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Difference between revisions of "Bifunctor"

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m (tex encoded by computer)
m (eqref)
 
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the morphism
 
the morphism
  
$$ \tag{1 }
+
\begin{equation} \label{eq1}
 
T( \alpha , \beta ) : \  
 
T( \alpha , \beta ) : \  
 
T(A  ^  \prime  , B)  \rightarrow \  
 
T(A  ^  \prime  , B)  \rightarrow \  
 
T(A, B  ^  \prime  ).
 
T(A, B  ^  \prime  ).
$$
+
\end{equation}
  
 
The following conditions
 
The following conditions
  
$$ \tag{2 }
+
\begin{equation} \label{eq2}
T(1 _ {A} , 1 _ {B} )  = \
+
T(1 _ {A} , 1 _ {B} )  = 1 _ {T (A, B) }  ,
1 _ {T (A, B) }  ,
+
\end{equation}
$$
 
  
 
$$  
 
$$  
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$$
 
$$
  
must also be met. In such a case one says that the functor $ T $
+
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.
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 \eqref{eq1} and \eqref{eq2} replaced by
  
$$ \tag{1'}
+
\begin{equation} \label{eq1bis} \tag{1'}
 
T ( \alpha , \beta ): \  
 
T ( \alpha , \beta ): \  
 
T (A, B)  \rightarrow \  
 
T (A, B)  \rightarrow \  
 
T (A  ^  \prime  , B  ^  \prime  ),
 
T (A  ^  \prime  , B  ^  \prime  ),
$$
+
\end{equation}
  
$$ \tag{2'}
+
\begin{equation} \label{eq2bis}\tag{2'}
 
T ( \alpha  ^  \prime  \alpha , \beta  ^  \prime  \beta )  =  T ( \alpha
 
T ( \alpha  ^  \prime  \alpha , \beta  ^  \prime  \beta )  =  T ( \alpha
 
  ^  \prime  , \beta  ^  \prime  ) \circ T ( \alpha , \beta ).
 
  ^  \prime  , \beta  ^  \prime  ) \circ T ( \alpha , \beta ).
$$
+
\end{equation}
  
 
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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR>
 +
</table>

Latest revision as of 19:02, 29 March 2024


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

\begin{equation} \label{eq1} T( \alpha , \beta ) : \ T(A ^ \prime , B) \rightarrow \ T(A, B ^ \prime ). \end{equation}

The following conditions

\begin{equation} \label{eq2} T(1 _ {A} , 1 _ {B} ) = 1 _ {T (A, B) } , \end{equation}

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

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 \eqref{eq1} and \eqref{eq2} replaced by

\begin{equation} \label{eq1bis} \tag{1'} T ( \alpha , \beta ): \ T (A, B) \rightarrow \ T (A ^ \prime , B ^ \prime ), \end{equation}

\begin{equation} \label{eq2bis}\tag{2'} T ( \alpha ^ \prime \alpha , \beta ^ \prime \beta ) = T ( \alpha ^ \prime , \beta ^ \prime ) \circ T ( \alpha , \beta ). \end{equation}

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=55688
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article