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''multi-place functor''

A function of several arguments, defined on categories, taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precisely, let  $  n $
categories  $  \mathfrak K _ {1} \dots \mathfrak K _ {n} $
be given. Construct the Cartesian product category  $  \mathfrak K = \overline{\mathfrak K}\; _ {1} \times \dots \times \overline{\mathfrak K}\; _ {n} $, 
where each category  $  \overline{\mathfrak K}\; _ {i} $
is either  $  \mathfrak K _ {i} $
or the opposite category  $  \mathfrak K _ {i}  ^ {*} $. 
A one-place covariant functor  $  F $
from  $  \mathfrak K $
with values in a category  $  \mathfrak C $
is called an  $  n $-
place functor on  $  \mathfrak K _ {1} \dots \mathfrak K _ {n} $
with values in  $  \mathfrak C $. 
The functor  $  F $
is covariant in those arguments which correspond to the factors  $  \mathfrak K _ {i} $
in  $  \mathfrak K $, 
and contravariant in the remaining arguments.

The conditions which must be satisfied by a mapping  $  F :  \mathfrak K \rightarrow \mathfrak C $
are given below (in the case  $  n = 2 $, 
with the first argument contravariant and the second covariant). The functor  $  F :  \mathfrak K _ {1}  ^ {*} \times \mathfrak K _ {2} \rightarrow \mathfrak C $
associates to each pair of objects  $  ( A , B ) $, 
$  A \in  \mathop{\rm Ob}  \mathfrak K _ {1} $, 
$  B \in  \mathop{\rm Ob}  \mathfrak K _ {2} $, 
an object  $  F ( A , B ) \in  \mathop{\rm Ob}  \mathfrak C $
and to each pair of morphisms  $  ( \alpha , \beta ) $, 
where

$$ 
\alpha :  A  \rightarrow  A _ {1}  \in \ 
 \mathop{\rm Mor}  \mathfrak K _ {1} ,\ \ 
\beta :  B  \rightarrow  B _ {1}  \in \ 
 \mathop{\rm Mor}  \mathfrak K _ {2} ,
$$

a morphism

$$ 
F ( \alpha , \beta ) :  F ( A _ {1} , B )  \rightarrow  F
( A , B _ {1} )  \in   \mathop{\rm Mor}  \mathfrak C ,
$$

in such a way that the following conditions are satisfied:

1)  $  F ( 1 _ {A} , 1 _ {B} ) = 1 _ {F ( A , B ) }  $
for any pair of objects  $  A , B $;

2) if  $  \alpha :  A \rightarrow A _ {1} $, 
$  \alpha _ {1} :  A _ {1} \rightarrow A _ {2} $, 
$  \alpha , \alpha _ {1} \in  \mathop{\rm Mor}  \mathfrak K _ {1} $, 
$  \beta :  B \rightarrow B _ {1} $, 
$  \beta _ {1} :  B _ {1} \rightarrow B _ {2} $, 
$  \beta , \beta _ {1} \in  \mathop{\rm Mor}  \mathfrak K _ {2} $, 
then

$$ 
F ( \alpha _ {1} \alpha , \beta _ {1} \beta )  = \ 
F ( \alpha , \beta _ {1} ) F ( \alpha _ {1} , \beta ) .
$$

Examples of multi-functors.

A) Let  $  \mathfrak K $
be a category with finite products. Then the product of  $  n $
objects can be considered as an  $  n $-
place functor that is covariant in all its arguments, defined on  $  \mathfrak K  ^ {n} = \mathfrak K \times \dots \times \mathfrak K $(
$  n $
times) and taking values in  $  \mathfrak K $. 
Similar functors can be constructed for coproducts, etc.

B) Let  $  \mathfrak K $
be an arbitrary category. Associate with each pair of objects  $  A , B $
from  $  \mathfrak K $
the set of morphisms  $  H _ {\mathfrak K }  ( A , B ) $
and with each pair of morphism  $  \alpha :  A \rightarrow A _ {1} $, 
$  \beta :  B \rightarrow B _ {1} $
the mapping  $  H _ {\mathfrak K }  ( \alpha , \beta ) :  H _ {\mathfrak K} ( A _ {1} , B ) \rightarrow H _ {\mathfrak K} ( A , B _ {1} ) $
given as follows: if  $  \phi :  A _ {1} \rightarrow B $
then  $  H _ {\mathfrak K} ( \alpha , \beta ) ( \phi ) = \beta \phi \alpha $. 
This construction gives a two-place functor from  $  \mathfrak K  ^ {*} \times \mathfrak K $
into the category of sets that is contravariant in its first argument and covariant in its second.

If  $  \mathfrak K $
is an [[Additive category|additive category]], then this functor can be regarded as taking values in the category of Abelian groups.

C) Let  $  \mathfrak K $
be a category with finite products. Consider the product as a two-place functor  $  \times :  \mathfrak K \times \mathfrak K \rightarrow \mathfrak K $. 
Then by combining Examples A) and B) it is possible to construct three-place functors  $  H _ {\mathfrak K} ( A , B \times C ) $
and  $  H _ {\mathfrak K} ( A \times B , C ) $. 
The first functor is naturally equivalent to the functor  $  H _ {\mathfrak K} ( A , B ) \times H _ {\mathfrak K} ( A , C ) $. 
If  $  \mathfrak C $
is the category of sets (cf. [[Sets, category of|Sets, category of]]), the second functor is naturally equivalent to the functor  $  H _ {\mathfrak C} ( A , H _ {\mathfrak C} ( B , C ) ) $.

D) Let  $  \theta $
be a [[Small category|small category]] and let  $  F ( \theta , \mathfrak C ) $
be the category of diagrams over the category of sets  $  \mathfrak C $
with scheme  $  \theta $, 
that is, the category of one-place covariant functors and their natural transformations. A two-place functor  $  E :  \theta \times F ( \theta , \mathfrak C ) \rightarrow \mathfrak C $
which is covariant in both arguments is constructed as follows: If  $  A \in  \mathop{\rm Ob}  \theta $
and  $  F \in  \mathop{\rm Ob}  F ( \theta , \mathfrak C ) $, 
then  $  E ( A , F ) = F ( A) $; 
if  $  \alpha :  A \rightarrow B \in  \mathop{\rm Mor}  \theta $
and  $  \sigma :  F \rightarrow G $
is a natural transformation, then  $  E ( \alpha , \sigma ) = \sigma _ {B} F ( \alpha ) = G ( \alpha ) \sigma _ {A} $. 
The functor  $  E $
is called the  "evaluation functorevaluation functor" . This functor is naturally equivalent to the functor  $   \mathop{\rm Nat} ( H _ {A} , F ) :  \theta \times F ( \theta , \mathfrak C ) \rightarrow \mathfrak C $, 
which associates with an object  $  A \in \theta $
and a functor  $  F :  \theta \rightarrow \mathfrak C $
the set of natural transformations of the [[Representable functor|representable functor]]  $  H _ {A} $
into  $  F $(
Yoneda's lemma).

====Comments====
A two-place functor is often called a [[bifunctor]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,   "Theory of categories" , Acad. Press  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>