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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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 object 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
   • CommentTimeNov 3rd 2010
   • (edited Nov 3rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDomenico, Jim and myself have been trying to write out in LaTeX some of the things that we have been discussing at [[infinity-Chern-Weil theory]]. There is now a pdf document: Domenico, Urs, Jim, [[schreiber:Cocycles for differential characteristic classes]] Comments are welcome.

   Domenico, Jim and myself have been trying to write out in LaTeX some of the things that we have been discussing at infinity-Chern-Weil theory.

   There is now a pdf document:

   Domenico, Urs, Jim, Cocycles for differential characteristic classes (schreiber)

   Comments are welcome.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 3rd 2010
   • (edited Nov 3rd 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexTypo: Chern-Smimons, page 53 as of the moment. And mathrm without backslash in 4.2.12. Kan compex in 4.2.19.

   Typo: Chern-Smimons, page 53 as of the moment. And mathrm without backslash in 4.2.12. Kan compex in 4.2.19.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeNov 3rd 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere is an interesting thing with Poisson-Lie structure relevant for physics. A Lie group is Poisson-Lie if it is a Poisson manifold and the multiplication is a Poisson map. At infinitesimal level there is a notion of Lie bialgebra, the tangent bialgebra to a Poisson-Lie group is a Lie bialgebra. Now the Lie bialgebra here can be considered as a Lie algebra with a additional cocycle with values in the tensor square. Now, conversely, if we want to integrate a Lie bialgebra, then we can choose a connected, simply connected Lie group integrating a Lie algebra and the cocycle with integrate as well. But suppose that we start with a not-simply connected Lie group with the cocycle on the Lie algebra. Then one can not do this in general. I am a bit uneasy with this, and think that some sort of higher categorical repair could do something about the questionable integration of the cocycle, if I understand it in a bit generalized way.

   There is an interesting thing with Poisson-Lie structure relevant for physics. A Lie group is Poisson-Lie if it is a Poisson manifold and the multiplication is a Poisson map. At infinitesimal level there is a notion of Lie bialgebra, the tangent bialgebra to a Poisson-Lie group is a Lie bialgebra. Now the Lie bialgebra here can be considered as a Lie algebra with a additional cocycle with values in the tensor square. Now, conversely, if we want to integrate a Lie bialgebra, then we can choose a connected, simply connected Lie group integrating a Lie algebra and the cocycle with integrate as well. But suppose that we start with a not-simply connected Lie group with the cocycle on the Lie algebra. Then one can not do this in general. I am a bit uneasy with this, and think that some sort of higher categorical repair could do something about the questionable integration of the cocycle, if I understand it in a bit generalized way.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Typo: Chern-Smimons, page 53 as of the moment. And mathrm without backslash in 4.2.12. Thanks, Zoran! I fixed these in our local version for the moment.

   Typo: Chern-Smimons, page 53 as of the moment. And mathrm without backslash in 4.2.12.

   Thanks, Zoran! I fixed these in our local version for the moment.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Now the Lie bialgebra here can be considered as a Lie algebra with a additional cocycle with values in the tensor square. That's a nice fact. Could you record that in some nLab entry? > But suppose that we start with a not-simply connected Lie group with the cocycle on the Lie algebra. Then one can not do this in general. I am a bit uneasy with this, and think that some sort of higher categorical repair could do something about the questionable integration of the cocycle, if I understand it in a bit generalized way. good question. That's a special case of a general question that I don't have a general good answer for. The Lie integration of $L_\infty$-algebra cocycles lives naturally on some higher connected covers. The general question is: to which extent does that push down again to one of the quotients? I need to think more about this.

   Now the Lie bialgebra here can be considered as a Lie algebra with a additional cocycle with values in the tensor square.

   That’s a nice fact. Could you record that in some nLab entry?

   But suppose that we start with a not-simply connected Lie group with the cocycle on the Lie algebra. Then one can not do this in general. I am a bit uneasy with this, and think that some sort of higher categorical repair could do something about the questionable integration of the cocycle, if I understand it in a bit generalized way.

   good question. That’s a special case of a general question that I don’t have a general good answer for. The Lie integration of $L_\infty$-algebra cocycles lives naturally on some higher connected covers. The general question is: to which extent does that push down again to one of the quotients? I need to think more about this.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeNov 3rd 2010
   • (edited Nov 3rd 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> I need to think more about this. Probably I would not be of much help, but I would like to know how to possibly start thinking about this. I see a question and have some intuition about the answer, but not a cue of the strategy... Yes, I will put the quoted fact in some entry. Today or soon. I added above: typo Kan compex in 4.2.19.

   I need to think more about this.

   Probably I would not be of much help, but I would like to know how to possibly start thinking about this. I see a question and have some intuition about the answer, but not a cue of the strategy…

   Yes, I will put the quoted fact in some entry. Today or soon.

   I added above: typo Kan compex in 4.2.19.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2010
   • (edited Nov 3rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> typo Kan compex in 4.2.19. Thanks, have fixed that in the local copy. > I would like to know how to possibly start thinking about this. I might need a more quiet moment to recollect my thoughts on this, but a first observaiton is this: we are asking for the reverse process of forming fractional characteristic maps: traditionally we start with a cocycle $\mathbf{B}G \to \mathbf{B}^n (\mathbb{Z} \to \mathbb{R})$ and find that after refining to a higher cover $\mathbf{B}\hat G$ it can be quotiented $$ \array{ \mathbf{B}\hat G &\stackrel{\frac{1}{k} c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow^{\cdot k} \\ \mathbf{B}G &\stackrel{c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) } \,. $$ So now we instead start with a diagram $$ \array{ \mathbf{B}\hat G &\stackrel{c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \\ \downarrow \\ \mathbf{B}G } $$ (or similar) and ask if we can complete it. If we put ourselves in the context of an $\infty$-topos, then of course we have the universal completion by the pushout. So the question is: can we usefully project out from the pushout an abelian class $$ \mathbf{B}G \coprod_{\mathbf{B}\hat G} \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \to A \,. $$

   typo Kan compex in 4.2.19.

   Thanks, have fixed that in the local copy.

   I would like to know how to possibly start thinking about this.

   I might need a more quiet moment to recollect my thoughts on this, but a first observaiton is this:

   we are asking for the reverse process of forming fractional characteristic maps: traditionally we start with a cocycle $\mathbf{B}G \to \mathbf{B}^n (\mathbb{Z} \to \mathbb{R})$ and find that after refining to a higher cover $\mathbf{B}\hat G$ it can be quotiented

   $$\array{ \mathbf{B}\hat G &\stackrel{\frac{1}{k} c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow^{\cdot k} \\ \mathbf{B}G &\stackrel{c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) } \,.$$

   So now we instead start with a diagram

   $$\array{ \mathbf{B}\hat G &\stackrel{c}{\to}& \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \\ \downarrow \\ \mathbf{B}G }$$

   (or similar) and ask if we can complete it.

   If we put ourselves in the context of an $\infty$-topos, then of course we have the universal completion by the pushout. So the question is: can we usefully project out from the pushout an abelian class

   $$\mathbf{B}G \coprod_{\mathbf{B}\hat G} \mathbf{B}^n (\mathbb{Z} \to \mathbb{R}) \to A \,.$$
8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeNov 3rd 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThanks, this is a useful point of view. Thanks for writing it down clearly. I added few references into [[Pontrjagin class]].

   Thanks, this is a useful point of view. Thanks for writing it down clearly.

   I added few references into Pontrjagin class.