Localization of Universal Problems. Local Colimits

The notion of the root of the category, which is a minimal (in a precise sense) weakly coreflective subcategory, is introduced in view of defining local solutions of universal problems: If U is a functor from C to C and c an object of C, the root of the comma-category c|U is called a U-universal root generated by c; when it exists, it is unique (up to isomorphism) and determines a particular form of the locally free diagrams defined by Guitart and Lair. In this case, the analogue of an adjoint functor is an adjoint-root functor of U, taking its values in the category of pro-objects of C. Local colimits are obtained if U is the insertion from a category into its category of ind-objects; they generalize Diers' multicolimits. Applications to posets and Galois theory are given.