Subject: categories: Adjoint cylinders

From: Michael Barr <[email-masked]>
Date: Sun, 5 Nov 2000 05:23:28 -0500 (EST)

In this discussion of adjoint cylinders, I haven't noticed anyone
pointing out that when L --| F --| R, then FL = Id (I use = for natural
equivalence here) iff FR = Id.  Dusko told me earlier this year that it
was in his thesis, but I would not be surprised if it were older than
that.  Here is a simple proof.  Let eta:  Id --> FL be the inner
adjunction of L --| F and epsilon:  FR --> Id be the outer adjunction of
F --| R. Then if alpha F means apply F,the composites

           alpha F              Hom(eta-,F-)
 Hom(L-,-) -------> Hom(FL-,F-) ------------> Hom(-,F-)

           alpha F              Hom(F-,epsilon-)
 Hom(-,R-) -------> Hom(F-,FR-) ----------------> Hom(F-,-)

are isomorphisms.  Then there is a commutative square (both composites
are Hom(eta-,epsilon -) o alpha F):

                        Hom(eta-,FR-) o alpha F
             Hom(L-,R-) ------------------------> Hom(-,FR-)
                 |                                    |
                 |                                    |
                 |                                    |
                 |                                    |
  Hom(FL-,epsilon-) o alpha F                  Hom(-,epsilon-)
                 |                                    |
                 |                                    |
                 |                                    |
                 v           Hom(eta,-)               v
             Hom(FL,-) -------------------------> Hom(-,-)

in which the upper and left hand maps are isomorphisms and it follows
that the bottom arrow is an isomorphism iff the right hand one is.

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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Sun, 5 Nov 2000 15:58:26 +0000 (GMT)

Mike Barr wrote:

> that.  Here is a simple proof....

Here is an even simpler one: FL is left adjoint to FR, by composability 
of adjoints; the identity functor is left adjoint to itself.

Peter Johnstone

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From: Dr Anne Heyworth <[email-masked]>
Date: Mon, 6 Nov 2000 08:40:01 +0000 (GMT)

I had an idea and I'd like to know whether you like it and some advice on
making it work if you do. 

There are quite a few interesting tales about category theorists around. I
think it will be a shame to lose this more human history as time passes
and us newer category theorists can't completely remember the stories....
So I'd like to set up a kind of archive - a sort of myths and legends of
the category theorists (but preferably more truth than myth!).

This is the plan so far (assuming you think this is not a terrible idea).

Format:
An email-style archive with tales/histories/mini-biographies submitted by
email in plain text and linked to from a page of titles (titles would
include rough dates, authors and characters - which would make it easily
searchable so you can easily read about your particular heroes!).

Procedure:
Email sent to me which would have to be accepted by every living character
in the story (for obvious reasons). Some procedure should be agreed for
the acceptance of stories about people no longer alive.
The best stories, I guess, will be those people write about themselves.

This is just an idea, I hope you will let me know whether you approve.

--Anne.

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Have you visited   www.thehungersite.com 
             and   www.therainforestsite.com   today?

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