REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The notion of an effective (co)descent morphism is one of the main notions of Grothendieck descent theory. Its applications in algebraic geometry, algebraic number theory and related issues from ring and module theories were known long ago. Later its connection with the definability theorems in mathematical logic was discovered. It is well-known that descent theory is closely related also to Galois theory. The role of a Galois extension in categorical Galois theory is played by an arbitrary effective descent morphism in the ground category. This together with the Grothendieck cohomology construction suggests to look for satisfactory characterizations of effective descent morphisms in categories of interest. We present a new approach to this issue. The important role in this approach are played by the amalgamation property and the notion of a factorization system. We apply this approach to various concrete categories of both algebraic and geometric nature. In particular, we describe effective codescent morphisms in categories of topological spaces, Hausdorff spaces, compact Hausdorff spaces, normal spaces, Banach spaces, metric spaces, complete metric spaces, compact metric spaces. Moreover, we characterize effective codescent morphisms of groups. Further, we describe effective codescent morphisms in abstract varieties of universal algebras with the amalgamation property. In particular, we give the necessary and sufficient condition for a codescent morphism of such a variety to be effective, formulated in a syntactical form. Applying this result we obtain a sufficient condition for all codescent morphisms of a variety to be effective. Using this result we prove that all codescent morphisms are effective in varieties with the so-called normal forms for elements from amalgamated free products (the so-called normal form theorem). Among such varieties are varieties of quasigroups, loops, magmas and left/right-closed magmas.