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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
Beautiful!
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)
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 ?
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 . 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.
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
Thanks, I’ll have a look at the article. Have you already absorbed this?
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.
Added link to YouTube of a recording of Lawvere’s Como lecture.
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)?
Enjoy
Great! Thanks a lot.
for future reference, I have made available my stash of Lawvere’s corpus
Wow!! That’s fantastic; thanks so much Matt.
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.)
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