View source for Universal problems

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A concept in [[Category|category]] theory. Let  $  {\mathcal G} :  {\mathcal C} \rightarrow {\mathcal D} $
be a [[Functor|functor]] between categories  $  {\mathcal C} $
and  $  {\mathcal D} $, 
and let  $  D \in {\mathcal D} $. 
The universal problem defined by this setup requires one to find a  "best approximation"  of  $  D $
in  $  {\mathcal C} $, 
i.e. a universal solution  $  ( C, \iota ) $
consisting of an object  $  C \in {\mathcal C} $
and a morphism  $  \iota :  {\mathcal D} \rightarrow {\mathcal G} ( C) $
in  $  {\mathcal D} $
such that for every object  $  C  ^  \prime  \in {\mathcal C} $
and every morphism  $  f:  D \rightarrow {\mathcal G} ( C  ^  \prime  ) $
there is a unique morphism  $  g:  C \rightarrow C  ^  \prime  $
such that

$$

commutes.

A universal solution exists if and only if the functor  $  {\mathcal D} ( D, {\mathcal G} - ) :  {\mathcal C} \rightarrow  \mathop{\rm Sets} $
is representable (by  $  C $, 
cf. [[Representable functor|Representable functor]]). There is a universal solution for each choice of  $  D $
if and only if the functor  $  {\mathcal G} $
has a left [[Adjoint functor|adjoint functor]]  $  {\mathcal F} :  {\mathcal D} \rightarrow {\mathcal C} $. 
A universal solution of a universal problem is unique up to an isomorphism.

===Examples.===

1) For  $  {\mathcal G} $
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  $  X $, 
the universal solution is the [[Free algebra|free algebra]] over  $  X $.

2) For  $  {\mathcal G} $
the functor which associates a Lie algebra  $   \mathop{\rm Lie} ( A) $
with every associative unitary algebra  $  A $
by  $  [ a, b] = ab- ba $
and for a Lie algebra  $  \mathfrak g $, 
the universal solution is  $  U( \mathfrak g ) $, 
the [[Universal enveloping algebra|universal enveloping algebra]] of  $  \mathfrak g $.

3) For the imbedding  $  {\mathcal G} :   \mathop{\rm comm}.Groups \rightarrow fnnem Groups $
and a group  $  G $, 
the universal solution is the commutator factor group of  $  G $(
cf. [[Commutator subgroup|Commutator subgroup]]).

4) In general, for every underlying (forgetful) functor  $  {\mathcal G} $
between categories of equationally defined algebras the corresponding universal problems have universal solutions, i.e. there are relatively free objects for any such functor  $  {\mathcal G} $.

5) For  $  {\mathcal G} :  {\mathcal C} \rightarrow {\mathcal C} \times {\mathcal C} $
the diagonal functor and  $  ( A, B) \in {\mathcal C} \times {\mathcal C} $, 
the universal problem can be stated in this way: Find an object  $  C= A \amalg B $
in  $  {\mathcal C} $
and a pair of morphisms  $  ( \iota _ {A} :  A \rightarrow C, \iota _ {B} :  B \rightarrow C) $
in  $  {\mathcal C} \times {\mathcal C} $
such that for any object  $  C  ^  \prime  \in {\mathcal C} $
and any pair  $  ( f _ {A} :  A \rightarrow C  ^  \prime  , f _ {B} :  B \rightarrow C  ^  \prime  ) $
there exists a unique morphism  $  f :  C \rightarrow C  ^  \prime  $
such that

$$

commutes. The universal solution is the [[Coproduct|coproduct]] of  $  A $
and  $  B $.

6) By considering the dual situation, i.e. by using the categories dual to  $  {\mathcal C} $
and  $  {\mathcal D} $, 
one obtains the dual notions. For  $  {\mathcal G} :  {\mathcal C} \rightarrow {\mathcal C} \times {\mathcal C} $
the diagonal functor and  $  ( A, B) \in {\mathcal C} \times {\mathcal C} $, 
the universal solution of the dual universal problem is the (categorical) product  $  A \times B $
of  $  A $
and  $  B $.

7) In general, limits and colimits can be obtained as universal solutions of appropriate universal problems.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Pareigis,   "Categories and functors" , Acad. Press  (1970)</TD></TR></table>