In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, profunctors are a generalization of
relations and also of
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
s.
Definition
A profunctor (also named
distributor
A distributor is an enclosed rotating switch used in spark-ignition internal combustion engines that have mechanically timed ignition. The distributor's main function is to route high voltage current from the ignition coil to the spark plugs ...
by the French school and module by the Sydney school)
from a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
to a category
, written
:
,
is defined to be a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
:
where
denotes the
opposite category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
of
and
denotes the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
. Given morphisms
respectively in
and an element
, we write
to denote the actions.
Using the
cartesian closure of
, the
category of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-cat ...
, the profunctor
can be seen as a functor
:
where
denotes the category
of
presheaves over
.
A correspondence from
to
is a profunctor
.
Profunctors as categories
An equivalent definition of a profunctor
is a category whose objects are the disjoint union of the objects of
and the objects of
, and whose morphisms are the morphisms of
and the morphisms of
, plus zero or more additional morphisms from objects of
to objects of
. The sets in the formal definition above are the hom-sets between objects of
and objects of
. (These are also known as het-sets, since the corresponding morphisms can be called ''heteromorphisms''.
[heteromorphism]) The previous definition can be recovered by the restriction of the hom-functor
to
.
This also makes it clear that a profunctor can be thought of as a relation between the objects of
and the objects of
, where each member of the relation is associated with a set of morphisms. A functor is a special case of a profunctor in the same way that a function is a special case of a relation.
Composition of profunctors
The composite
of two profunctors
:
and
is given by
:
where
is the left
Kan extension
Kan extensions are Universal property, universal constructs in category theory, a branch of mathematics. They are closely related to Adjoint functors, adjoints, but are also related to Limit (category theory), limits and End (category theory), ends ...
of the functor
along the
Yoneda functor
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
of
(which to every object
of
associates the functor
).
It can be shown that
:
where
is the least equivalence relation such that
whenever there exists a morphism
in
such that
:
and
.
Equivalently, profunctor composition can be written using a
coend
:
The bicategory of profunctors
Composition of profunctors is associative only up to isomorphism (because the product is not strictly associative in Set). The best one can hope is therefore to build a
bicategory
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomor ...
Prof whose
* 0-cells are
small categories,
* 1-cells between two small categories are the profunctors between those categories,
* 2-cells between two profunctors are the
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s between those profunctors.
Properties
Lifting functors to profunctors
A functor
can be seen as a profunctor
by postcomposing with the Yoneda functor:
:
.
It can be shown that such a profunctor
has a right adjoint. Moreover, this is a characterization: a profunctor
has a right adjoint if and only if
factors through the
Cauchy completion
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
of
, i.e. there exists a functor
such that
.
References
*
*
*
*
* {{nlab, id=heteromorphism, title=Heteromorphism
Functors