category theory

# Contents

## 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 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 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

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

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.

Revised on September 2, 2012 05:15:23 by Todd Trimble (67.81.93.25)