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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2010
    • (edited Feb 3rd 2010)

    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.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeNov 9th 2012

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

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 9th 2012

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2012
    • (edited Nov 9th 2012)

    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 ΣX\Sigma \to X out of a manifold Σ\Sigma with boundary and with boundary data held fixed (this includes all the variational problems in classical physics) the domain is the pullback

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

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

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeNov 9th 2012

    I added the Perugia reference to variational calculus.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2012

    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.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 13th 2012

    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.