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
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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).
Reflection of an object of a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection_of_an_object_of_a_category&oldid=16715