In a paper circulated since end of last year Dana Scott has proposed a category PEQU of algebraic lattices with partial equivalence relations as objects and equivariant Scott continuous maps as morphisms. One nice thing about PEQU is that Scott has shown it to be equivalent to the category EQU of T_0 spaces with (total) equivalence relations and equivariant continuous maps. It has been shown (by students of Scott) that PEQU is not even well-powered despite being regular and locally cartesian closed (in particular, it is a logos and thus a model for FOL). If one considers $\omega$-PEQU, the category of $\omega$-algebraic lattices with partial equivalence relations as objects and equivariant Scott continuous maps as morphisms, then it can be shown easily to be equivalent to PER(P $\omega$), the category of pers over the good old P $\omega$ model of lambda calculus, which embeds into Ass(P $\omega$), the category of assemblies over P $\omega$ and finally into RT(P $\omega$), the realisability topos over P $\omega$. All these full inclusions are reflective. This realisability model has been studied by Wesley Phoa in his Thesis (1990) and in his paper "Building Domains from Graph Models" (MSCS, 1992). The reason for the equivalence of $\omega$-PEQU and PER(P $\omega$) is that any $\omega$-algebraic lattice L is contained in P $\omega$ via an inflationary retract e : L -> P $\omega$ , p : P $\omega$ -> L. Now if R is a per on L then P $\omega$ may be endowed with the per e[R] := { (e(x),e(y)) | x R y } and the pair e,p of equivariant Scott continuous maps establishes an isomorphism between (L,R) and (P $\omega$ , e[R]). This observation does not hinge on countable cardinality. For any cardinal $\kappa$ we may consider $\kappa$-PEQU (where the bases of the algebraic lattices are required to have cardinality less or equal $\kappa$) and show it to be equivalent to PER(P $\kappa$) by the same argument as above. The reason why this is possible is that any infinite set M contains List(M) x M as a retract and therefore can be endowed with a graph model structure -- as emphasised by Scott. The benefits of this (simple) observation are the following (1) $\kappa$-PEQU does not admit a generic (regular) mono (= weak s.o. classifier) as for any partial combinatory algebra A the category PER(A) does not admit such (as each homset has cardinality less or equal |A| and there are 2^{|A|} (regular) subobjects of (A, \Delta(A))) but, never mind, the required weak / strong classifiers live in Ass(A) / RT(A) , respectively (2) Phoa already has started to investigate SDT in PER(P $\omega$), one may take as a dominance (S , \Delta(S)) where S is Sierpinski space together with [t] : ({*} , \Delta({*})) -> (S , \Delta(S)) where t(*) = \top. The same may be taken as a dominance for PEQU without any size restrictions. Anyway, PEQU appears as an amalgamation of ALL realisability models over graph models of all size. Assuming a Grothendieck universe U we may consider (the fairly big) graph model P U appearing most naturally so as we have List(U) x U \subseteq U. Thus PER(P U) contains PEQU, i.e. EQU, when one restricts the underlying algebraic lattices / T_0 spaces of PEQU / EQU to live in universe U. But notice that (P U , \Delta(P U)) lives outside this PEQU. Anyway, PER(P U), Ass(P U), RT(P U) appear as the appropriate models of logic in which to study Scott's recently proposed categories.
From the point of view of mathematical practice, however, the restriction to cardinality $\omega$ does not seem to be too restrictive as most spaces in "real" mathematics have a countable base (e.g. separable metric spaces).
Thomas Streicher