In mathematics, a full subcategoryA of a categoryB 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 Bthere exists an A-object and a B-morphism such that for each B-morphism to an A-object there exists a unique A-morphism with. The pair is called the A-reflection of B. The morphism is called the A-reflection arrow.. This is equivalent to saying that the embedding functor is a right adjoint. The left adjoint functor is called the reflector. The map is the unit of this adjunction. The reflector assigns to the A-object and for a B-morphism is determined by the commuting diagram If all A-reflection arrows are epimorphisms, then the subcategory A is said to be epireflective. Similarly, it is bireflective if all reflection arrows are bimorphisms. All these notions are special case of the common generalization—-reflective subcategory, where is a class of morphisms. The -reflective hull of a classA of objects is defined as the smallest -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, coreflective subcategory, coreflective hull, anti-coreflective subcategory.
Examples
Algebra
The category of abelian groupsAb is a reflective subcategory of the category of groups, Grp. The reflector is the functor which sends each group to its abelianization. In its turn, the category of groups is a reflective subcategory of the category of inverse semigroups.
Dually, the category of anti-commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the anti-commutator ideal. This is used in the construction of the exterior algebra from the tensor algebra.
The category of abelian torsion groups is a coreflective subcategory of the category of abelian groups. The coreflector is the functor sending each group to its torsion subgroup.
The categories of elementary abelian groups, abelian p-groups, and p-groups are all reflective subcategories of the category of groups, and the kernels of the reflection maps are important objects of study; see focal subgroup theorem.
The category of groups is a coreflective subcategory of the category of monoids: the right adjoint maps a monoid to its group of units.
The category of completely regular spacesCReg is a reflective subcategory of Top. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.
