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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2013
   • (edited May 3rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt just occurred to me that there is an immediate axiomatization of the [[Liouville-Poincaré 1-form]] (the canonical differential 1-form on a cotangent bundle) in [[differential cohesion]]. In fact, it is the special case of a much more general notion: for $A$ any type in differential cohesion the total space $\underset{X}{\sum} \mathcal{O}_X(A)$ of the $A$-valued structure sheaf over any $X$ carries a canonical $A$-cocycle. For $A = \Omega^1$ the sheaf of 1-forms and $X$ a manifold, this is the traditional Liouville-Poincaré 1-form on $T^* X$. I made a quick note on that at _[differential cohesion -- Liouville-Poincaré cocycle](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#PoincareCocycle)_. Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the [[BV-BRST complex]]. I still need to write that out. Not today though.

   It just occurred to me that there is an immediate axiomatization of the Liouville-Poincaré 1-form (the canonical differential 1-form on a cotangent bundle) in differential cohesion.

   In fact, it is the special case of a much more general notion: for $A$ any type in differential cohesion the total space $\underset{X}{\sum} \mathcal{O}_X(A)$ of the $A$-valued structure sheaf over any $X$ carries a canonical $A$-cocycle.

   For $A = \Omega^1$ the sheaf of 1-forms and $X$ a manifold, this is the traditional Liouville-Poincaré 1-form on $T^* X$.

   I made a quick note on that at differential cohesion – Liouville-Poincaré cocycle.

   Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the BV-BRST complex. I still need to write that out. Not today though.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 17th 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> traditional Liouville-Poincaré 1-form Google does not give any hits with phrase Liouville-Poincaré.

   traditional Liouville-Poincaré 1-form

   Google does not give any hits with phrase Liouville-Poincaré.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2013
   • (edited Jan 17th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAll sources on that 1-form say it is called the _Liouville form_ or the _Poincaré-form_. I thought if it's really due to both, one should give credit to both.

   All sources on that 1-form say it is called the Liouville form or the Poincaré-form. I thought if it’s really due to both, one should give credit to both.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJan 17th 2013
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   Author: zskoda
   Format: MarkdownItexThanks. I do know it as canonical form, indeed, <http://en.wikipedia.org/wiki/Tautological_one-form>. Liouville-Poincaré sounds OK a priori, but it may repel some of the outside users -- one has to be careful with new mixed names as they are expected to be different from the one with one name; for example Euler equation (in fluid dynamics) is completely different from say Euler-Lagrange equation (in variational calculus). I was just told be a student yesterday that he does not read $n$Lab any more, because the $n$Lab is like machine -- always gives a correct answer but unrecognizable and hi-brow even when he thinks he knows the notion. This was sad to hear because it was about a good student.

   Thanks. I do know it as canonical form, indeed, http://en.wikipedia.org/wiki/Tautological_one-form. Liouville-Poincaré sounds OK a priori, but it may repel some of the outside users – one has to be careful with new mixed names as they are expected to be different from the one with one name; for example Euler equation (in fluid dynamics) is completely different from say Euler-Lagrange equation (in variational calculus). I was just told be a student yesterday that he does not read $n$Lab any more, because the $n$Lab is like machine – always gives a correct answer but unrecognizable and hi-brow even when he thinks he knows the notion. This was sad to hear because it was about a good student.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 17th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe somebody know or enjoys to dig out the original sources? Chances are that the idea is originally neither to Liouville nor to Poincaré! :-)

   Maybe somebody know or enjoys to dig out the original sources? Chances are that the idea is originally neither to Liouville nor to Poincaré! :-)

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeJan 17th 2013
   • PermaLink
   Author: zskoda
   Format: MarkdownItexYes, likely :) it could be instructive, how they were lead there, it is always instructive when it is about such great minds :)

   Yes, likely :) it could be instructive, how they were lead there, it is always instructive when it is about such great minds :)