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.

Examples

• The category of abelian groups Ab in a reflective subcategory of the category of groups, Grp. The reflector is the functor which sends each group to its abelianization.
• The category of all compact Hausdorff spaces in a reflective subcategory of the category of all Tychonoff spaces. The reflector is given by the Stone-Čech compactification.

Adjoint functors