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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2011
    • (edited Apr 28th 2011)

    I have written an “exegesis” of Lawvere’s Some Thoughts on the Future of Category Theory (see that link).

    A version of this I have also posted in reply to this MO question

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeApr 29th 2011

    Beautiful!

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeApr 29th 2011

    One detail: adjoint triple induces a monad left adjoint to a comonad. It does induce 2 monads and two comonads – as discussed in adjoint monad a right adjoint of a monad is a comonad. Do you really need/mean instead the monad part on the right adjoint when you say that it induces an “adjoint pair of monads” ? (it exists but is not conjugate to the first in any sense)

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 29th 2011

    By the way, under Examples in related entry local geometric morphism there is a place where it says focal point. Is it meant local point ?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2011

    Beautiful!

    Thanks

    One detail: adjoint triple induces a monad left adjoint to a comonad. It does induce 2 monads and two comonads – as discussed in adjoint monad a right adjoint of a monad is a comonad. Do you really need/mean instead the monad part on the right adjoint when you say that it induces an “adjoint pair of monads” ?

    RIght, I mean comonadmonadcomonad \dashv monad. So I have changed “adjoint pair of monads” to adjoint monad.

    I also added to adjoint triple the statement that every such induces two adjoint monads, hence a total of 4 (co)monads. We had mentioned this previously at adjoint qudruple but not at the triple entry.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2011

    As far as smoothness, for smooth morphism of schemes there is a base change formula, and another similar for proper morphisms, which can be abstracted in the language of adjunctions and fibered categories. This approach to “smooth functors” is in recent Georges Maltsiniotis’s work

    • G. Maltsiniotis, Structures d’asphéricité, foncteurs lisses, et fibrations, Ann. Math. Blaise Pascal, 12, pp. 1-39 (2005), ps.
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2011

    Thanks, I’ll have a look at the article. Have you already absorbed this?

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2011

    No, I had some conversation with the author when in Seville, 2009, but I did not have enough background at the time to understand it quickly.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 24th 2014
    • (edited Aug 24th 2014)

    Added link to YouTube of a recording of Lawvere’s Como lecture.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 25th 2016

    My library doesn’t give me access to ’Some thoughts on the future of category theory’. Could someone mail me the file please (d.corfield@kent.ac.uk)?

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 25th 2016

    Enjoy

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 25th 2016

    Great! Thanks a lot.

    • CommentRowNumber13.
    • CommentAuthorMatt Earnshaw
    • CommentTimeAug 25th 2016

    for future reference, I have made available my stash of Lawvere’s corpus

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2016

    Wow!! That’s fantastic; thanks so much Matt.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2016

    I have added a link to Matt’s repository to the page William Lawvere (I put Matt’s name within double brackets, so there is now also a gray link there; obviously that can be removed if desired.)