Reflection of an object of a category

reflector of an object of a category

Let be a subcategory of a category ; an object is called a reflection of an object in , or a -reflection, if there exists a morphism such that for any object of the mapping

is bijective. In other words, for any morphism there is a unique morphism such that . A -reflection of an object is not uniquely defined, but any two -reflections of an object are isomorphic. The -reflection of an initial object of is an initial object in . The left adjoint of the inclusion functor (if it exists), i.e. the functor assigning to an object of its reflection in , is called a reflector.

Examples. In the category of groups the quotient group of an arbitrary group by its commutator subgroup is a reflection of in the subcategory of Abelian groups. For an Abelian group , the quotient group by its torsion subgroup is a reflection of in the full subcategory of torsion-free Abelian groups. The injective hull of the group is a reflection of the groups and in the subcategory of full torsion-free Abelian groups.

Reflections are usually examined in full subcategories. A full subcategory of a category in which there are reflections for all objects of is called reflective (cf. Reflexive category).

Comments

The reflection of an object solves a universal problem (cf. Universal problems).