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.
[1]: 91 This adjoint is sometimes called a reflector, or localization.
[2] 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.
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
there exists a unique A-morphism
is called the A-reflection arrow.
(Although often, for the sake of brevity, we speak about
This is equivalent to saying that the embedding functor
is the unit[broken anchor] of this adjunction.
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—
-reflective hull of a class A of objects is defined as the smallest
[citation needed] Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.