We know from a previous paper that the reticulation of coherent quantale A is bounded distributive lattice L ( ) whose prime spectrum homeomorphic to m -prime . This studies how can be used for transferring some properties lattices quantales and vice versa. shall illustrate this thesis by proving several results on pure w –pure elements means In particular, we investigate σ –ideals transferred ...

We introduce and present results about a class of quantales, the topological relational quantales, that can be associated with tuples (X,R) such that X is a topological space and R is a lower-semicontinuous equivalence relation on X (see Definition 1 below). Our motivating case studies and examples of topological relational quantales include concrete quantales such as the relational quantales o...

In this first of two papers on ∞-category theory, we present information and resources on the basic prerequisites to parse the Riehl-Verity theory of ∞-cosmoi, the main topic of, e.g, [RV16b]. We provide basic definitions of monoidal categories, enriched categories, model categories, simplicial sets, and quasicategories, leading up to the main definition of [RV17], that of an ∞cosmos. Examples ...

For a completely distributive quantale L, L-fuzzy strongest postcondition predicate transformers are introduced, and it is shown that, under reasonable assumptions, they are linear or affine continuous mappings between continuous L-idempotent semimodules of L-fuzzy monotonic predicates. © 2012 Elsevier B.V. All rights reserved.

A 2-form between two sup-lattices L and R is defined to be a suplattice bimorphism L×R → 2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*-algebras.

Abstract We introduce a general definition for coloured cyclic operads over symmetric monoidal ground category, which has several appealing features. The forgetful functor from to both adjoints, each of is relatively simple. Explicit formulae these adjoints allow us lift the Cisinski–Moerdijk model structure on category enriched in simplicial sets sets.

We prove that for each locally $\alpha$-presentable category $\mathcal K$ there exists a regular cardinal $\gamma$ such any $\alpha$-accessible functor out of (into another category) is continuous if and only it preserves $\gamma$-small limits; as consequence we obtain new adjoint theorem specific to the functors K$. Afterwards generalize these results enriched setting deduce, among other thing...

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if categorical structure question is sufficiently weak (such as monoidal, but not strict categories) then 2-category accessible. Furthermore, explore flexible limits such possess interaction with filtered colimits.

Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of effects is irrelevant. Understanding such systems in terms of a lattice of effects grounds understanding of the essential issues, and provides guidance when ...