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''of an object of a category''
 
''of an object of a category''
  
A concept generalizing the corresponding concepts in algebra and topology. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816801.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816802.png" /> is called a retract of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816803.png" /> if there exist morphisms
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A concept generalizing the corresponding concepts in algebra and topology. An object $R$ of a category $\mathfrak{K}$ is called a retract of an object $A$ if there exist morphisms
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$$
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\mu : R \rightarrow A \ \ \ \text{and}\ \ \  \nu : A \rightarrow R
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$$
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such that $\nu\mu = 1_R$. The morphism $\mu$ in this case is a [[monomorphism]] and, moreover, the equalizer of the pair of morphisms $1_A$, $\mu\nu$. Dually, the morphism $\nu$ is an [[epimorphism]] and also the co-equalizer of the pair of morphisms $1_A$, $\mu\nu$. $\mu$ is sometimes known as a ''section'' and $\nu$ as a ''retraction''.
  
<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/r081/r081680/r0816804.png" /></td> </tr></table>
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If $R$ is a retract of an object $A$ and an object $R'$ is isomorphic to $R$, then $R'$ is a retract of $A$. Therefore an isomorphism class of retracts forms a single subobject of $A$. Each retract of $A$, defined by morphisms $\mu : R \rightarrow A$ and $\nu : A \rightarrow R$, corresponds to an idempotent morphism $\phi=\mu\nu : A \rightarrow A$. Two retracts $R$ and $R'$  of an object $A$ belong to the same subobject if and only if they correspond to the same idempotent. The retracts of any object of an arbitrary category form a set.
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816805.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816806.png" /> in this case is a [[Monomorphism|monomorphism]] and, moreover, the equalizer of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816807.png" />. Dually, the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816808.png" /> is an [[Epimorphism|epimorphism]] and also the co-equalizer of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r0816809.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168010.png" /> is sometimes known as a section and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168011.png" /> as a retraction.
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168012.png" /> is a retract of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168013.png" /> and an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168014.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168016.png" /> is a retract of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168017.png" />. Therefore an isomorphism class of retracts forms a single subobject of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168018.png" />. Each retract of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168019.png" />, defined by morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168021.png" />, corresponds to an idempotent morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168022.png" />. Two retracts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168024.png" /> of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081680/r08168025.png" /> belong to the same subobject if and only if they correspond to the same idempotent. The retracts of any object of an arbitrary category form a set.
 
  
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====Comments====
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The last sentence above is true only if one assumes that all categories involved are locally small (i.e.  "have small hom-sets" ) (cf. also [[Small category]]).
  
 
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{{TEX|done}}
====Comments====
 
The last sentence above is true only if one assumes that all categories involved are locally small (i.e.  "have small hom-sets" ) (cf. also [[Small category|Small category]]).
 

Revision as of 18:18, 1 October 2017

of an object of a category

A concept generalizing the corresponding concepts in algebra and topology. An object $R$ of a category $\mathfrak{K}$ is called a retract of an object $A$ if there exist morphisms $$ \mu : R \rightarrow A \ \ \ \text{and}\ \ \ \nu : A \rightarrow R $$ such that $\nu\mu = 1_R$. The morphism $\mu$ in this case is a monomorphism and, moreover, the equalizer of the pair of morphisms $1_A$, $\mu\nu$. Dually, the morphism $\nu$ is an epimorphism and also the co-equalizer of the pair of morphisms $1_A$, $\mu\nu$. $\mu$ is sometimes known as a section and $\nu$ as a retraction.

If $R$ is a retract of an object $A$ and an object $R'$ is isomorphic to $R$, then $R'$ is a retract of $A$. Therefore an isomorphism class of retracts forms a single subobject of $A$. Each retract of $A$, defined by morphisms $\mu : R \rightarrow A$ and $\nu : A \rightarrow R$, corresponds to an idempotent morphism $\phi=\mu\nu : A \rightarrow A$. Two retracts $R$ and $R'$ of an object $A$ belong to the same subobject if and only if they correspond to the same idempotent. The retracts of any object of an arbitrary category form a set.


Comments

The last sentence above is true only if one assumes that all categories involved are locally small (i.e. "have small hom-sets" ) (cf. also Small category).

How to Cite This Entry:
Retract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Retract&oldid=41988
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article