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

this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but

. Each full reflective subcategory X of a finitely-complete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass -- or rather a subcategory -- of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...

introduction

The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of Bunge-Funk (1996, 1998).

We give a form of the Van Kampen Theorem involving covering morphisms in a lextensive category. This includes the usual results for covering maps of locally connected spaces, for light maps of compact Hausdorff spaces, and for locally strong separable algebras. @ 1997 Elsevier Science B.V. 1991 Muth. Sut~j. Cluss.: lSB99,57MlO