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)
How to Cite This Entry:
Universal problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_problems&oldid=49092
This article was adapted from an original article by B. Pareigis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article