Subject: categories: Regular embedding

From: "Ockham's stubble" <[email-masked]>
Date: Mon, 25 May 1998 17:00:07 -0400

Would the following (aside from the first) help to prepare a more
digestible proof of Barr's full regular embedding theorem?

1) Say that an object U of a (but not just any) category E is open if U
   is a finitely presentable object of the opposite of E (i.e., if it
   satisfies the analogue of the filter convergence characterization
   of open sets). An object P of E is strict if it is the projective
   limit of a diagram of opens such that each limit projection is an
   effective epimorphism; P is called saturated if for every U -->> V
   of opens, each P -->> V lifts to some P -->> U; finally, P is called
   weakly projective if it is projective with respect to effective
   epimorphisms of opens. Every strict, saturated object is  weakly
   projective.   
2) Given P in Pro(C), where C is a regular category, let P' denote the
   fibred product of all X -->> P which are opens of Pro(C)/P, and P*
   the projective limit of: P <<-- P' <<-- P'' <<-- P''' ...; one
   shows that the projection P* --> P is an effective epimoprhism, that 
   P* is a strict, saturated object of Pro(C)/P, and therefore that P*
   is a weakly projective object of Pro(C). Theorem 14 of Barr's '86
   JPAA paper (i.e., step three in his message of 21 May) applies.
3) The construction is (to my knowledge) due to Joyal (c. 1972, or
   earlier). Objects of the form X*, where X is an object of C, are 
   principal prime models, a notion Makkai (Full continuous embeddings
   of toposes, Trans AMS, 1982) distilled from Barr's original proof
   (Joyal called them absolutely flasque). 
cheers, 
-b
PS:Without the terminology, that Pro(C) (=~ Lex(C,Set)^op) is regular 
   if C is is suggested in SGA4, Expose I, 8.9 (especially cf. exercise
   8.9.9(b)).

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From: Michael Barr <[email-masked]>
Date: Tue, 26 May 1998 14:03:07 -0400 (EDT)

Well, I am not trying to be snide, but for me, my argument is more
understandable.  But if you prefer to see it in the way you put it, I have
no objection.  A couple more comments.

In 1986 (or, really, 1984), I did not really understand that this argument
was *really* the same argument, just dressed up nicer, using functors
instead of diagrams.  

In 1970, I was fresh from homological algebra.  The construction struck me
(and still does) as reminiscent of the argument you use in showing that if
you have the beginnings of a map from a projective complex to an acyclic
one, you can continue it one more step.  The main difference was that this
was a well-founded poset instead of an inter-indexed chain.  And if anyone
ever wondered why I called these A diagrams and P diagrams, it had nothing
to do with A&P and everything to do with acyclic and projective.  Thus the
embedding theorem is, like many other things I have done, a form of
acyclic models.