We categorify the adjunction between locales and topological spaces, this amounts to an (generalized) bounded ionads topoi. show that is idempotent. relate Scott adjunction, which was discussed from a more categorical point of view in [Liba]. hint $0$-dimensional inhabits categorified one.

We give a short and elementary proof of the Schröder-Simpson Theorem, viz., that the space of all continuous maps from a given space X, to the non-negative reals with their Scott topology, is the cone-theoretic dual of the probabilistic powerdomain on X.

For a poset P, let σ(P) and Γ(P) respectively denote the lattice of its Scott open subsets closed ordered by inclusion, set ΣP=(P,σ(P)). In this paper, we discuss lower Vietoris topology on give some sufficient conditions to make two topologies equal. We build an adjunction between σ(Γ(P)) prove that ΣP is core-compact iff ΣΓ(P) sober, locally compact σ(Γ(P))=υ(Γ(P)) (the topology). This answer...

A representation of continuous and prime-continuous lattices via formal topology is found. This representation stems from special examples of formal topologies in constructive analysis that give rise to the definition of the classes of locally Stone and locally Scott formal topologies. As an application, a representation theorem for locally compact spaces is obtained.

The notion of sobriety is extended to the realm topological spaces valued in a commutative and unital quantale, via an adjunction between category quantale modules quantale-valued spaces. Relations such sober domains based on flat ideals are investigated. In particular, it shown that if unit element coincides with its top element, then every domain (relative ideals) Scott topology space.

We introduce, comment and develop the Scott adjunction, mostly from point of view a category theorist. Besides its technical conceptual aspects, in nutshell we provide categorification topology over posets with directed suprema. From establish an adjunction between accessible categories colimits Grothendieck topoi. show that bicategory topoi is enriched $2$-category it has tensors respect to th...

Many token-based, distributed mutual exclusion algorithms can be generalized by a single algorithm. The algorithm’s performance is dependent upon the logical topology imposed on the nodes and the policy used to forward requests. This paper extends the generalized algorithm to support prioritized, real-time requests in a generalized fashion and presents models that can be used to analyze the per...

If a posets lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with Scott-continuous functions as morphisms. The same applies for the quantitative version ...