Difference between revisions of "Reflection of an object of a category"
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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. | 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. | ||
− | 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. | + | 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. |
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]]). | 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]]). |
Revision as of 20:29, 30 October 2016
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