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A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552801.png" /> be a [[Category|category]] with null (or zero) morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552802.png" /> is called a kernel of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552803.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552804.png" /> and if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552805.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552806.png" /> can be uniquely represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552807.png" />. A kernel of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552808.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k0552809.png" />.
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A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a [[Category|category]] with null (or zero) morphisms. A morphism $\mu : K \to A$ is called a kernel of a morphism $\alpha : A \to B$ if $\mu \, \alpha = 0$ and if every morphism $\phi$ for which $\phi \, \alpha = 0$ can be uniquely represented as $\phi = \psi \, \mu$. A kernel of a morphism $\alpha$ is denoted by $\ker \alpha$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528011.png" /> are both kernels of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528013.png" /> for a unique [[Isomorphism|isomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528014.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528015.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528016.png" /> is an isomorphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528017.png" /> is a kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528018.png" />. Thus, the kernels of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528019.png" /> form a subobject of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528020.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528021.png" />.
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If $\mu$ and $\mu '$ are both kernels of $\alpha$, then $\mu ' = \xi \, \mu$ for a unique [[Isomorphism|isomorphism]] $\xi$. Conversely, if $\mu = \ker \alpha$ and if $\xi$ is an isomorphism, then $\mu ' = \xi \, \mu$ is a kernel of $\alpha$. Thus, the kernels of a morphism $\alpha$ form a subobject of $A$, denoted by $\operatorname{Ker} \alpha$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528023.png" /> is a [[Monomorphism|monomorphism]]. In general, the converse is not true; a monomorphism which occurs as a kernel is called a [[Normal monomorphism|normal monomorphism]]. The kernel of the null morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528024.png" /> is the identity morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528025.png" />. The kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528026.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528027.png" /> contains a null object (cf. [[Null object of a category|Null object of a category]]).
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If $\mu = \ker \alpha$, then $\mu$ is a [[Monomorphism|monomorphism]]. In general, the converse is not true; a monomorphism which occurs as a kernel is called a [[Normal monomorphism|normal monomorphism]]. The kernel of the null morphism $0 : A \to B$ is the identity morphism $\mathbf{1}_A$. The kernel of $\mathbf{1}_A$ exists if and only if $\mathfrak{K}$ contains a null object (cf. [[Null object of a category|Null object of a category]]).
  
Kernels do not always exist in a category with null morphisms. On the other hand, in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528028.png" /> with a null object a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528029.png" /> has a kernel if and only if a pullback of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528031.png" /> exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528032.png" />.
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Kernels do not always exist in a category with null morphisms. On the other hand, in a category $\mathfrak{K}$ with a null object a morphism $\alpha : A \to B$ has a kernel if and only if a pullback of $\alpha$ and $0 : 0 \to B$ exists in $\mathfrak{K}$.
  
 
The concept of the  "kernel of a morphism"  is the dual to that of the  "cokernel of a morphism" .
 
The concept of the  "kernel of a morphism"  is the dual to that of the  "cokernel of a morphism" .
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====Comments====
 
