Re: Fibrations in a 2-category

[JeanBenabou <jean.benabou@wanadoo.fr>]

Dear Ross,

Thank you for answering me and agreeing with me about the difference  
between internal fibrations and you call "representable" ones.I had  
started to write a few comments about what you said on the  
description of the Eilenberg- Moore category as sheaves on the  
Kleisli category for a generalized topology. I shall give a single  
answer to the two mails because they have some common features.
I apologize if this answer is very sketchy. Each of these questions  
would deserve a long development, which I can't give for two reasons.

1- Long mails are not accepted on the official list. (I don't want  
our moderator to think that any criticism is hinted at by this  
remark, but some questions, especially those dealing with  
"foundations", do need long developments if they are to be discussed  
seriously. Thus the official list does not permit such discussions.  
Can anybody tell me where they can take place publicly?)

2- I'm supposed to give a 2 hours lecture on Feb. 5 on "Transcendent  
" methods in Category Theory. The audience is quite "mixed":  
mathematicians, philosophers, linguists, and even... musicians! Quite  
a challenge since some of them have only the faintest notions about  
Category Theory. Thus most of my time and energy are devoted to its  
preparation.

§1- EILENBERG-MOORE VESUS KLEISLI

1.1.  I have no objection to the terminology; "sheaves for a  
generalized topoloogy". You could even drop "generalized" provided  
you indicate precisely what you mean by "topology" and "sheaf". After  
all Grothendieck did precisely that when he used these two words for  
his "topologies' on categories which were indeed "generalized" from  
"usual" topology.

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)"

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?

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",
Let me look at simple example, namely the "notion" of group.
In the case of sets there are many closely related notions, let me  
describe a few ones.
(i) The category Grp of groups which is monadic over Set
(ii)The Kleisli category of this monad
(iii) The Lawvere theory of groups, say Th(Grp)

If we replace Set by a category S with finite products, and denote by  
Grp(S) the category of internal groups of S; what is (part of ) the  
general picture?
(a) Grp(S) is the category of product preserving functors Th(Grp) -->  
S, which you can view as "S-valued sheaves on Th(Grp) for the obvious  
(generalized) topology . It is equipped with the forgetful functor U;  
Grp(S) --> S "evaluation at 1 (I apologize for such trivialities)
(b) Suppose U has a left adjoint F and let T be the associated monad.
(b.1) Is it obvious that Grp(S) is monadic for the monad T?
(b.2) What is the precise relation between Th(Grp) and the Kleisli  
category Kl(T) of T ?
(b.3) Can Grp(S) be interpreted as S valued sheaves on K(T) for a  
suitable topology?

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?

§2 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.It  
is very easy to generalize important notions of Category Theory to 2- 
categories by making them "representable" but to me the real problem  
is to "internalize" these notions,( that is easy by using the  
internal language which I introduced for  precisely that purpose) and  
to STUDY THE PROPERTIES of these internalized notions; I am long past  
believing that ,apart from size conditions, ZF, with or without  
Universes or AC is enough to express all mathematical possibilities.  
As an,example is the important notion of "definability" which took me  
a long time to understand, by going "outside" of ZF

In your mail you say:

"An internal fibration between groups in a topos E is a group  
morphism whose underlying morphism in E is an epimorphism; for a  
representable fibration, it is a split epimorphism in E. Jack Duskin  
alerted me to this many years ago."

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 talked many times of this example in my seminar as an  
illustration of what internal languages could do. And Duskin attended  
my seminar for a whole year, and many other times for shorter periods  
But let's forget about this "detail" and concentrate about more  
important things.

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.
I contend that indexed categories are not the same even as cloven  
fibrations. They are the same IN SETS. But if you you go back to §1,  
you'll immediately see the difference. Fibrations and cloven  
fibrations can be internalized, e.g. in a topos (although this  
assumption is much too strong); Indexed categories cannot. And even  
if by some very complicated construction one could, in some special  
case, internalize them, unless you add some very strong artificial  
assumptions they would not coincide with internal fibrations, not  
even cloven ones.

Let me give a final trivial example to try again to convince you and  
a few other ones. Let me call for short "surjective" morphism of  
groups a morphism of groups such the the underlying morphism is an  
epi (better be a regular epi). Obviously they are stable under  
composition. How would you formulate this in terms of "internal  
indexed categories", assuming you have defined such notion?

There is a lot more I could say but the mail is already very long,  
and I hope it will be forwarded.
Thanks for reading me.

Best wishes,
Jean 
    

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