A concept expressing the universality and naturalness of
many important mathematical constructions, such as a free universal
algebra, various completions, and direct and inverse limits.
Let
be a covariant functor in one argument from a category
into a category
.
induces a functor
where
is the category dual to
,
is the category of sets, and
is the basic set-valued functor. The functor
is contravariant in the first argument and covariant
in the second. Similarly, any covariant functor
induces a functor
which is also contravariant in the first argument
and covariant in the second. The functors
and
are
adjoint,
or form an
adjoint pair,
if
and
are isomorphic, that is, if there is a natural transformation
that establishes a one-to-one correspondence between the sets of morphisms
and
for all objects
and
.
The transformation
is called the
adjunction of
with
,
is called the
left adjoint
of
and
the
right adjoint
of
(this is written
,
or simply
).
The transformation
is called the
co-adjunction.
Let
.
For all
and
,
let
The families of morphisms
and
define natural transformations
and
,
called the
unit
and
co-unit
of the adjunction
.
They satisfy the following equations:
In general, a pair of natural transformations
and
leads to an adjoint pair (or adjunction) if the following equations hold:
for all objects
and
.
A natural transformation
is the unit of some adjunction if and only if for any morphism
in
there is a unique morphism
in
such that
.
This property expresses the fact that
is a free object over
with respect to the functor
in the sense of the following definition. An object
together with a morphism
is
free over an object
if every morphism
can be written uniquely in the form
for some morphism
.
A functor
has a left adjoint if and only if for every
there is an object
that is free over
with respect to
.
Examples of adjoint functors.
1)
If
,
where
is the category of sets, then
has a left adjoint only if it is representable. A representable functor
has a left adjoint if and only if all co-products
exist in
,
where
and
for all
.
2)
In the category
of sets, for any set
the basic functor
is the right adjoint of the functor
.
3)
In the category of Abelian groups, the functor
is the right adjoint of the functor
of tensor multiplication by
,
and the imbedding functor of the full subcategory of torsion
groups is the left adjoint of the functor of
taking the torsion part of any Abelian group.
4)
Let
be the forgetful functor from an arbitrary variety
of universal algebras into the category of sets. The functor
has a left adjoint
,
which assigns to every set
the free algebra of the variety
with
as set of free generators.
5)
The imbedding functor
of an arbitrary reflective subcategory
of a category
is the right adjoint of the
-reflector
(cf. also
Reflexive subcategor).
In particular, the imbedding functor of the category of Abelian groups
in the category of groups has a left adjoint, which assigns to every group
its quotient group by the commutator subgroup.
Properties of adjoint functors.
The left adjoint functor of a given functor
is uniquely determined up to isomorphism of functors. Left
adjoints commute with co-limits (e.g. co-products) and send null objects
and null morphism into null objects and null morphisms, respectively.
Let
and
be categories that are complete on the left and locally small on the left. A functor
has a left adjoint
if and only if the following conditions hold: a)
commutes with limits; b) for every
,
at least one of the sets
,
,
is non-empty; and c) for every
,
there is a set
such that every morphism
is representable in the form
,
where
,
,
.
By passing to dual categories, one may establish a duality between the
concepts of a
"left adjoint functor"
and a
"right adjoint functor" ;
this enables one to deduce the properties of right adjoints from those of left adjoints.
The concept of an adjoint functor is directly connected with the concept of a
triple
(or monad) in a category.