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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2011
   • (edited Apr 28th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](http://mathoverflow.net/questions/13167/lawveres-some-thoughts-on-the-future-of-category-theory)

   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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeApr 29th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexBeautiful!

   Beautiful!

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 29th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexOne 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)

   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)

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 29th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexBy the way, under Examples in related entry [[local geometric morphism]] there is a place where it says [[focal point]]. Is it meant [[local point]] ?

   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 ?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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 $comonad \dashv monad$. 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.

   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 $comonad \dashv monad$. 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.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeApr 30th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAs 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](http://people.math.jussieu.fr/~maltsin/ps/asphbl.ps).

   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.
7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, I'll have a look at the article. Have you already absorbed this?

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

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeApr 30th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNo, 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 24th 2014
   • (edited Aug 24th 2014)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAdded link to YouTube of a [recording](http://www.youtube.com/playlist?list=PLRIsuk3ZOitzDqeVKOZhe0MrzEUtRNAh5) of Lawvere's Como lecture.

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

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexMy 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)?

    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)?

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexEnjoy

    Enjoy

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexGreat! Thanks a lot.

    Great! Thanks a lot.

13. • CommentRowNumber13.
    • CommentAuthorMatt Earnshaw
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: Matt Earnshaw
    Format: MarkdownItexfor future reference, I have made available my [stash of Lawvere's corpus](https://github.com/mattearnshaw/lawvere)

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

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexWow!! That's fantastic; thanks so much Matt.

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

15. • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 25th 2016
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexI 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.)

    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.)