Subject: categories: Preprints: On Mackey topologies ... and On *-autonomous ...

From: Michael Barr <[email-masked]>
Date: Thu, 12 Oct 2000 10:22:38 -0400 (EDT)

I have just posted two papers to ftp.math.mcgill.ca/pub/barr.  The first
is:
 mackey
     M. Barr and H. Kleisli, On Mackey topologies in topological abelian
     groups.  Submitted to TAC.

The main result can be described as follows:  Let C be a full subcategory
of topological abelian groups that includes the circle group T and closed
under subobjects and projects.  For a group A, let A^ denote the abstract
group of all its characters, that is continuous homomorphisms to the
circle. If A is an object of C, say that A' has a compatible topology if
it has the same elements and the same characters are A and say that A' has
a Mackey topology if it has the finest compatible topology.  Say that C
has Mackey coreflections if the inclusion of the full subcategory of
Mackey groups has a right adjoint.  Then we show, among other things, that
the existence of Mackey coreflections is equivalent to the injectivity of
the circle in C and then the Mackey category is equivalent to chu(Ab,T).

The second is:
 tvs
     On *-autonomous categories of topological vector spaces.  To appear
     in Cahiers de Topologie et Geometrie Differentielle Categorique.
 
The main result here is that the category of Mackey spaces (defined
similarly in terms of continuous linear functionals and proved by Mackey
to be coreflective) among the locally convex TV spaces is equivalent to
chu(Vect,K), where K is the field of scalars (either R or C) and is
therefore *-autonomous.  Another fact adduced is that--unusually--
chu(Vect,K) is a *-autonomous subcategory of Chu(Vect,K).  

Sometime in the next week or so, I will be posting 
derfun
Effaceable resolutions and derived functors.  Submitted to HHA.

It is well known that to define the derived functors, say, of Tor, it is
sufficient to use flat resolutions.  The usual proofs of this fact use the
existence of projective resolutions to do this.  For example, to infer
functoriality, you use the possibility of lifting resolutions.  Suppose,
for example, you are in a category of modules over a sheaf of rings in a
topos.  Generally, there are no projectives (save 0), but the sheaves
associated to representable functors are flat.  More generally, suppose
that T: A --> C is a right exact functor between abelian categories.  Call
an object E T-effaceable if whenever 0 --> A --> B --> E -->0 is exact,
then TA --> TB is monic.  Assume as a hypothesis that the kernel of an
epimorphism between eeffaceables is effaceable and that every object is
the target of an epimorphism from an effaceable.  Then derived functors
are definable using effaceable resolutions.