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(Comment: Natural transformation common term, cite MacLane (1998))
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''natural transformation''
 
''natural transformation''
  
An analogue of the concept of a homomorphism of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors themselves play the role of the modules). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421502.png" /> are one-place covariant functors from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421503.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421504.png" />. A functorial morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421505.png" /> associates to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421507.png" /> a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421508.png" />, in such a way that for every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421509.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215010.png" /> the following diagram is commutative:
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An analogue of the concept of a homomorphism of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors themselves play the role of the modules). Suppose that $  F _ {1} $
 +
and $  F _ {2} $
 +
are one-place covariant functors from a category $  \mathfrak K $
 +
into a category $  \mathfrak C $.  
 +
A functorial morphism $  \phi : F _ {1} \rightarrow F _ {2} $
 +
associates to each object $  A $
 +
of $  \mathfrak K $
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a morphism $  \phi _ {A} : F _ {1} ( A) \rightarrow F _ {2} ( A) $,  
 +
in such a way that for every morphism $  \alpha : A \rightarrow B $
 +
in $  \mathfrak K $
 +
the following diagram is commutative:
  
<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/f/f042/f042150/f04215011.png" /></td> </tr></table>
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$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215012.png" />, then, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215013.png" />, one obtains the so-called identity morphism of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215016.png" /> are two functorial morphisms, then, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215017.png" />, one obtains the functorial morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215018.png" />, called the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215020.png" />. Composition of functorial morphisms is associative. Therefore, for a small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215021.png" />, all functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215023.png" /> and their functorial morphisms form a so-called functor category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215024.png" />, or a category of diagrams with scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215025.png" />.
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\begin{array}{llr}
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F _ {1} ( A)  &\rightarrow ^ { {F _ 1} ( \alpha ) }  &F _ {1} ( B)  \\
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{\phi _ {A} } \downarrow  &{}  &\downarrow  {\phi _ {B} }  \\
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F _ {2} ( A)  & \mathop \rightarrow \limits _ { {F _ {2} ( \alpha ) }}  &F _ {2} ( B). \\
 +
\end{array}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215026.png" /> be a functorial morphism and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215028.png" /> be two functors. The formulas
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$$
  
<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/f/f042/f042150/f04215029.png" /></td> </tr></table>
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If  $  F _ {1} = F _ {2} $,
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then, setting  $  \phi _ {A} = 1 _ {F _ {1}  ( A) } $,
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one obtains the so-called identity morphism of the functor  $  F _ {1} $.
 +
If  $  \phi : F _ {1} \rightarrow F _ {2} $
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and  $  \psi : F _ {2} \rightarrow F _ {3} $
 +
are two functorial morphisms, then, setting  $  ( \phi \psi ) _ {A} = \phi _ {A} \psi _ {A} $,
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one obtains the functorial morphism  $  \phi \psi : F _ {1} \rightarrow F _ {3} $,
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called the product of  $  \phi $
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and  $  \psi $.  
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Composition of functorial morphisms is associative. Therefore, for a small category  $  \mathfrak K $,
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all functors from  $  \mathfrak K $
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into  $  \mathfrak C $
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and their functorial morphisms form a so-called functor category  $  \mathop{\rm Funct} ( \mathfrak K , \mathfrak C ) $,
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or a category of diagrams with scheme  $  \mathfrak K $.
  
<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/f/f042/f042150/f04215030.png" /></td> </tr></table>
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Let  $  \phi : F _ {1} \rightarrow F _ {2} : \mathfrak K \rightarrow \mathfrak C $
 +
be a functorial morphism and let  $  G: \mathfrak M \rightarrow \mathfrak K $
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and  $  H: \mathfrak C \rightarrow \mathfrak N $
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be two functors. The formulas
  
define functorial morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215032.png" />, respectively. Then for any functorial morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215034.png" />, the following relationship holds:
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$$
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\forall B \in  \mathop{\rm Ob} \mathfrak M :\
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( G \star \phi ) _ {B}  = \phi _ {G ( B) }  ,
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$$
  
<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/f/f042/f042150/f04215035.png" /></td> </tr></table>
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$$
 +
\forall A \in  \mathop{\rm Ob} \mathfrak R : \
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( \phi \star H) _ {A}  = H ( \phi _ {A} )
 +
$$
  
A functorial morphism is also called a natural transformation of functors. Functorial morphisms of many-place functors are defined by analogy with functorial morphisms of one-place functors.
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define functorial morphisms  $  G \star \phi :  GF _ {1} \rightarrow GF _ {2} $
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and  $  \phi \star H:  F _ {1} H \rightarrow F _ {2} H $,
 +
respectively. Then for any functorial morphisms $  \phi :  F _ {1} \rightarrow F _ {2} :  \mathfrak K \rightarrow \mathfrak C $
 +
and  $  \psi :  H _ {1} \rightarrow H _ {2} :  \mathfrak C \rightarrow \mathfrak N $,
 +
the following relationship holds:
  
