Universal problems

A concept in category theory. Let $ {\mathcal G} : {\mathcal C} \rightarrow {\mathcal D} $ be a 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. ==='"`UNIQ--h-0--QINU`"'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 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

[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)