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).
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