====Comments====
The concept  "kernel of a pair of morphismskernel of a pair of morphisms"  (not to be confused with  "kernel pair of a morphism" ) is also frequently used. In English, the usual name for this concept is an equalizer. An equalizer of a parallel pair of morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528033.png" /> is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528035.png" /> and such that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528036.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528037.png" /> factors uniquely through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528038.png" />. Kernels are a special case of equalizers: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528039.png" /> is a kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528040.png" /> if and only if it is an equalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528042.png" />. Conversely, in an [[Additive category|additive category]] an equalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528044.png" /> is the same thing as a kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055280/k05528045.png" />; but in general the notion of equalizer is more widely applicable, since it does not require the existence of null morphisms. A monomorphism which occurs as an equalizer is called a regular monomorphism.
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The concept  "kernel of a pair of morphismskernel of a pair of morphisms"  (not to be confused with  "kernel pair of a morphism" ) is also frequently used. In English, the usual name for this concept is an equalizer. An equalizer of a parallel pair of morphism $\alpha, \beta : A \to B$ is a morphism $\mu : E \to A$ such that $\mu \, \alpha = \mu \, \beta$ and such that every $\phi$ satisfying $\phi \, \alpha = \phi \, \beta$ factors uniquely through $\mu$. Kernels are a special case of equalizers: $\mu$ is a kernel of $\alpha$ if and only if it is an equalizer of $\alpha$ and $0 : A \to B$. Conversely, in an [[Additive category|additive category]] an equalizer of $\alpha$ and $\beta$ is the same thing as a kernel of $\alpha - \beta$; but in general the notion of equalizer is more widely applicable, since it does not require the existence of null morphisms. A monomorphism which occurs as an equalizer is called a regular monomorphism.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Adámek,  "Theory of mathematical structures" , Reidel  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Adámek,  "Theory of mathematical structures" , Reidel  (1983)</TD></TR></table>
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Revision as of 02:56, 14 January 2017

A concept generalizing that of the kernel of a linear transformation of vector spaces, the kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a category with null (or zero) morphisms. A morphism $\mu : K \to A$ is called a kernel of a morphism $\alpha : A \to B$ if $\mu \, \alpha = 0$ and if every morphism $\phi$ for which $\phi \, \alpha = 0$ can be uniquely represented as $\phi = \psi \, \mu$. A kernel of a morphism $\alpha$ is denoted by $\ker \alpha$.

If $\mu$ and $\mu '$ are both kernels of $\alpha$, then $\mu ' = \xi \, \mu$ for a unique isomorphism $\xi$. Conversely, if $\mu = \ker \alpha$ and if $\xi$ is an isomorphism, then $\mu ' = \xi \, \mu$ is a kernel of $\alpha$. Thus, the kernels of a morphism $\alpha$ form a subobject of $A$, denoted by $\operatorname{Ker} \alpha$.

If $\mu = \ker \alpha$, then $\mu$ is a monomorphism. In general, the converse is not true; a monomorphism which occurs as a kernel is called a normal monomorphism. The kernel of the null morphism $0 : A \to B$ is the identity morphism $\mathbf{1}_A$. The kernel of $\mathbf{1}_A$ exists if and only if $\mathfrak{K}$ contains a null object (cf. Null object of a category).

Kernels do not always exist in a category with null morphisms. On the other hand, in a category $\mathfrak{K}$ with a null object a morphism $\alpha : A \to B$ has a kernel if and only if a pullback of $\alpha$ and $0 : 0 \to B$ exists in $\mathfrak{K}$.

The concept of the "kernel of a morphism" is the dual to that of the "cokernel of a morphism" .


Comments

The concept "kernel of a pair of morphismskernel of a pair of morphisms" (not to be confused with "kernel pair of a morphism" ) is also frequently used. In English, the usual name for this concept is an equalizer. An equalizer of a parallel pair of morphism $\alpha, \beta : A \to B$ is a morphism $\mu : E \to A$ such that $\mu \, \alpha = \mu \, \beta$ and such that every $\phi$ satisfying $\phi \, \alpha = \phi \, \beta$ factors uniquely through $\mu$. Kernels are a special case of equalizers: $\mu$ is a kernel of $\alpha$ if and only if it is an equalizer of $\alpha$ and $0 : A \to B$. Conversely, in an additive category an equalizer of $\alpha$ and $\beta$ is the same thing as a kernel of $\alpha - \beta$; but in general the notion of equalizer is more widely applicable, since it does not require the existence of null morphisms. A monomorphism which occurs as an equalizer is called a regular monomorphism.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
[a2] J. Adámek, "Theory of mathematical structures" , Reidel (1983)
How to Cite This Entry:
Kernel of a morphism in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_morphism_in_a_category&oldid=40178
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article