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''multi-place functor''
 
''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651801.png" /> categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651802.png" /> be given. Construct the Cartesian product category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651803.png" />, where each category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651804.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651805.png" /> or the opposite category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651806.png" />. A one-place covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651807.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651808.png" /> with values in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m0651809.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518011.png" />-place functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518012.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518013.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518014.png" /> is covariant in those arguments which correspond to the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518016.png" />, and contravariant in the remaining arguments.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518017.png" /> are given below (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518018.png" />, with the first argument contravariant and the second covariant). The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518019.png" /> associates to each pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518022.png" />, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518023.png" /> and to each pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518024.png" />, where
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518025.png" /></td> </tr></table>
+
$$
 +
\alpha : A  \rightarrow  A _ {1}  \in \
 +
\mathop{\rm Mor}  \mathfrak K _ {1} ,\ \
 +
\beta : B  \rightarrow  B _ {1}  \in \
 +
\mathop{\rm Mor}  \mathfrak K _ {2} ,
 +
$$
  
 
a morphism
 
a morphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518026.png" /></td> </tr></table>
+
$$
 +
F ( \alpha , \beta ) : F ( A _ {1} , B )  \rightarrow  F
 +
( A , B _ {1} )  \in  \mathop{\rm Mor}  \mathfrak C ,
 +
$$
  
 
in such a way that the following conditions are satisfied:
 
in such a way that the following conditions are satisfied:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518027.png" /> for any pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518028.png" />;
+
1) $  F ( 1 _ {A} , 1 _ {B} ) = 1 _ {F ( A , B ) }  $
 +
for any pair of objects $  A , B $;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518034.png" />, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518035.png" /></td> </tr></table>
+
$$
 +
F ( \alpha _ {1} \alpha , \beta _ {1} \beta )  = \
 +
F ( \alpha , \beta _ {1} ) F ( \alpha _ {1} , \beta ) .
 +
$$
  
 
Examples of multi-functors.
 
Examples of multi-functors.
  
A) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518036.png" /> be a category with finite products. Then the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518037.png" /> objects can be considered as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518038.png" />-place functor that is covariant in all its arguments, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518040.png" /> times) and taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518041.png" />. Similar functors can be constructed for coproducts, etc.
+
A) Let $  \mathfrak K $
 
+
be a category with finite products. Then the product of $  n $
B) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518042.png" /> be an arbitrary category. Associate with each pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518043.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518044.png" /> the set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518045.png" /> and with each pair of morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518047.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518048.png" /> given as follows: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518049.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518050.png" />. This construction gives a two-place functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518051.png" /> into the category of sets that is contravariant in its first argument and covariant in its second.
+
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 $(
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518052.png" /> is an [[Additive category|additive category]], then this functor can be regarded as taking values in the category of Abelian groups.
+
$  n $
 +
times) and taking values in $  \mathfrak K $.  
 +
Similar functors can be constructed for coproducts, etc.
  
C) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518053.png" /> be a category with finite products. Consider the product as a two-place functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518054.png" />. Then by combining Examples A) and B) it is possible to construct three-place functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518056.png" />. The first functor is naturally equivalent to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518057.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518058.png" /> is the category of sets (cf. [[Sets, category of|Sets, category of]]), the second functor is naturally equivalent to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518059.png" />.
+
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.
  
D) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518060.png" /> be a [[Small category|small category]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518061.png" /> be the category of diagrams over the category of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518062.png" /> with scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518063.png" />, that is, the category of one-place covariant functors and their natural transformations. A two-place functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518064.png" /> which is covariant in both arguments is constructed as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518067.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518069.png" /> is a natural transformation, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518070.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518071.png" /> is called the "evaluation functorevaluation functor" . This functor is naturally equivalent to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518072.png" />, which associates with an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518073.png" /> and a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518074.png" /> the set of natural transformations of the [[Representable functor|representable functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518075.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065180/m06518076.png" /> (Yoneda's lemma).
+
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====
 
====Comments====
A two-place functor is often called a bifunctor.
+
A two-place functor is often called a [[bifunctor]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 18:59, 6 August 2020


multi-place functor

A function of several arguments, defined on categories, taking values in a category and giving a one-place 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, 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), 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 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 $ H _ {A} $ into $ F $( Yoneda's lemma).

Comments

A two-place functor is often called a bifunctor.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
[a2] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Multi-functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-functor&oldid=18045
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article