Bifunctor
A mapping , defined on the Cartesian product of two categories
and
with values in
, which assigns to each pair of objects
,
some object
, and to each pair of morphisms
![]() |
the morphism
![]() | (1) |
The following conditions
![]() | (2) |
![]() |
must also be met. In such a case one says that the functor 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
![]() | (1prm) |
![]() | (2prm) |
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) |
Bifunctor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bifunctor&oldid=46056