In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint. Informally, a reflector acts as a kind of completion operation. It adds in any "missing" pieces of the structure in such a way that reflecting it again has no further effect.

Definition

A full subcategory A of a category B is said to be reflective in B if for each B-object B there exists an A-object A_B and a B-morphism such that for each B-morphism to an A-object A there exists a unique A-morphism with. The pair (A_B,r_B) is called the A-reflection of B. The morphism r_B is called the A-reflection arrow. (Although often, for the sake of brevity, we speak about AB only as being the A-reflection of B). This is equivalent to saying that the embedding functor is a right adjoint. The left adjoint functor is called the reflector. The map r_B is the unit of this adjunction. The reflector assigns to B the A-object AB and Rf for a B-morphism f is determined by the commuting diagram If all A-reflection arrows are (extremal) epimorphisms, then the subcategory A is said to be (extremal) epireflective. Similarly, it is bireflective if all reflection arrows are bimorphisms. All these notions are special case of the common generalization—E-reflective subcategory, where E is a class of morphisms. The E-reflective hull of a class A of objects is defined as the smallest E-reflective subcategory containing A. Thus we can speak about reflective hull, epireflective hull, extremal epireflective hull, etc. An anti-reflective subcategory is a full subcategory A such that the only objects of B that have an A-reflection arrow are those that are already in A. Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.

Examples

Algebra

Topology

Functional analysis

Category theory

Properties