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''of an object in a category''
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A concept analogous to the concept of a substructure of a mathematical structure. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909801.png" /> be any [[Category|category]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909802.png" /> be a fixed object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909803.png" />. In the class of all monomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909804.png" /> with codomain (target) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909805.png" />, one may define a pre-order relation (the relation of divisibility from the right): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909806.png" /> precedes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909807.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909808.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s0909809.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098010.png" />. In fact, the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098011.png" /> is uniquely determined because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098012.png" /> is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098013.png" />: The monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098015.png" /> are equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098017.png" />. An equivalence class of monomorphisms is called a subobject of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098018.png" />. A subobject with representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098019.png" /> is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098020.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098021.png" />. It is also possible to use Hilbert's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098022.png" />-symbol to select representatives of subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098023.png" /> and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the induced topology.
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The pre-order relation between the monomorphisms with codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098024.png" /> induces a partial order relation between the subobjects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098025.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098027.png" />. This relation is analogous to the inclusion relation for subsets of a given set.
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''of an object in a category''
  
If the monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098028.png" /> is regular (cf. [[Normal monomorphism|Normal monomorphism]]), then any monomorphism equivalent to it is also regular. One can therefore speak of the regular subobjects of any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098029.png" />. In particular, the subobject represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098030.png" /> is regular. In categories with zero morphisms one similarly introduces normal subobjects. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098031.png" /> possesses a [[Bicategory(2)|bicategory]] structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098032.png" />, then a subobject <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098033.png" /> of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098034.png" /> is called admissible (with respect to this bicategory structure) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098035.png" />.
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A concept analogous to the concept of a substructure of a mathematical structure. Let  $  \mathfrak K $
 +
be any [[Category|category]] and let  $  A $
 +
be a fixed object in  $  \mathfrak K $.
 +
In the class of all monomorphisms of  $  \mathfrak K $
 +
with codomain (target) $  A $,  
 +
one may define a pre-order relation (the relation of divisibility from the right):  $  \mu :  X \rightarrow A $
 +
precedes  $  \sigma :  Y \rightarrow A $,
 +
or  $  \mu \prec \sigma $,
 +
if  $  \mu = \mu  ^  \prime  \sigma $
 +
for some  $  \mu  ^  \prime  : X \rightarrow Y $.  
 +
In fact, the morphism  $  \mu  ^  \prime  $
 +
is uniquely determined because  $  \sigma $
 +
is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain  $  A $:
 +
The monomorphisms  $  \mu :  X \rightarrow A $
 +
and  $  \sigma : Y \rightarrow A $
 +
are equivalent if and only if  $  \mu \prec \sigma $
 +
and  $  \sigma \prec \mu $.  
 +
An equivalence class of monomorphisms is called a subobject of the object  $  A $.  
 +
A subobject with representative  $  \mu : X \rightarrow A $
 +
is sometimes denoted by  $  ( \mu :  X \rightarrow A ] $
 +
or by  $  ( \mu ] $.  
 +
It is also possible to use Hilbert's $  \tau $-
 +
symbol to select representatives of subobjects of  $  A $
 +
and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the [[induced topology]].
  
 +
The pre-order relation between the monomorphisms with codomain  $  A $
 +
induces a partial order relation between the subobjects of  $  A $:
 +
$  ( \mu ] \leq  ( \sigma ] $
 +
if  $  \mu \prec \sigma $.
 +
This relation is analogous to the inclusion relation for subsets of a given set.
  
 +
If the monomorphism  $  \mu $
 +
is regular (cf. [[Normal monomorphism|Normal monomorphism]]), then any monomorphism equivalent to it is also regular. One can therefore speak of the regular subobjects of any object  $  A $.
 +
In particular, the subobject represented by  $  1 _ {A} $
 +
is regular. In categories with zero morphisms one similarly introduces normal subobjects. If  $  \mathfrak K $
 +
possesses a [[Bicategory(2)|bicategory]] structure  $  ( \mathfrak K , \mathfrak L , \mathfrak M ) $,
 +
then a subobject  $  ( \mu :  X \rightarrow A ] $
 +
of an object  $  A $
 +
is called admissible (with respect to this bicategory structure) if  $  \mu \in \mathfrak M $.
  
 
====Comments====
 
====Comments====
The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090980/s09098036.png" /> used in this article (and elsewhere in this Encyclopaedia) is not standard. Most authors do not bother to distinguish notationally between a subobject and a monomorphism which represents it.
+
The notation $  ( \mu ] $
 +
used in this article (and elsewhere in this Encyclopaedia) is not standard. Most authors do not bother to distinguish notationally between a subobject and a monomorphism which represents it.
  
 
For references see [[Category|Category]].
 
For references see [[Category|Category]].

Latest revision as of 08:24, 6 June 2020


of an object in a category

A concept analogous to the concept of a substructure of a mathematical structure. Let $ \mathfrak K $ be any category and let $ A $ be a fixed object in $ \mathfrak K $. In the class of all monomorphisms of $ \mathfrak K $ with codomain (target) $ A $, one may define a pre-order relation (the relation of divisibility from the right): $ \mu : X \rightarrow A $ precedes $ \sigma : Y \rightarrow A $, or $ \mu \prec \sigma $, if $ \mu = \mu ^ \prime \sigma $ for some $ \mu ^ \prime : X \rightarrow Y $. In fact, the morphism $ \mu ^ \prime $ is uniquely determined because $ \sigma $ is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain $ A $: The monomorphisms $ \mu : X \rightarrow A $ and $ \sigma : Y \rightarrow A $ are equivalent if and only if $ \mu \prec \sigma $ and $ \sigma \prec \mu $. An equivalence class of monomorphisms is called a subobject of the object $ A $. A subobject with representative $ \mu : X \rightarrow A $ is sometimes denoted by $ ( \mu : X \rightarrow A ] $ or by $ ( \mu ] $. It is also possible to use Hilbert's $ \tau $- symbol to select representatives of subobjects of $ A $ and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the induced topology.

The pre-order relation between the monomorphisms with codomain $ A $ induces a partial order relation between the subobjects of $ A $: $ ( \mu ] \leq ( \sigma ] $ if $ \mu \prec \sigma $. This relation is analogous to the inclusion relation for subsets of a given set.

If the monomorphism $ \mu $ is regular (cf. Normal monomorphism), then any monomorphism equivalent to it is also regular. One can therefore speak of the regular subobjects of any object $ A $. In particular, the subobject represented by $ 1 _ {A} $ is regular. In categories with zero morphisms one similarly introduces normal subobjects. If $ \mathfrak K $ possesses a bicategory structure $ ( \mathfrak K , \mathfrak L , \mathfrak M ) $, then a subobject $ ( \mu : X \rightarrow A ] $ of an object $ A $ is called admissible (with respect to this bicategory structure) if $ \mu \in \mathfrak M $.

Comments

The notation $ ( \mu ] $ used in this article (and elsewhere in this Encyclopaedia) is not standard. Most authors do not bother to distinguish notationally between a subobject and a monomorphism which represents it.

For references see Category.

How to Cite This Entry:
Subobject. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subobject&oldid=15026
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article