Reflective subcategory

In mathematics, a 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. 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 The pair $(A_{B},r_{B})$ is called the A-reflection of B. The morphism $r_{B}$ is called A-reflection arrow. (Although often, for the sake of brevity, we speak about $A_{B}$ only as about the A-reflection of B).

This is equivalent to saying that the embedding functor $E\colon \mathbf {A} \hookrightarrow \mathbf {B}$ is adjoint. The coadjoint functor $R\colon \mathbf {B} \to \mathbf {A}$ is called the reflector. The map $r_{B}$ is the unit of this adjunction. 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.

Dual notions to the above mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull.