Effective Descent Morphisms in Categories of Lax Algebras

In this paper we investigate effective descent morphisms in categories of reflexive and transitive lax algebras. We show in particular that open and proper maps are effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps. Introduction A morphism p : E → B in a category C with pullbacks is called effective descent if it allows a description of structures over the base B as algebras on structures over the extension E of B. Here the meaning of “structure over B” might depend on the category C; however, in this paper we define it simply to be a morphism with codomain B. In that particular case p : E → B is effective descent if and only if the pullback functor p∗ : (C ↓ B)→ (C ↓ E) is monadic. In locally cartesian closed categories effective descent morphisms are easy to describe: they are exactly the regular epimorphisms. Such a characterization is far from being true in an arbitrary category; in general it can be quite a hard problem to find necessary and sufficient conditions for a morphism to be effective descent (see, for instance, [12] for the topological case). In order to obtain such conditions, it is often useful to embed our category into a category which has an easy description of effective descent morphisms, and then apply the pullback criterion of Theorem 1.1 below; this will be the basic technique of this paper. Following a suggestion of George Janelidze, we investigate effective descent morphisms in categories of reflexive and transitive lax algebras Alg(T;V) when V is a lattice, providing this way a unified treatment of descent theory for various categories. In particular, we characterize effective descent morphisms between quasi-metric spaces and, moreover, show that (suitably defined) open and proper maps are effective descent in Alg(T;V), encompassing the results for topological spaces obtained by Moerdijk [10, 11] and Sobral [13]. 2000 Mathematics Subject Classification. 18D05, 18C20, 18D10.