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 and a B-morphism such that for each B to an A there exists a unique A.

is called the A-reflection of B is called the A-reflection arrow. only as being the A-reflection of B).

is a right adjoint. The left adjoint functor is called the reflector is the unit of this adjunction.

the A for a B 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 -reflective subcategory, is a class of morphisms.

The -reflective hull of a class A-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.

Topology

• The category of Kolmogorov spaces (T0 spaces) is a reflective subcategory of Top, the category of topological spaces, and the Kolmogorov quotient is the reflector.

• The category of completely regular spaces CReg is a reflective subcategory of Top. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.

• The category of all compact Hausdorff spaces is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces). The reflector is given by the Stone–Čech compactification.

• The category of all complete metric spaces with uniformly continuous mappings is a reflective subcategory of the category of metric spaces. The reflector is the completion of a metric space on objects, and the extension by density on arrows.

• The category of sheaves is a reflective subcategory of presheaves on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.

• The category Seq of sequential spaces is a coreflective subcategory of Top consisting of all sequentially open (that is, complements of sequentially closed sets).

Functional analysis

• The category of Banach spaces is a reflective subcategory of the category of normed spaces and bounded linear operators. The reflector is the norm completion functor.

Category theory

• For any Grothendieck site (C, J), the topos of sheaves on (C, J) is a reflective subcategory of the topos of presheaves on C, with the special further property that the reflector functor is left exact. The reflector is the sheafification functor a : Presh(C) → Sh(C, J), and the adjoint pair (a, i) is an important example of a geometric morphism in topos theory.

Properties

• The components of the counit are isomorphisms.

• If D is a reflective subcategory of C, then the inclusion functor D → C creates all limits that are present in C.

• A reflective subcategory has all colimits that are present in the ambient category.

• The monad induced by the reflector/localization adjunction is idempotent.

Notes

References