them now.

   Okay. You need to remember including the relevant redirects when you create an entry! I’ll inlcude them now.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 17th 2011
   • (edited Jun 17th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWell, I am not sure. Maybe we indeed want to have [[D-scheme]] separate from [[diffiety]]. It is almost the same, but formally not entirely. I had that in mind then. But I stopped the activity on the [[D-geometry]] at the time, abruptly because there was no support/interest from the rest of the n-community at the time. Edit: why did you add toc for higher geometry -- this is a notion of 1-categorical algebraic geometry. It can be viewed higher categorical, but it is optional. I would be much happier to consider it part of the cluster [[D-geometry]] as most of the practioners (cited references) do.

   Well, I am not sure. Maybe we indeed want to have D-scheme separate from diffiety. It is almost the same, but formally not entirely. I had that in mind then. But I stopped the activity on the D-geometry at the time, abruptly because there was no support/interest from the rest of the n-community at the time.

   Edit: why did you add toc for higher geometry – this is a notion of 1-categorical algebraic geometry. It can be viewed higher categorical, but it is optional. I would be much happier to consider it part of the cluster D-geometry as most of the practioners (cited references) do.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have kept the entries separately, just cross-linked them. I had already renamed the toc to just "Geometry". We just have a single such toc. I used to call it "Higher geoemtry", but it should just be called "Geometry". Around here everything is higher by default

   I have kept the entries separately, just cross-linked them.

   I had already renamed the toc to just “Geometry”. We just have a single such toc. I used to call it “Higher geoemtry”, but it should just be called “Geometry”.

   Around here everything is higher by default

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexStill, I will once replace this by lower circle D-geometry once. It is more closely related circle, but I have abosolutely no time to work tyhese days on it.

   Still, I will once replace this by lower circle D-geometry once. It is more closely related circle, but I have abosolutely no time to work tyhese days on it.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn any case, [[crystal]] of qcoh modules is equivalent to a D-module and crystal of schemes to a D-scheme. See the exposition by Lurie from Gaitsgory's seminar linked at [[crystal]].

   In any case, crystal of qcoh modules is equivalent to a D-module and crystal of schemes to a D-scheme. See the exposition by Lurie from Gaitsgory’s seminar linked at crystal.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, Thanks. I know Lurie's notes, but apparently I didn't read them thoroughly. So I was proud to have figured out that the Jet scheme is just the change of base along the de Rham space projection. But sure enough that's what Lurie says on p. 6 there. Thanks.

   Ah, Thanks. I know Lurie’s notes, but apparently I didn’t read them thoroughly.

   So I was proud to have figured out that the Jet scheme is just the change of base along the de Rham space projection. But sure enough that’s what Lurie says on p. 6 there. Thanks.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexYou know, now that it is officially not my original idea, I can tell you my opinion about it without being constrained by modesty: this is great! :-) This is way better than what BD write. This makes D-geometry entirely general abstract and work in vast generality away from the original examples.

    You know, now that it is officially not my original idea, I can tell you my opinion about it without being constrained by modesty: this is great! :-) This is way better than what BD write. This makes D-geometry entirely general abstract and work in vast generality away from the original examples.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added to [[D-scheme]] pointers to Lurie's statement/proof of the assertion that $\mathcal{D}_X$-schemes are just schemes over $\mathbf{\Pi}_{inf}(X)$.

    I have added to D-scheme pointers to Lurie’s statement/proof of the assertion that $\mathcal{D}_X$-schemes are just schemes over $\mathbf{\Pi}_{inf}(X)$.

12. • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexThis looks like somewhat related to the relative connections in the sense of [[p-connection]]. The idea of jet scheme as a crystal of schemes is I think from Grothendieck's work on de Rham cohomology for alegbraic varieties. About de Rham space there is also fundamental idea in a paper of Kapranov in late 1980s where he had the idea for categories of complexes.

    This looks like somewhat related to the relative connections in the sense of p-connection. The idea of jet scheme as a crystal of schemes is I think from Grothendieck’s work on de Rham cohomology for alegbraic varieties. About de Rham space there is also fundamental idea in a paper of Kapranov in late 1980s where he had the idea for categories of complexes.

13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexI am not user if I remember right but I think it is about this article * M.M.Kapranov, ON DG-MODULES OVER THE DE RHAM COMPLEX AND THE VANISHING CYCLES FUNCTOR I have the article, but I am not very familiar with it, I looked just at few related details at the time when I looked at that subject. Edit: after taking a look, maybe not quite that. Some other apsects related to the subject. > This makes D-geometry entirely general abstract and work in vast generality away from the original examples. It makes sense whenever you have a resolution of diagonal.

    I am not user if I remember right but I think it is about this article

    • M.M.Kapranov, ON DG-MODULES OVER THE DE RHAM COMPLEX AND THE VANISHING CYCLES FUNCTOR

    I have the article, but I am not very familiar with it, I looked just at few related details at the time when I looked at that subject.

    Edit: after taking a look, maybe not quite that. Some other apsects related to the subject.

    This makes D-geometry entirely general abstract and work in vast generality away from the original examples.

    It makes sense whenever you have a resolution of diagonal.