Multi-functor

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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 categories be given. Construct the Cartesian product category , where each category is either or the opposite category . A one-place covariant functor from with values in a category is called an -place functor on with values in . The functor is covariant in those arguments which correspond to the factors in , and contravariant in the remaining arguments.

The conditions which must be satisfied by a mapping are given below (in the case , with the first argument contravariant and the second covariant). The functor associates to each pair of objects , , , an object and to each pair of morphisms , where

a morphism

in such a way that the following conditions are satisfied:

1) for any pair of objects ;

2) if , , , , , , then

Examples of multi-functors.

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

B) Let be an arbitrary category. Associate with each pair of objects from the set of morphisms and with each pair of morphism , the mapping given as follows: if then . This construction gives a two-place functor from into the category of sets that is contravariant in its first argument and covariant in its second.

If is an additive category, then this functor can be regarded as taking values in the category of Abelian groups.

C) Let be a category with finite products. Consider the product as a two-place functor . Then by combining Examples A) and B) it is possible to construct three-place functors and . The first functor is naturally equivalent to the functor . If is the category of sets (cf. Sets, category of), the second functor is naturally equivalent to the functor .

D) Let be a small category and let be the category of diagrams over the category of sets with scheme , that is, the category of one-place covariant functors and their natural transformations. A two-place functor which is covariant in both arguments is constructed as follows: If and , then ; if and is a natural transformation, then . The functor is called the "evaluation functorevaluation functor" . This functor is naturally equivalent to the functor , which associates with an object and a functor the set of natural transformations of the representable functor into (Yoneda's lemma).