This article is about functors on product categories. For morphisms between bicategories see 2-functor and pseudofunctor.
By a bifunctor (short for binary functor, that is $2$-ary) or functor of two variables is simply a functor whose domain is a product category:
For $C_1$, $C_2$ and $D$ categories, a functor
is also called a bifunctor from $C_1$ and $C_2$ to $D$.
(terminology)
While the term bicategories is used for (weak) 2-categories, the terminology “bifunctor” is not used in this context, instead one speaks of pseudo-functors or 2-functors between bicategories/2-categories.
In fact, even for the sense of a functor of 2 variable, the term “bifunctor” may not be used as frequently anymore as it used to, except maybe in parts of computer science and in model category-theory (cf. Quillen bifunctor).
Famous bifunctors are
the hom functor
on any locally small category $C$, or if $C$ is a closed category, the internal hom functor
on every monoidal category $(C, \otimes)$ the tensor product functor
A bifunctor of the form $D^{op} \times C \to Set$ is called a profunctor.
bifunctor, two-variable adjunction, Quillen bifunctor
In the generality of enriched category theory (hence for enriched functors on enriched product categories):
Last revised on August 23, 2023 at 08:34:57. See the history of this page for a list of all contributions to it.