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 finds the best possible approximation of an object of the category B in the subcategory A, in the sense that it does not lose any information about morphisms to objects of the subcategory A.
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 {\displaystyle A_{B}} and a B-morphism r B : B → A B {\displaystyle r_{B}\colon B\to A_{B}} such that for each B-morphism f : B → A {\displaystyle f\colon B\to A} to an A-object A {\displaystyle A} there exists a unique A-morphism f ¯ : A B → A {\displaystyle {\overline {f}}\colon A_{B}\to A} with f ¯ ∘ r B = f {\displaystyle {\overline {f}}\circ r_{B}=f} .
The pair ( A B , r B ) {\displaystyle (A_{B},r_{B})} is called the A-reflection of B. The morphism r B {\displaystyle r_{B}} is called the A-reflection arrow. (Although often, for the sake of brevity, we speak about A B {\displaystyle A_{B}} only as being the A-reflection of B). This is equivalent to saying that the embedding functor E : A ↪ B {\displaystyle E\colon \mathbf {A} \hookrightarrow \mathbf {B} } is a right adjoint. The left adjoint functor R : B → A {\displaystyle R\colon \mathbf {B} \to \mathbf {A} } is called the reflector. The map r B {\displaystyle r_{B}} is the unit of this adjunction. The reflector assigns to B {\displaystyle B} the A-object A B {\displaystyle A_{B}} and R f {\displaystyle Rf} for a B-morphism f {\displaystyle 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 cases of the common generalization E {\displaystyle E} -reflective subcategory, where E {\displaystyle E} is a class of morphisms. The E {\displaystyle E} -reflective hull of a class A of objects is defined as the smallest E {\displaystyle E} -reflective subcategory containing A. Thus we can speak about the 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 The category of abelian groups Ab is a reflective subcategory of the category of groups, Grp. The reflector is the functor that sends each group to its abelianization. In its turn, the category of groups is a reflective subcategory of the category of inverse semigroups. Similarly, the category of commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the commutator ideal. This is used in the construction of the symmetric algebra from the tensor algebra. 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 fields is a reflective subcategory of the category of integral domains (with injective ring homomorphisms as morphisms). The reflector is the functor that sends each integral domain to its field of fractions. 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.
… excerpt ends here. Continue reading the full article.
