Universal problems
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.
References
| [a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |
| [a2] | B. Pareigis, "Categories and functors" , Acad. Press (1970) |
Universal problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_problems&oldid=49092