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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2010
   • (edited Feb 3rd 2010)
   • PermaLink
   Author: Urs
   Format: Markdownstarted adding list of references to the page [[Bill Lawvere]] not that I made it very far -- just three items so far :-) I was really looking for an online copy of "Categorical dynamics" as referenced at [[synthetic differential geometry]] and [[generalized smooth algebra]], but haven't found it yet. I was thinking that the "Toposes of laws of motion" that I do reference must be something close. But I don't know.

   started adding list of references to the page Bill Lawvere

   not that I made it very far -- just three items so far :-)

   I was really looking for an online copy of "Categorical dynamics" as referenced at synthetic differential geometry and generalized smooth algebra, but haven't found it yet. I was thinking that the "Toposes of laws of motion" that I do reference must be something close. But I don't know.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI added a link to a web page that has a copy of Lawvere's "Perugia notes".

   I added a link to a web page that has a copy of Lawvere’s “Perugia notes”.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 9th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOh, fantastic, Mike! I've long wanted to have a look at those.

   Oh, fantastic, Mike! I’ve long wanted to have a look at those.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 9th 2012
   • (edited Nov 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the pointer. I hadn't seen that before. Happy to find that it matches well with _[variational calculus -- In terms of smooth spaces](http://ncatlab.org/nlab/show/variational%20calculus#InTermsOfSmoothSpaces)_. Concerning the pullback alluded to on the [very last page](http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf#page=14) which constrains the domain of variation, I take that as occasion to advertize what I find a pleasant effect of formalized cohesion on variational calculus: for the typical variational calculus on functionals on spaces of maps $\Sigma \to X$ out of a manifold $\Sigma$ with boundary and with _boundary data held fixed_ (this includes all the variational problems in classical physics) the domain is the pullback $$ (X^\Sigma)_{var} \coloneqq X^\Sigma \underset{X^{\partial \Sigma}}{\times} \flat (X^{\partial \Sigma}) \,, $$ where "$\flat$" is the modality of discrete cohesion.

   Thanks for the pointer. I hadn’t seen that before. Happy to find that it matches well with variational calculus – In terms of smooth spaces.

   Concerning the pullback alluded to on the very last page which constrains the domain of variation, I take that as occasion to advertize what I find a pleasant effect of formalized cohesion on variational calculus:

   for the typical variational calculus on functionals on spaces of maps $\Sigma \to X$ out of a manifold $\Sigma$ with boundary and with boundary data held fixed (this includes all the variational problems in classical physics) the domain is the pullback

   $$(X^\Sigma)_{var} \coloneqq X^\Sigma \underset{X^{\partial \Sigma}}{\times} \flat (X^{\partial \Sigma}) \,,$$

   where “$\flat$” is the modality of discrete cohesion.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeNov 9th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI added the Perugia reference to [[variational calculus]].

   I added the Perugia reference to variational calculus.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Zoran. I had wanted to do that last night, but then my battery died and my train stopped :-) I have expanded the comment line above the reference as follows: > A perspective on variational calculus in terms of internal hom mapping spaces in a context of cohesion along the lines discussed above has been indicated in... I have also tried to organize the References-section a bit more. But it's always hard to do so.

   Thanks, Zoran. I had wanted to do that last night, but then my battery died and my train stopped :-)

   I have expanded the comment line above the reference as follows:

   A perspective on variational calculus in terms of internal hom mapping spaces in a context of cohesion along the lines discussed above has been indicated in…

   I have also tried to organize the References-section a bit more. But it’s always hard to do so.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 13th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexThe Perugia notes are really nice! His introduction to his philosophy of sets is one of the strongest argument I've seen for structural foundations.

   The Perugia notes are really nice! His introduction to his philosophy of sets is one of the strongest argument I’ve seen for structural foundations.