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''reflector of an object of a category''
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805301.png" /> be a subcategory of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805302.png" />; an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805303.png" /> is called a reflection of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805304.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805305.png" />, or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805307.png" />-reflection, if there exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805308.png" /> such that for any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r0805309.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053010.png" /> the mapping
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053011.png" /></td> </tr></table>
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''reflector of an object of a category''
  
is bijective. In other words, for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053012.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053014.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053015.png" />-reflection of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053016.png" /> is not uniquely defined, but any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053017.png" />-reflections of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053018.png" /> are isomorphic. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053019.png" />-reflection of an initial object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053020.png" /> is an initial object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053021.png" />. The left adjoint of the inclusion functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053022.png" /> (if it exists), i.e. the functor assigning to an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053023.png" /> its reflection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053024.png" />, is called a reflector.
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Let  $  \mathfrak C $
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be a subcategory of a category  $  \mathfrak K $;
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an object $  B \in \mathfrak C $
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is called a reflection of an object $  A \in \mathfrak K $
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in  $  \mathfrak C $,
 +
or a  $  \mathfrak C $-
 +
reflection, if there exists a morphism  $  \pi : A \rightarrow B $
 +
such that for any object $  X $
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of $  \mathfrak C $
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the mapping
  
Examples. In the category of groups the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053025.png" /> of an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053026.png" /> by its commutator subgroup is a reflection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053027.png" /> in the subcategory of Abelian groups. For an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053028.png" />, the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053029.png" /> by its torsion subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053030.png" /> is a reflection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053031.png" /> in the full subcategory of torsion-free Abelian groups. The [[injective hull]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053032.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053033.png" /> is a reflection of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053035.png" /> in the subcategory of full torsion-free Abelian groups.
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$$
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H _ {X} ( \pi ) : H _ {\mathfrak C} ( B, X)  \rightarrow  H _ {\mathfrak K} ( A, X)
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$$
  
Reflections are usually examined in full subcategories. A full subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053036.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053037.png" /> in which there are reflections for all objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080530/r08053038.png" /> is called reflective (cf. [[Reflective subcategory|Reflexive category]]).
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is bijective. In other words, for any morphism  $  \alpha :  A \rightarrow X $
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there is a unique morphism  $  \alpha  ^  \prime  :  B \rightarrow X \in \mathfrak C $
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such that  $  \alpha = \pi \alpha  ^  \prime  $.  
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A  $  \mathfrak C $-
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reflection of an object  $  A $
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is not uniquely defined, but any two  $  \mathfrak C $-
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reflections of an object  $  A $
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are isomorphic. The  $  \mathfrak C $-
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reflection of an initial object of  $  \mathfrak K $
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is an initial object in  $  \mathfrak C $.  
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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 $
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its reflection in $  \mathfrak C $,
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is called a reflector.
  
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Examples. In the category of groups the quotient group  $  G/G  ^  \prime  $
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of an arbitrary group  $  G $
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by its commutator subgroup is a reflection of  $  G $
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in the subcategory of Abelian groups. For an Abelian group  $  A $,
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the quotient group  $  A/T( A) $
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by its torsion subgroup  $  T( A) $
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is a reflection of  $  A $
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in the full subcategory of torsion-free Abelian groups. The [[injective hull]]  $  \widetilde{A}  $
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of the group  $  A/T( A) $
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is a reflection of the groups  $  A $
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and  $  A/T( A) $
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in the subcategory of full torsion-free Abelian groups.
  
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Reflections are usually examined in full subcategories. A full subcategory  $  \mathfrak C $
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of a category  $  \mathfrak K $
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in which there are reflections for all objects of  $  \mathfrak K $
 +
is called reflective (cf. [[Reflective subcategory|Reflexive category]]).
  
 
====Comments====
 
====Comments====
 
The reflection of an object solves a universal problem (cf. [[Universal problems|Universal problems]]).
 
The reflection of an object solves a universal problem (cf. [[Universal problems|Universal problems]]).

Latest revision as of 08:10, 6 June 2020


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. Reflexive category).

Comments

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

How to Cite This Entry:
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=39564
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article