Quantitative semantics, topology, and possibility measures

AbstractGiven a topological space X and a complete lattice L, we study the space of L-predicatesFL(X) = [X → Lop]op, continuous maps from X to Lop in its Scott-topology. It yields a functor FL(·) from TOP-L, a full subcategory of TOP subsuming continuous domains, to SUP, the category of complete sub-lattices and maps preserving suprema. Elements of F2(X)are continuous predicates (= closed sets), and elements of F[0,1](X) may be viewed as probabilistic predicates. Alternatively, one may consider the complete sup-lattice PL(X) = O(X) −0 L of maps μ:O(X) → L preserving suprema (= possibility measures), which results in another functor PL(·) from TOP to SUP. We show that these functors are equivalent for two restrictions. First, we leave SUP unchanged and restrict TOP-L to CONT, the category of continuous domains in their Scott-topology; second, we fix TOP but restrict L to co-continuous lattices. Conversely, if FL(X) and PL(X) are isomorphic for all topological spaces X then L is indeed co-continuous. Further, if X is a sober space and FL(X) and PL(X) are isomorphic for all complete lattices L then X is a continuous domain and its topology is the Scott-topology. Possibility measures have extensions to the upper powerdomain κX, or to the full power set P(X), which are defined similarly to outer measures. Utilizing the notion of sup-semirings, we employ such extensions and the isomorphism FL(X) ≅ PL(X) to show that the sup-primes of PL(X) are exactly scalar multiples of point valuations with sup-primes as scalars for an underlying sober space X. Combining this with classical results in the theory of continuous lattices, we restate the notion of cones in our setting and show that the space of possibility measures PL(X) of a continuous domain X is the free L-module over X for the sup-semiring L = [0, ∞]