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''of a morphism in a category''
 
''of a morphism in a category''
  
The concept dual to the concept of the [[Kernel of a morphism in a category|kernel of a morphism in a category]]. In categories of vector spaces, groups, rings, etc. it describes a largest [[Quotient object|quotient object]] of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231901.png" /> that annihilates the image of a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231902.png" />.
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The concept dual to the concept of the [[Kernel of a morphism in a category|kernel of a morphism in a category]]. In categories of vector spaces, groups, rings, etc. it describes a largest [[Quotient object|quotient object]] of an object $  B $
 +
that annihilates the image of a homomorphism $  \alpha : A \rightarrow B $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231903.png" /> be a category with null morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231904.png" /> is called a cokernel of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231905.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231906.png" /> and if any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231907.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231908.png" /> can be expressed in unique way as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c0231909.png" />. A cokernel of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319010.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319011.png" />.
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Let $  \mathfrak K $
 +
be a category with null morphisms. A morphism $  \nu : B \rightarrow C $
 +
is called a cokernel of a morphism $  \alpha : A \rightarrow B $
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if $  \alpha \nu = 0 $
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and if any morphism $  \phi $
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such that $  \alpha \phi = 0 $
 +
can be expressed in unique way as $  \phi = \nu \psi $.  
 +
A cokernel of a morphism $  \alpha $
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is denoted by $  \mathop{\rm coker}  \alpha $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319013.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319014.png" /> for a unique isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319015.png" />.
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If $  \nu = \mathop{\rm coker}  \alpha $
 +
and $  \nu  ^  \prime  = \mathop{\rm coker}  \alpha $
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then $  \nu  ^  \prime  = \nu \xi $
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for a unique isomorphism $  \xi $.
  
Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319017.png" /> is an isomorphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319018.png" /> is a cokernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319019.png" />. Thus, all cokernels of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319020.png" /> form a quotient object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319021.png" />, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319024.png" /> is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319025.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319026.png" />. The cokernel of the unit morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319027.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319028.png" /> contains a zero object.
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Conversely, if $  \nu = \mathop{\rm coker}  \alpha $
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and $  \xi $
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is an isomorphism, then $  \nu  ^  \prime  = \nu \xi $
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is a cokernel of $  \alpha $.  
 +
Thus, all cokernels of a morphism $  \alpha $
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form a quotient object of $  B $,  
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which is denoted by $  \mathop{\rm Coker}  \alpha $.  
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If $  \nu = \mathop{\rm coker}  \alpha $,  
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then $  \nu $
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is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism 0: A \rightarrow B $
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is $  1 _ {B} $.  
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The cokernel of the unit morphism $  1 _ {A} $
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exists if and only if $  \mathfrak K $
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contains a zero object.
  
In a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319029.png" /> with a zero object, a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319030.png" /> has a cokernel if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319031.png" /> contains a co-Cartesian square with respect to the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319033.png" />. This condition is satisfied, in particular, for any morphism of a right locally small category with a zero object and products.
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In a category $  \mathfrak K $
 +
with a zero object, a morphism $  \alpha : A \rightarrow B $
 +
has a cokernel if and only if $  \mathfrak K $
 +
contains a co-Cartesian square with respect to the morphisms $  \alpha $
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and  $  0: A \rightarrow 0 $.  
 +
This condition is satisfied, in particular, for any morphism of a right [[locally small category]] with a zero object and products.
  
 +
====Comments====
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The co-Cartesian square, or fibred sum or pushout, of two morphisms  $  f:  S \rightarrow A $,
 +
$  g:  S \rightarrow B $
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is (if it exists) a commutative diagram
  
 +
$$
  
====Comments====
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\begin{array}{rcc}
The co-Cartesian square, or fibred sum or pushout, of two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319035.png" /> is (if it exists) a commutative diagram
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S  & \stackrel{f}{\rightarrow}    & A  \\
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{ g } \downarrow  &{}  & \downarrow { {f _ {1} } }  \\
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B  & \stackrel{g_1}{\rightarrow}  &B \amalg _ {S} A  \\
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\end{array}
  
<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/c/c023/c023190/c02319036.png" /></td> </tr></table>
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$$
  
such that for any two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319039.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319040.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023190/c02319042.png" />.
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such that for any two morphisms $  a: A \rightarrow Y $,  
 +
$  b: B \rightarrow Y $
 +
such that $  af = bg $
 +
there exists a unique morphism $  h: B \amalg _ {S} A \rightarrow Y $
 +
for which $  a = hf _ {1} $,  
 +
$  b = hg _ {1} $.

Latest revision as of 16:16, 5 June 2020


of a morphism in a category

The concept dual to the concept of the kernel of a morphism in a category. In categories of vector spaces, groups, rings, etc. it describes a largest quotient object of an object $ B $ that annihilates the image of a homomorphism $ \alpha : A \rightarrow B $.

Let $ \mathfrak K $ be a category with null morphisms. A morphism $ \nu : B \rightarrow C $ is called a cokernel of a morphism $ \alpha : A \rightarrow B $ if $ \alpha \nu = 0 $ and if any morphism $ \phi $ such that $ \alpha \phi = 0 $ can be expressed in unique way as $ \phi = \nu \psi $. A cokernel of a morphism $ \alpha $ is denoted by $ \mathop{\rm coker} \alpha $.

If $ \nu = \mathop{\rm coker} \alpha $ and $ \nu ^ \prime = \mathop{\rm coker} \alpha $ then $ \nu ^ \prime = \nu \xi $ for a unique isomorphism $ \xi $.

Conversely, if $ \nu = \mathop{\rm coker} \alpha $ and $ \xi $ is an isomorphism, then $ \nu ^ \prime = \nu \xi $ is a cokernel of $ \alpha $. Thus, all cokernels of a morphism $ \alpha $ form a quotient object of $ B $, which is denoted by $ \mathop{\rm Coker} \alpha $. If $ \nu = \mathop{\rm coker} \alpha $, then $ \nu $ is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism $ 0: A \rightarrow B $ is $ 1 _ {B} $. The cokernel of the unit morphism $ 1 _ {A} $ exists if and only if $ \mathfrak K $ contains a zero object.

In a category $ \mathfrak K $ with a zero object, a morphism $ \alpha : A \rightarrow B $ has a cokernel if and only if $ \mathfrak K $ contains a co-Cartesian square with respect to the morphisms $ \alpha $ and $ 0: A \rightarrow 0 $. This condition is satisfied, in particular, for any morphism of a right locally small category with a zero object and products.

Comments

The co-Cartesian square, or fibred sum or pushout, of two morphisms $ f: S \rightarrow A $, $ g: S \rightarrow B $ is (if it exists) a commutative diagram

$$ \begin{array}{rcc} S & \stackrel{f}{\rightarrow} & A \\ { g } \downarrow &{} & \downarrow { {f _ {1} } } \\ B & \stackrel{g_1}{\rightarrow} &B \amalg _ {S} A \\ \end{array} $$

such that for any two morphisms $ a: A \rightarrow Y $, $ b: B \rightarrow Y $ such that $ af = bg $ there exists a unique morphism $ h: B \amalg _ {S} A \rightarrow Y $ for which $ a = hf _ {1} $, $ b = hg _ {1} $.

How to Cite This Entry:
Cokernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cokernel&oldid=15734
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article