By a multi-pointed category we mean a category C equipped with a so called ideal of null morphisms, i.e. a class N of morphisms satisfying f ∈ N ∨ g ∈ N ⇒ fg ∈ N for any composable pair f, g of morphisms. Such categories are precisely the categories enriched in the category of pairs X = (X,N) where X is a set and N is a subset of X, whereas a pointed category has the same enrichment, but restri...

The aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. All values aretaken in a fixed frame $L$. A definition of (strongly) completelydistributive $L$-ordered sets is introduced. The main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $L$-ordereds...

We continue the study of Capretta’s delay monad as a means of introducing non-termination from iteration into Martin-Löf type theory. In particular, we explain in what sense this monad provides a canonical solution. We discuss a class of monads that we call ω-complete pointed classifying monads. These are monads whose Kleisli category is an ωcomplete pointed restriction category where pure maps...

A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination relevant comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets...

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associato...

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. Mathematics Subject Classification: 18C20, 18D15, 18A05, 18B30, 18B35.

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi.

We show that, in any Mal’tsev (and a fortiori protomodular) category E, not only the fibre GrdXE of internal groupoids above the object X is a naturally Mal’tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a ne...

RÉSUMÉ. Cet article étudie l’enveloppe réflective engendrée par une sous-catégorie pleine d’une catégorie complète. Il s’agit de la plus petite sous-catégorie qui est pleine et réflective. Comme application nous obtenons l’enveloppe coréflective de la sous-catégorie pleine de cubes pointés dans la catégorie des espaces topologiques pointés. Par la suite nous déterminons l’enveloppe réflective d...

We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a ...