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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>
| + | ''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.
| + | 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 |
| | | |
− | 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.
| + | $$ |
| + | H _ {X} ( \pi ) : H _ {\mathfrak C} ( B, X) \rightarrow H _ {\mathfrak K} ( A, X) |
| + | $$ |
| | | |
− | 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]]).
| + | 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==== | | ====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]]). |
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).
The reflection of an object solves a universal problem (cf. Universal problems).