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 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).
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 |
Multi-functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-functor&oldid=47918