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