Date: Mon, 7 Oct 1996 16:34:59 -0300 (ADT)
Subject: Local smallness in Gabriel-Zisman

Date: Mon, 7 Oct 1996 15:19:49 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>

I have been looking at Gabriel-Zisman.  It seems clear from things that
they say that their categories are intended to be locally small.  On the
other hand, they say nothing about the local smallness of the categories
of fractions they construct.  Am I missing something or is there a
problem there.  Even when there is a calculus of fractions, it is
not obvious.  I am especially interested in the case of chain complexes
with homology isomorphisms inverted.

Michael


Date: Tue, 8 Oct 1996 08:36:12 -0300 (ADT)
Subject: Re: Local smallness in Gabriel-Zisman

Date: Mon, 7 Oct 1996 17:51:57 -0400 (EDT)
From: Dan Christensen <jdchrist@math.mit.edu>

Michael Barr wrote:

| I have been looking at Gabriel-Zisman.  It seems clear from things that
| they say that their categories are intended to be locally small.  On the
| other hand, they say nothing about the local smallness of the categories
| of fractions they construct.  Am I missing something or is there a
| problem there.  Even when there is a calculus of fractions, it is
| not obvious.

There is definitely a problem, one that they chose (consciously, I'm
sure) to ignore.

| I am especially interested in the case of chain complexes
| with homology isomorphisms inverted.

If you take chain complexes of R-modules then there is no problem.
(In fact, any AB 5 abelian category will do in place of R-modules, I
believe; see Weibel's book on homological algebra.)  More generally,
in any closed model category Quillen proved that you can invert the
weak equivalences and get a locally small category.  Sometimes the
entire point of a closed model structure is to show that the category
of fractions is locally small; a good example is Bousfield's proof
that you can invert the h_*-isomorphisms in the category of spaces,
where h_*(-) is any generalized homology theory.

Dan


Date: Tue, 8 Oct 1996 08:37:49 -0300 (ADT)
Subject: Re: Local smallness in Gabriel-Zisman

Date: Tue, 8 Oct 1996 12:12:28 +1000
From: Murray Adelman <murray@zeus.mpce.mq.edu.au>

I looked at this problem long ago when I was considering quotients of
the free abelian category (which is something like chain complexes mod
homotopy.)  Here is what I remember.

Even if you have a calculus of fractions and even if the category is
locally small, the maps in the quotient are bounded from below by the
relations.  If the category is not well-powered, there is no reason to
expect a (small) set of relations.

All this doesn't apply to your category because it doesn't have finite
limits, but I think that Peter Freyds proof that the homotopy category
is not concrete might be modifiable into an example where a caluclus
of fractions is not small.

All of this, of course doesn't say anything about the case where
homology isos are inverted, but I think it does show that it needs
proving.

        Murray


Date: Tue, 8 Oct 1996 08:36:59 -0300 (ADT)
Subject: Local smallness in Gabriel-Zisman

Date: Tue, 8 Oct 1996 11:46:01 +1000
From: Ross Street <street@mpce.mq.edu.au>

Michael Batanin and I have been discussing the question of
Michael Barr: when is the category of fractions of a locally
small category locally small?

Let  C  be a locally small category and  S  a (possibly large)
set of arrows in  C.  In general, the category  F  of fractions
of  C  wrt  S  is not locally small.  If  S  admits a calculus of
left fractions then the hom  F(a,b)  is a colimit of homs  C(a,s)
where  b --> s  runs over the filtered category  b/S.  What we
need is a small final subcategory of  b/S.  Sometimes this can be
found using a well-copowered factorization system in  C.

But this does not work when  C  is chain complexes in a locally small
abelian category  A  and  S  consists of chain maps inverted by homology
(although  S  does admit a calculus of both left and right fractions).
Then  F = K(A)  is called the derived category of  A  which is probably
not locally small in general.  However, if  A  has enough projectives,
K(A)  is equivalent to the category of projective chain complexes in
A  with homotopy classes of chain maps.  This is an aspect of the general
theory of Quillen model structures.

So, if  A  is locally small abelian and has enough projectives (or
enough injectives), then  K(A)  is locally small.

Regards,
Ross


Date: Thu, 10 Oct 1996 21:56:20 -0300 (ADT)
Subject: Derived local smallness (corrections)

Date: Thu, 10 Oct 1996 11:25:56 +1000
From: Ross Street <street@mpce.mq.edu.au>

Since no-one has corrected me by now, I had better do it.

The derived category D(A) (not K(A)) of an abelian category
A  is the category of fractions obtained from the category of
chain complexes in  A  by inverting the chain maps inverted by
homology. But I am not sure what is needed on  A  for there to be a
Quillen model structure on chain complexes with  D(A)  as the
homotopy category (perhaps finite homological dimension will do?).

If you take  A  to have enough projectives and restrict to chain
complexes bounded below, you do get a model structure (see SLNM 43).

Now it seems to me that, since every chain complex is a filtered colimit
of chain complexes bounded below, we can use this to deduce the
result I claimed before: that the derived category  D(A)  is locally
small if  A  is locally small abelian with enough projectives.

Regards,
Ross