A non empty class C of connected spaces is said to be a closed class if it is closed under weak equivalences and pointed homotopy colimits. A closed class can be characterized as a non empty class of connected spaces which is closed under weak equivalences and is closed under certain simple operations: arbitrary wedges, homotopy push-outs and homotopy sequential colimits. The notion of a closed...

In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in Pos} posets, with action preserving monotone maps between them, some categorical properties are considered. particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers etc. category. Also, several kinds epimorphisms monomorphisms characterized ...

Modular speciications: constructions with nite colimits, diagrams, isomorphisms Catherine ORIAT R esum e : La composition de sp eciications modulaires peut ^ etre mod elis ee, dans le forma-lisme des cat egories, par des colimites de diagrammes. Les sommes amalgam ees permettent en particulier de d ecrire l'assemblage de deux sp eciications qui ont une partie commune. Ce travail etend cette id ...

Let K be a locally finitely presentable category. If K is abelian and the sequence 0 K // X // k // C c // // 0 // is short exact, we show that 1) K is finitely generated⇔ c is finitely presentable; 2) k is finitely presentable⇔ C is finitely presentable. The “⇐” directions fail for semi-abelian varieties. We show that all but (possibly) 2)(⇐) follow from analogous properties which hold in all ...

There exists a KZ-doctrine on the 2-category of the locally small categories whose algebras are exactly the categories which admits all the colimits indexed by ω-chains. The paper presents a wide survey of this topic. In addition, we show that this chain cocompletion KZ-doctrine lifts smoothly to KZ-doctrines on (many variations of) the 2-categories of monoidal and symmetric monoidal categories...

In this section, C denotes a fixed category, which is locally U-small by convention. Our goal here is to define locally presentable categories, which are, very roughly, categories determined by colimits of nice diagrams in a prescribed U-small full subcategory. Under this interpretation as being completely determined by a U-small category, it is not surprising that the theory of localizations o...

We show that every λm-injectivity class (i.e., the class of all the objects injective with respect to some class of λ-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of λ-presentable morphisms. This was known for small-injectivity classes, and referred to as the “small object argument”. An analogous ...