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...

A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and ‘order-cocomplete’. In fact, tensors and cotensors in a Q-category determine, and are determin...

A setoid is a set together with a constructive representation of an equivalencerelation on it. Here, we give category theoretic support to the notion. Wefirst define a category Setoid and prove it is cartesian closed with coproducts.We then enrich it in the cartesian closed category Equiv of sets and classicalequivalence relations, extend the above results, and prove that Setoid...

For a small quantaloid Q, a Q-closure space is a small category enriched in Q equipped with a closure operator on its presheaf category. We investigate Q-closure spaces systematically with specific attention paid to their morphisms and, as preordered fuzzy sets are a special kind of quantaloid-enriched categories, in particular fuzzy closure spaces on fuzzy sets are introduced as an example. By...

We describe categorical models of a circuit-based (quantum) functional pro- gramming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics ...

We define the phrase ‘category enriched in an fc-multicategory’ and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a b...

The theory of enriched accessible categories over a suitable base category V is developed. It is proved that these enriched accessible categories coincide with the categories of flat functors, but also with the categories of models of enriched sketches. A particular attention is devoted to enriched locally presentable categories and enriched functors.

The theory of enriched accessible categories over a suitable base category V is developed. It is proved that these enriched accessible categories coincide with the categories of at functors, but also with the categories of models of enriched sketches. A particular attention is devoted to enriched locally presentable categories and enriched functors.

This paper is about a generalization of Scott’s domain theory in such a way that its definitions and theorems become meaningful in quasimetric spaces. The generalization is achieved by a change of logic: the fundamental concepts of original domain theory (order, way-below relation, Scott-open sets, continuous maps, etc.) are interpreted as predicates that are valued in an arbitrary completely d...