this paper we show that this is the case if and only if the ground category is "lextensive" (see section 3). Moreover, this form of Van Kampen Theorem holds for many other classes of morphisms, in particular for separable (= decidable) morphisms in the sense of [4]. It turns out that the main condition in the general Van Kampen Theorem is that the canonical morphism B 1 +B 2 ! B must be an effective descent morphism. This is satisfied in the topological situation described above. On the other hand this condition is natural, because Grothendieck's original idea of descent arose from the gluing construction for sheaves similar to that used in the Van Kampen Theorem for coverings. Note that our theorem does not include all known results in full generality, such as the topological Van Kampen Theorem under certain homotopical conditions [3, 8.4.2], and Magid's Van Kampen Theorem for (strongly) separable algebras [12] (however, it provides a Van Kampen Theorem for all separable algebras -- which is a new result). 1 A general setting for the Van Kampen Theorems

A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordant-dissonant and inseparable-separable.

A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder. 1.

Dedicated to Jiˇrí Rosick´y at the occasion of his sixtieth birthday Abstract. In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) overK 2. The link with existing notions in terms of morphism classes is given via the respective Eilenberg– Moore categories. 1.

In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properties for them, such as Hausdorff separation, compactness and local compactness, and we describe important topological constructions, such as the compact-open topology for function spaces and the Stone-Čech compactification. Of course, in a categorical setting, spaces are not investigated “directly ” in terms of their points and neighbourhoods, as in the traditional set-theoretic setting; rather, one exploits the fact that the relations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the set-theoretic setting, leave alone the much greater level of generality and applicability. The idea of providing a category with some kind of topological structure is certainly

For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of so-called weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.

Dedicated to Dominique Bourn on the occasion of his sixtieth birthday

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(communicated by George Janelidze) We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure