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
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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
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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=19255