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A concept in [[Category|category]] theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957001.png" /> be a [[Functor|functor]] between categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957003.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957004.png" />. The universal problem defined by this setup requires one to find a  "best approximation"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957005.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957006.png" />, i.e. a universal solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957007.png" /> consisting of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957008.png" /> and a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u0957009.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570010.png" /> such that for every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570011.png" /> and every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570012.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570013.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570014.png" /></td> </tr></table>
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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.
 
commutes.
  
A universal solution exists if and only if the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570015.png" /> is representable (by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570016.png" />, cf. [[Representable functor|Representable functor]]). There is a universal solution for each choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570017.png" /> if and only if the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570018.png" /> has a left [[Adjoint functor|adjoint functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570019.png" />. A universal solution of a universal problem is unique up to an isomorphism.
+
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.===
 
===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 $.
  
1) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570021.png" />, the universal solution is the [[Free algebra|free algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570022.png" />.
+
2) For $  {\mathcal G} $
 
+
the functor which associates a Lie algebra $  \mathop{\rm Lie} ( A) $
2) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570023.png" /> the functor which associates a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570024.png" /> with every associative unitary algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570025.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570026.png" /> and for a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570027.png" />, the universal solution is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570028.png" />, the [[Universal enveloping algebra|universal enveloping algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570029.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570030.png" /> and a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570031.png" />, the universal solution is the commutator factor group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570032.png" /> (cf. [[Commutator subgroup|Commutator subgroup]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570033.png" /> between categories of equationally defined algebras the corresponding universal problems have universal solutions, i.e. there are relatively free objects for any such functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570034.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570035.png" /> the diagonal functor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570036.png" />, the universal problem can be stated in this way: Find an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570038.png" /> and a pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570040.png" /> such that for any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570041.png" /> and any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570042.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570043.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570044.png" /></td> </tr></table>
+
$$
  
commutes. The universal solution is the [[Coproduct|coproduct]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570046.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570048.png" />, one obtains the dual notions. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570049.png" /> the diagonal functor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570050.png" />, the universal solution of the dual universal problem is the (categorical) product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570051.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095700/u09570053.png" />.
+
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.
 
7) In general, limits and colimits can be obtained as universal solutions of appropriate universal problems.

Latest revision as of 08:27, 6 June 2020


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