nLab bifunctor


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 22-ary) or functor of two variables is simply a functor whose domain is a product category:

For C 1C_1, C 2C_2 and DD categories, a functor

F:C 1×C 2D F \;\colon\; C_1 \times C_2 \longrightarrow D

is also called a bifunctor from C 1C_1 and C 2C_2 to DD.


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

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


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.