Locally convex cones and the Schröder-Simpson Theorem

At MFPS 23 at Birmingham, Schröder and Simpson had announced a result that I did not absorb at the time. A special case of that theorem for stably compact spaces was announced by Cohen, Escardo and the author shortly afterwards. I did not believe in the theorem of Schröder and Simpson until I saw a proof that they showed to me in detail during a visit to Edinburgh. At the New Interactions Workshop at Birmingham in January 2009 Alex Simpson presented a simplified proof of their result. This proof is elementary, although technically involved, whilst our proof for the special case of stably compact spaces used functional analytic methods. After the Birmingham workshop together with Achim Jung we thought about a conceptually more transparent proof. I continued to think about it when visiting Hans-Peter Künzi in Cape Town for two weeks in February/March 2009. And I finally came up with a more conceptual proof. This paper paper has two goals: Firstly, I want to present the conceptual proof of the Schröder-Simpson Theorem. The Schröder-Simpson Theorem is stated in terms of Domain Theory and uses directed complete partially ordered cones and Scott-continuous maps. These structures are used to model probabilistic phenomena in denotational semantics. The proof presented here relies on another generalization of vector spaces to an asymmetric setting – cones with convex quasiuniform structures – which has not been used in the semantic community until now. It is the second goal of this paper to introduce these two parallel developments of asymmetric generalizations of topological vector spaces in the style of a survey. There are the order theoretical and topological point of view on one hand, and the quasiuniform aspect on the other hand. Both developments had been pursued in parallel until now. We want to point out the close connection of these developments to the community working on asymmetric normed and asymmetric locally convex spaces. For this reason, all concepts are explicitly introduced in this paper and the proofs of all results that we use from the literature are included.