Multi-functor

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