Reflections and Coreflections

As is well-known topological spaces can be related to each other by means of continuous maps. More generally, objects in a category can be related to each other by means of morphisms. There is an analogous relationship between categories via so-called functors. The classical definition of universal maps in the sense of N. Bourbaki [18] corresponds to a categorical one which utilizes a functor. The existence of all universal maps with respect to a given functor F is related to a pair of adjoint functors (G, F), where G (resp. F) is called a left adjoint (resp. right adoint). The relationships between these functors are described by means of natural transformations u and v (which occur as “maps” between functors). Thus, an adjoint situation (G, F, u, v) is obtained. In the first part of this chapter adjoint situations are studied together with some examples. In the second part an important special case of adjoint situations (G, F, u, v) is investigated, namely the case where F is an inclusion functor I from a subcategory A of a category C to C (the notion of inclusion functor corresponds to the notion of inclusion map in classical mathematics). Then G is called a reflector from C to A and A is called reflective. If the morphisms belonging to all universal maps with respect to I are epimorphisms, extremal epimorphisms or bimorphisms, then G is called an epireflector, extremal epireflector or bireflector respectively and we say epireflective, extremal epireflective or bireflective subcategory rather than reflective subcategory. The famous characterization theorem for epireflective (and extremal epireflective) subcategories is proved and the results are applied to bire-flective subconstructs of topological constructs.