Contents

category theory

# Contents

## Definition

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

$F \;\colon\; C_1 \times C_2 \longrightarrow D$

is also called a bifunctor from $C_1$ and $C_2$ to $D$.

###### Remark

(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).

## Examples

Famous bifunctors are

• the hom functor

$Hom(-,-) : C^{op} \times C \to Set$

on any locally small category $C$, or if $C$ is a closed category, the internal hom functor

$[-,-] : C^{op} \times C \to C \,.$
• on every monoidal category $(C, \otimes)$ the tensor product functor

$\otimes : C \times C \to C$

A bifunctor of the form $D^{op} \times C \to Set$ is called a profunctor.

In the generality of enriched category theory (hence for enriched functors on enriched product categories):