Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the res...

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...

Behavioural preorders and equivalences for higher-order languages have been extensively studied in the last decades, often leading to rich and satisfactory theories. However, in presence of effectful computations ordinary behavioural relations can be too discriminating, as highlighted by probabilistic higher-order languages. An elegant way to avoid such problem is to move from qualitative, bool...

Let Sp be the category of symmetric spectra, regarded as having the stable model structure. We prove that the category of small categories enriched over Sp admits a model category structure. The method of proof applies to other categories than symmetric spectra as well, as long as these categories are linked to the category of simplicial sets via a certain strong monoidal Quillen pair.

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Qmodule transforms, is defined. Such operators — that turn out to be precisely the homomorphisms between free objects in those categories — find concrete applications in two different branches of image processing, namely fuzz...

This paper continues the development of a simplicial theory of weak ω-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the 3-categorical Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier ...