Abstracts
Tuesday, 19 September 2000
2:30 - 4:00 M Barr
Mackey topologies and injectives
Abstract: Let C be a full subcateory of topological abelian groups
(TAGs) that is complete and contains the circle group T (1-torus). A
character is a continuous homomorphism to T. Denote the (discrete) group
of characters of A by A^. It is clear that for any group A there is a
weakest topology \sigma A that has the same character group as A,
namely the topology induced by the natural embedding of A into T^{A^}.
Depending on the nature of C, there may or may not be a strongest such
topology (among topologies that belong to C). If there is and it is
functorial, we denote the functor by \tau and say that C admits Mackey
coreflections. Assume that every object of C has enough characters to
separate points (called DS). Then C has Mackey reflections iff T is
injective (called DE) in C. The "if" direction is sort of known
although not quite stated explicitly in the main source. The other direction
seems not have been suspected. The main tool used in the proof is
(insert blare of trumpets here) the category chu(Ab,T).
(Joint work with H Kleisli)
Tuesday, 26 September 2000
2:30 - 4:00 M Barr
On HSP completions of categories of TAGs
Abstract:
For a category C of TAGs, let PC, SP and HP denote the closure
of C under products, subobjects, and Hausdorff quotients, resp. A
category of TAGs is said to satisfy
- DS (dually separated) if every group has a separating family of
characters;
- DE (dually embedded) if every character on a toplogically
embedded
subgroup can be extended to the full group;
- DC (dually closed) if every closed (topologically embedded)
subgroup is the intersection of the kernels of all the characters
that vanish
on it. Equivalently if the quotient is DS.
Proposition: HSPC is closed under products, subobjects, and Hausdorff
quotients.
Theorem 1: Let C have finite products and satisfy DS, DE, and DC.
Then so do PC, SC, and HC.
Theorem 2: Let C satisfy DS, DE, and DC. Then so does the category of
all groups of the form A x D, where A is in C and D is discrete.
It follows from the first talk that HSPC in that case has Mackey
coreflections. Examples will be given.