Reflective subcategory

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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.

[edit] Examples

• The category of abelian groups Ab is 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 is a reflective subcategory of the category of all Tychonoff spaces. The reflector is given by the Stone-Čech compactification.