A concept in
category
theory. Let
be a
functor
between categories
and
,
and let
.
The universal problem defined by this setup requires
one to find a
"best approximation"
of
in
,
i.e. a universal solution
consisting of an object
and a morphism
in
such that for every object
and every morphism
there is a unique morphism
such that
commutes.
A universal solution exists if and only if the functor
is representable (by
,
cf.
Representable functor).
There is a universal solution for each choice of
if and only if the functor
has a left
adjoint functor
.
A universal solution of a universal problem is unique up to an isomorphism.
Examples.
1)
For
the underlying (or forgetful) functor from a category
of equationally defined algebras (such as associative algebras,
commutative associative algebras, Lie algebras, vector spaces, groups) to
the category of sets and for a set
,
the universal solution is the
free algebra
over
.
2)
For
the functor which associates a Lie algebra
with every associative unitary algebra
by
and for a Lie algebra
,
the universal solution is
,
the
universal enveloping algebra
of
.
3)
For the imbedding
and a group
,
the universal solution is the
commutator factor group
of
(cf.
Commutator subgroup).
4)
In general, for every underlying (forgetful) functor
between categories of equationally defined algebras the corresponding
universal problems have universal solutions, i.e. there are
relatively free objects
for any such functor
.
5)
For
the diagonal functor and
,
the universal problem can be stated in this way: Find an object
in
and a pair of morphisms
in
such that for any object
and any pair
there exists a unique morphism
such that
commutes. The universal solution is the
coproduct
of
and
.
6)
By considering the dual situation, i.e. by using the categories dual to
and
,
one obtains the dual notions. For
the diagonal functor and
,
the universal solution of the dual universal problem is the (categorical)
product
of
and
.
7)
In general,
limits
and
colimits
can be obtained as universal solutions of appropriate universal problems.