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.

Countable Lawvere theories model computational effects such as exceptions, side-effects, interactive input/output, nondeterminism and probabilistic nondeterminism. The category of countable Lawvere theories has sums, tensors, and distributive tensors, modelling natural combinations of such effects. It is also closed under taking images. Enrichment in a category such as Cpo allows one to extend ...

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. Here I generalize this to a k–fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schwänzl and Vogt but with the addition of making visible the coherent associators α. The symmetric case can easily be recovered. The introduction of...

We observe that the Reedy model structure on a diagram category can be constructed by iterating an operation of “bigluing” model structures along a pair of functors and a natural transformation. This yields a new explanation of the definition of Reedy categories: they are almost exactly those small categories for which the category of diagrams and its model structure can be constructed by itera...

Grothendieck toposes are studied via the process of taking the associated Sl-enriched category of relations. It is shown that this process is adjoint to that of taking the topos of sheaves of an abstract category of relations. As a result, pullback and comma toposes are calculated in a new way. The calculations are used to give a new characterization of localic morphisms and to derive interpola...