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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161901.png" />, defined on the Cartesian product of two categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161903.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161904.png" />, which assigns to each pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161906.png" /> some object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b0161907.png" />, and to each pair of morphisms
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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/b/b016/b016190/b0161908.png" /></td> </tr></table>
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A mapping  $  T: \mathfrak A \times \mathfrak B \rightarrow \mathfrak C $,
 +
defined on the Cartesian product of two categories  $  \mathfrak A $
 +
and  $  \mathfrak B $
 +
with values in  $  \mathfrak C $,
 +
which assigns to each pair of objects  $  A \in \mathfrak A $,
 +
$  B \in \mathfrak B $
 +
some object  $  C \in \mathfrak C $,
 +
and to each pair of morphisms
 +
 
 +
$$
 +
\alpha : A  \rightarrow  A  ^  \prime  ,\ \
 +
\beta : B  \rightarrow  B  ^  \prime
 +
$$
  
 
the morphism
 
the morphism
  
<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/b/b016/b016190/b0161909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
T( \alpha , \beta ) : \
 +
T(A  ^  \prime  , B)  \rightarrow \
 +
T(A, B  ^  \prime  ).
 +
$$
  
 
The following conditions
 
The following conditions
  
<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/b/b016/b016190/b01619010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
 
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T(1 _ {A} , 1 _ {B} ) = \
<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/b/b016/b016190/b01619011.png" /></td> </tr></table>
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1 _ {T (A, B) }  ,
 
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$$
must also be met. In such a case one says that the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016190/b01619012.png" /> is contravariant with respect to the first argument and covariant with respect to the second.
 
  
 +
$$
 +
T ( \alpha  ^  \prime  \circ \alpha , \beta  ^  \prime  \circ \beta )  =  T (
 +
\alpha , \beta  ^  \prime  ) \circ T ( \alpha  ^  \prime  , \beta ),
 +
$$
  
 +
must also be met. In such a case one says that the functor  $  T $
 +
is contravariant with respect to the first argument and covariant with respect to the second.
  
 
====Comments====
 
====Comments====
 
What is described above is a bifunctor contravariant in its first argument and covariant in its second. A bifunctor covariant in both arguments, the more fundamental notion ([[#References|[a1]]]), has (1) and (2) replaced by
 
What is described above is a bifunctor contravariant in its first argument and covariant in its second. A bifunctor covariant in both arguments, the more fundamental notion ([[#References|[a1]]]), has (1) and (2) replaced by
  
<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/b/b016/b016190/b01619013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
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$$ \tag{1'}
 +
T ( \alpha , \beta ): \
 +
T (A, B)  \rightarrow \
 +
T (A  ^  \prime  , B  ^  \prime  ),
 +
$$
  
<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/b/b016/b016190/b01619014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2prm)</td></tr></table>
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$$ \tag{2'}
 +
T ( \alpha  ^  \prime  \alpha , \beta  ^  \prime  \beta )  = T ( \alpha
 +
^  \prime  , \beta  ^  \prime  ) \circ T ( \alpha , \beta ).
 +
$$
  
 
Similarly one can define bifunctors contravariant in both arguments and covariant in the first and contravariant in the second argument.
 
Similarly one can define bifunctors contravariant in both arguments and covariant in the first and contravariant in the second argument.

Revision as of 10:59, 29 May 2020


A mapping $ T: \mathfrak A \times \mathfrak B \rightarrow \mathfrak C $, defined on the Cartesian product of two categories $ \mathfrak A $ and $ \mathfrak B $ with values in $ \mathfrak C $, which assigns to each pair of objects $ A \in \mathfrak A $, $ B \in \mathfrak B $ some object $ C \in \mathfrak C $, and to each pair of morphisms

$$ \alpha : A \rightarrow A ^ \prime ,\ \ \beta : B \rightarrow B ^ \prime $$

the morphism

$$ \tag{1 } T( \alpha , \beta ) : \ T(A ^ \prime , B) \rightarrow \ T(A, B ^ \prime ). $$

The following conditions

$$ \tag{2 } T(1 _ {A} , 1 _ {B} ) = \ 1 _ {T (A, B) } , $$

$$ T ( \alpha ^ \prime \circ \alpha , \beta ^ \prime \circ \beta ) = T ( \alpha , \beta ^ \prime ) \circ T ( \alpha ^ \prime , \beta ), $$

must also be met. In such a case one says that the functor $ T $ is contravariant with respect to the first argument and covariant with respect to the second.

Comments

What is described above is a bifunctor contravariant in its first argument and covariant in its second. A bifunctor covariant in both arguments, the more fundamental notion ([a1]), has (1) and (2) replaced by

$$ \tag{1'} T ( \alpha , \beta ): \ T (A, B) \rightarrow \ T (A ^ \prime , B ^ \prime ), $$

$$ \tag{2'} T ( \alpha ^ \prime \alpha , \beta ^ \prime \beta ) = T ( \alpha ^ \prime , \beta ^ \prime ) \circ T ( \alpha , \beta ). $$

Similarly one can define bifunctors contravariant in both arguments and covariant in the first and contravariant in the second argument.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
Bifunctor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bifunctor&oldid=17051
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article