Effective domains and intrinsic structure

Topos theory is the categorical analog of constructive set theory; and conveniently, PERs (partial equivalence relations) do sit inside a topos-the category of PERs can be (loosely speaking) identified with the full subcategory of modest sets in Hyland´s effective topos. (The effective topos is the topos-theoretic version of recursive realizability.) Working in the effective topos is especially attractive since not only can set-theoretic reasoning be used, but one also has a lot of category-theoretic and topos-theoretic machinery at one´s disposal. That is the point of view taken in this research. The basic theory of Σ-spaces is discussed. A convex power domain is also presented. Modal operators are outlined. Parallelism and sheaves are examined. Finally, the fixed-point classifier is presented