This paper introduces a new approach to the theory of Ω-categories enriched by a frame. The approach combines ideas from various areas such as generalized ultrametric domains, Ω-categories, constructive analysis, and fuzzy mathematics. As the basic framework, we use the Wagner’s Ω-category [18,19] with a frame instead of a quantale with unit. The objects and morphisms in the category will be ca...

In this paper, we consider a non-posetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as balanced mon...

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 paper, we characterize local $T_{0}$ and $T_{1}$ quantale-valued gauge spaces, show how these concepts are related to each other and apply them to $mathcal{L}$-approach distance spaces and $mathcal{L}$-approach system spaces. Furthermore, we give the characterization of a closed point and $D$-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each o...

We provide a new characterization of the notion of sheaf on an étale groupoid G, in terms of a particular kind of Hilbert module on the quantale O(G) of the groupoid. All the theory is developed in the context of the more general class of quantales known as stable quantal frames, of which examples are easy to construct because their category is algebraic. The homomorphisms of our Hilbert module...

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic ...