View source for Reflection of an object of a category

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''reflector of an object of a category''

Let  $  \mathfrak C $
be a subcategory of a category  $  \mathfrak K $; 
an object  $  B \in \mathfrak C $
is called a reflection of an object  $  A \in \mathfrak K $
in  $  \mathfrak C $, 
or a  $  \mathfrak C $-
reflection, if there exists a morphism  $  \pi :  A \rightarrow B $
such that for any object  $  X $
of  $  \mathfrak C $
the mapping

$$ 
H _ {X} ( \pi ) :  H _ {\mathfrak C} ( B, X)  \rightarrow  H _ {\mathfrak K} ( A, X)
$$

is bijective. In other words, for any morphism  $  \alpha :  A \rightarrow X $
there is a unique morphism  $  \alpha  ^  \prime  :  B \rightarrow X \in \mathfrak C $
such that  $  \alpha = \pi \alpha  ^  \prime  $. 
A  $  \mathfrak C $-
reflection of an object  $  A $
is not uniquely defined, but any two  $  \mathfrak C $-
reflections of an object  $  A $
are isomorphic. The  $  \mathfrak C $-
reflection of an initial object of  $  \mathfrak K $
is an initial object in  $  \mathfrak C $. 
The left adjoint of the inclusion functor  $  \mathfrak C \rightarrow \mathfrak K $(
if it exists), i.e. the functor assigning to an object of  $  \mathfrak K $
its reflection in  $  \mathfrak C $, 
is called a reflector.

Examples. In the category of groups the quotient group  $  G/G  ^  \prime  $
of an arbitrary group  $  G $
by its commutator subgroup is a reflection of  $  G $
in the subcategory of Abelian groups. For an Abelian group  $  A $, 
the quotient group  $  A/T( A) $
by its torsion subgroup  $  T( A) $
is a reflection of  $  A $
in the full subcategory of torsion-free Abelian groups. The [[injective hull]]  $  \widetilde{A}  $
of the group  $  A/T( A) $
is a reflection of the groups  $  A $
and  $  A/T( A) $
in the subcategory of full torsion-free Abelian groups.

Reflections are usually examined in full subcategories. A full subcategory  $  \mathfrak C $
of a category  $  \mathfrak K $
in which there are reflections for all objects of  $  \mathfrak K $
is called reflective (cf. [[Reflective subcategory|Reflexive category]]).

====Comments====
The reflection of an object solves a universal problem (cf. [[Universal problems|Universal problems]]).