nLab bifunctor

Disambiguation

This article is about functors of two variables. Possibly the term ‘bifunctor’ has been used for a functor between bicategories (citation?), but such usage (if it exists) seems to be rare; the usual term for that is ‘pseudo functor’.¹

Definition

A bifunctor (short for binary functor, that is $2$-ary) or functor of two variables is simply a functor whose domain is the product of two categories.

For $C_1$, $C_2$ and $D$ categories, a functor

$$F : C_1 \times C_2 \to D$$

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

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.

Related concepts

• binary function, bilinear map, multilinear map

• binary morphism

• bifunctor, two-variable adjunction, Quillen bifunctor

  end/coend

• multifunctor

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1. Outside of certain computer science contexts, it is not clear that the term ‘bifunctor’ is used frequently nowadays, even for the sense of a functor of two variables. It is used more frequently in older texts. ↩