Dear Jean On 12/01/2011, at 5:50 PM, JeanBenabou wrote:
1.2. There is an ambiguity in your mail when you write.
"After all, I believe Linton's work aimed at generalizing to all monads the correspondence between monads of finite rank on Set and Lawvere theories, under which Eilenberg-Moore algebras become product-preserving presheaves on part of the Kleisli category)"
I agree there is ambiguity. Linton's paper is in a Lecture Note volume now in TAC Reprints for people to see.
As far as I remember Linton did not deal with "all" monads but with monads ON SETS.
The same ambiguity can be found e.g. in Lack's mail where he writes:
"Similarly, the Elienberg-Moore algebras can be seen as the presheaves on the Kleisli category which send certain diagrams to limits."
He never mentions the fact that the monad is ON SETS. I suppose your "sheaf-interpretation" holds only for monads on Set, am I wrong?
The comment about generalized topology was meant to be informal. The Proposition it followed gave a Yoneda structure version of Linton's pullback of the Yoneda embedding making the Eilenberg-Moore category a full subcategory of presheaves on the Kleisli category. This works for monads on any category.
What if we replace Set by another category, say S? I don't want to under estimate Lawvere's, or Linton's or your work, but; I apologize to theses authors this is a bit of "glorious past-time story",
Yes, we are talking about "the good old days". But that is what the original question was about.
A partial answer to these questions can be given when S is a topos with NNO, but,for me at least, even in that case, there remain many important questions which I can't answer. Has anybody been interested by the kind of questions raised in the previous subsection?
Yes, I have wondered about some of those matters. There are many questions I cannot answer.
ยง2 FIBRATIONS AND "REPRESENTABLE" FIBRATIONS.
Thank you Ross for agreeing with me about the difference between (internal) fibrations and what you call "representable" ones. Sorry if I disagree with you, but I tend to prefer the first ones.
I am not sure what you are disagreeing about. Are you disagreeing that I agree with you? I certainly believe you prefer the first ones. I would like to hear what you have to say about them. I had something to say about the representably-defined ones but believe the others are more important. You are looking at harder problems by studying them. At no time have I claimed that the problems I look at are the most important ones.
I do not know when Duskin "alerted" you. What I now is that I found the remark in the original paper of Grothendieck on fibrations (1961) and that I "internalized" it in 1970 when I introduced internal languages.
I was not fortunate enough to be around Grothendieck or to attend your 1970 lectures. So Jack Duskin was able to tell me some aspects of what he had heard from you. Similarly I didn't originally learn the group epimorphism example from reading Grothendieck; I saw it in Gray's LaJolla paper if my memory serves me correctly. Gray refers to Grothendieck whom I read later. I am glad that you have pointed out that you had lectured on the internal version of this in 1970.
For more than 20 years I have tried to convince people that fibrations and indexed categories are not "the same thing", even if we use AC and universes. For a long time I didn't convince you. I remember having offered 6 bottles of champagne to anyone who could prove, using only indexed categories, that the composite of two fibrations is a fibration. And I got an answer from you where you had to go through the Grothendieck construction for one of the indexed categories. Thus you didn't get the champagne. Of course, if you visit me in Paris, I'll be very glad to share with you a bottle, for old time's sake.
If you recall what I said at the time was that I would not bid for your champagne prize. I was not offering an answer. The reason is that it cannot be done. Fibrations cannot be composed using homomorphisms of bicategories from a category into Cat. As you know, the question doesn't make sense. However, homomorphisms (and morphisms) of bicategories are also useful concepts and I thank you for giving them to us. I wish you the best for your lecture preparation. I am sure the lectures will be inspiring. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]