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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 23rd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexpolished and expanded [[Ehresmann connection]]

   polished and expanded Ehresmann connection

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 30th 2011
   • (edited Nov 30th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI have improved (hopefully) the section on the definition via horizontal subspaces at [[Ehresmann connection]]. On the other hand, I think (and [wikipedia](http://en.wikipedia.org/wiki/Ehresmann_connection) agrees) that the statement about the terminology is wrong at two places. One is the statement in the entry that the Ehresmann connection must be on a principal bundle (but must be on a fiber bundle) to be called such and another is suspicious phrase "Cartan-Ehresmann connection", in my opinion [[Cartan connection]] is by the definition in a smaller generality then Ehresmann. Finally the Ehresmann connections on a principal and its associated bundles are in 1-1 correspondence: If $T^H P\subset T P$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P\times_G F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map $\rho_f : P\to P\times_G F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F$ defines the horizontal subspace $T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^H P\times_G F$ is a connection on the associated bundle $P\times_G F\to X$. I added now this reasoning to the entry as well.

   I have improved (hopefully) the section on the definition via horizontal subspaces at Ehresmann connection. On the other hand, I think (and wikipedia agrees) that the statement about the terminology is wrong at two places. One is the statement in the entry that the Ehresmann connection must be on a principal bundle (but must be on a fiber bundle) to be called such and another is suspicious phrase “Cartan-Ehresmann connection”, in my opinion Cartan connection is by the definition in a smaller generality then Ehresmann.

   Finally the Ehresmann connections on a principal and its associated bundles are in 1-1 correspondence: If $T^H P\subset T P$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P\times_G F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map $\rho_f : P\to P\times_G F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F$ defines the horizontal subspace $T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^H P\times_G F$ is a connection on the associated bundle $P\times_G F\to X$. I added now this reasoning to the entry as well.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2012
   • (edited Jun 28th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[Ehresmann connection]]_ a pointer to the formalization of flat Ehresmann connections in [[cohesive homotopy type theory]]. I have just posted a little more chat about this _[here](http://golem.ph.utexas.edu/category/2012/06/flat_ehresmann_connections_in.html)_ to the $n$Café.

   I have added to Ehresmann connection a pointer to the formalization of flat Ehresmann connections in cohesive homotopy type theory.

   I have just posted a little more chat about this here to the $n$Café.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 14th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a link to the PDF file of the original paper. <a href="https://ncatlab.org/nlab/revision/diff/Ehresmann+connection/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ehresmann+connection/19">v19</a>, <a href="https://ncatlab.org/nlab/show/Ehresmann+connection">current</a>

   Added a link to the PDF file of the original paper.

   diff, v19, current

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeFeb 16th 2021
   • PermaLink
   Author: zskoda
   Format: MarkdownItexUrs, 3: link to $n$Café seem not to work.

   Urs, 3: link to $n$Café seem not to work.

6. • CommentRowNumber6.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 16th 2021
   • (edited Feb 16th 2021)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI wrote: >Even if you go to the café and find that [day's entry](https://golem.ph.utexas.edu/category/2012/06/index.shtml), the follow up gives the same error. The problem is thus in the Café. but when I checked back the problem did not seem to exist any more. Strange.

   I wrote:

   Even if you go to the café and find that day’s entry, the follow up gives the same error. The problem is thus in the Café.

   but when I checked back the problem did not seem to exist any more. Strange.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeFeb 17th 2021
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIt does again. Though the main page works.

   It does again. Though the main page works.

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 17th 2021
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThere is no problem for me at the moment.

   There is no problem for me at the moment.