 +
$$
 +
( \phi \star H _ {1} ) ( F _ {2} \star \psi )  = \
 +
( F _ {1} \star \psi ) ( \phi \star H _ {2} ).
 +
$$
  
 +
A functorial morphism is also called a natural transformation of functors. Functorial morphisms of many-place functors are defined by analogy with functorial morphisms of one-place functors.
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
 
<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">[a1]</TD> <TD valign="top"> B. Mitchell, ''Theory of categories'', Acad. Press  (1965)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> Saunders Mac Lane, ''Categories for the Working Mathematician'', Graduate Texts in Mathematics '''5''', Springer (1998) ISBN 0-387-98403-8</TD></TR>
+
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Saunders Mac Lane, ''Categories for the Working Mathematician'', Graduate Texts in Mathematics '''5''', Springer (1998) {{ISBN|0-387-98403-8}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 16:56, 23 November 2023


natural transformation

An analogue of the concept of a homomorphism of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors themselves play the role of the modules). Suppose that $ F _ {1} $ and $ F _ {2} $ are one-place covariant functors from a category $ \mathfrak K $ into a category $ \mathfrak C $. A functorial morphism $ \phi : F _ {1} \rightarrow F _ {2} $ associates to each object $ A $ of $ \mathfrak K $ a morphism $ \phi _ {A} : F _ {1} ( A) \rightarrow F _ {2} ( A) $, in such a way that for every morphism $ \alpha : A \rightarrow B $ in $ \mathfrak K $ the following diagram is commutative:

$$ \begin{array}{llr} F _ {1} ( A) &\rightarrow ^ { {F _ 1} ( \alpha ) } &F _ {1} ( B) \\ {\phi _ {A} } \downarrow &{} &\downarrow {\phi _ {B} } \\ F _ {2} ( A) & \mathop \rightarrow \limits _ { {F _ {2} ( \alpha ) }} &F _ {2} ( B). \\ \end{array} $$

If $ F _ {1} = F _ {2} $, then, setting $ \phi _ {A} = 1 _ {F _ {1} ( A) } $, one obtains the so-called identity morphism of the functor $ F _ {1} $. If $ \phi : F _ {1} \rightarrow F _ {2} $ and $ \psi : F _ {2} \rightarrow F _ {3} $ are two functorial morphisms, then, setting $ ( \phi \psi ) _ {A} = \phi _ {A} \psi _ {A} $, one obtains the functorial morphism $ \phi \psi : F _ {1} \rightarrow F _ {3} $, called the product of $ \phi $ and $ \psi $. Composition of functorial morphisms is associative. Therefore, for a small category $ \mathfrak K $, all functors from $ \mathfrak K $ into $ \mathfrak C $ and their functorial morphisms form a so-called functor category $ \mathop{\rm Funct} ( \mathfrak K , \mathfrak C ) $, or a category of diagrams with scheme $ \mathfrak K $.

Let $ \phi : F _ {1} \rightarrow F _ {2} : \mathfrak K \rightarrow \mathfrak C $ be a functorial morphism and let $ G: \mathfrak M \rightarrow \mathfrak K $ and $ H: \mathfrak C \rightarrow \mathfrak N $ be two functors. The formulas

$$ \forall B \in \mathop{\rm Ob} \mathfrak M :\ ( G \star \phi ) _ {B} = \phi _ {G ( B) } , $$

$$ \forall A \in \mathop{\rm Ob} \mathfrak R : \ ( \phi \star H) _ {A} = H ( \phi _ {A} ) $$

define functorial morphisms $ G \star \phi : GF _ {1} \rightarrow GF _ {2} $ and $ \phi \star H: F _ {1} H \rightarrow F _ {2} H $, respectively. Then for any functorial morphisms $ \phi : F _ {1} \rightarrow F _ {2} : \mathfrak K \rightarrow \mathfrak C $ and $ \psi : H _ {1} \rightarrow H _ {2} : \mathfrak C \rightarrow \mathfrak N $, the following relationship holds:

$$ ( \phi \star H _ {1} ) ( F _ {2} \star \psi ) = \ ( F _ {1} \star \psi ) ( \phi \star H _ {2} ). $$

A functorial morphism is also called a natural transformation of functors. Functorial morphisms of many-place functors are defined by analogy with functorial morphisms of one-place functors.

Comments

The term "natural transformation" is common.

References

[a1] B. Mitchell, Theory of categories, Acad. Press (1965)
[a2] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer (1998) ISBN 0-387-98403-8
How to Cite This Entry:
Functorial morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functorial_morphism&oldid=34054
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article