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.