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A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.
 
A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108201.png" /> be a covariant functor in one argument from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108202.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108203.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108204.png" /> induces a functor
+
Let $  F : \mathfrak K \rightarrow \mathfrak C $
 +
be a covariant functor in one argument from a category $  \mathfrak K $
 +
into a category $  \mathfrak C $.  
 +
$  F $
 +
induces a functor
  
<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/a/a010/a010820/a0108205.png" /></td> </tr></table>
+
$$
 +
H  ^ {F} ( X , Y )  = \
 +
H _ {\mathfrak C }  ( F (X) , Y ) : \mathfrak K  ^ {*} \times
 +
\mathfrak C  \rightarrow  \mathfrak S ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108206.png" /> is the category dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108208.png" /> is the category of sets, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a0108209.png" /> is the basic set-valued functor. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082010.png" /> is contravariant in the first argument and covariant in the second. Similarly, any covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082011.png" /> induces a functor
+
where $  \mathfrak K  ^ {*} $
 +
is the category dual to $  \mathfrak K $,  
 +
$  \mathfrak S $
 +
is the category of sets, and $  H _ {\mathfrak C} ( X , Y ) : \mathfrak K  ^ {*} \times \mathfrak C \rightarrow \mathfrak S $
 +
is the basic set-valued functor. The functor $  H  ^ {F} $
 +
is contravariant in the first argument and covariant in the second. Similarly, any covariant functor $  G : \mathfrak C \rightarrow \mathfrak K $
 +
induces a functor
  
<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/a/a010/a010820/a01082012.png" /></td> </tr></table>
+
$$
 +
H _ {G} ( X , Y )  = \
 +
H _ {\mathfrak K }  ( X , G (Y) ) : \
 +
\mathfrak K  ^ {*} \times \mathfrak C  \rightarrow  \mathfrak S ,
 +
$$
  
which is also contravariant in the first argument and covariant in the second. The functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082014.png" /> are adjoint, or form an adjoint pair, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082016.png" /> are isomorphic, that is, if there is a natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082017.png" /> that establishes a one-to-one correspondence between the sets of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082019.png" /> for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082021.png" />. The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082022.png" /> is called the adjunction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082025.png" /> is called the left adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082027.png" /> the right adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082028.png" /> (this is written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082029.png" />, or simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082030.png" />). The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082031.png" /> is called the co-adjunction.
+
which is also contravariant in the first argument and covariant in the second. The functors $  F $
 +
and $  G $
 +
are adjoint, or form an adjoint pair, if $  H  ^ {F} $
 +
and $  H _ {G} $
 +
are isomorphic, that is, if there is a natural transformation $  \theta : H  ^ {F} \rightarrow H _ {G} $
 +
that establishes a one-to-one correspondence between the sets of morphisms $  H _ {\mathfrak C} ( F (X) , Y ) $
 +
and $  H _ {\mathfrak K} ( X , G (Y) ) $
 +
for all objects $  X \in  \mathop{\rm Ob}  \mathfrak K $
 +
and $  Y \in  \mathop{\rm Ob}  \mathfrak C $.  
 +
The transformation $  \theta $
 +
is called the adjunction of $  F $
 +
with $  G $,  
 +
$  F $
 +
is called the left adjoint of $  G $
 +
and $  G $
 +
the right adjoint of $  F $(
 +
this is written $  \theta : F \lTL G $,  
 +
or simply $  F \lTL G $).  
 +
The transformation $  \theta  ^ {-1} : H _ {G} \rightarrow H  ^ {F} $
 +
is called the co-adjunction.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082032.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082034.png" />, let
+
Let $  \theta : F \lTL G $.  
 +
For all $  X \in  \mathop{\rm Ob}  \mathfrak K $
 +
and $  Y \in  \mathop{\rm Ob}  \mathfrak C $,  
 +
let
  
<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/a/a010/a010820/a01082035.png" /></td> </tr></table>
+
$$
 +
\epsilon _ {X}  = \theta
 +
( 1 _ {F (X) }  ) ,\ \
 +
\eta _ {Y}  = \theta  ^ {-1}
 +
( 1 _ {G (Y) }  ) .
 +
$$
  
The families of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082037.png" /> define natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082039.png" />, called the unit and co-unit of the adjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082040.png" />. They satisfy the following equations:
+
The families of morphisms $  \{ \epsilon _ {X} \} $
 +
and $  \{ \eta _ {Y} \} $
 +
define natural transformations $  \epsilon :   \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $
 +
and $  \eta : F G \rightarrow  \mathop{\rm Id} _ {\mathfrak C} $,  
 +
called the unit and co-unit of the adjunction $  \theta $.  
 +
They satisfy the following equations:
  
<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/a/a010/a010820/a01082041.png" /></td> </tr></table>
+
$$
 +
G ( \eta _ {Y} )
 +
\epsilon _ {G (Y) }  = \
 +
1 _ {G (Y) }  ,\ \
 +
\eta _ {F (X) }
 +
F ( \epsilon _ {X} )  = \
 +
1 _ {F (X) }  .
 +
$$
  
In general, a pair of natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082043.png" /> leads to an adjoint pair (or adjunction) if the following equations hold:
+
In general, a pair of natural transformations $  \phi :   \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $
 +
and $  \psi : F G \rightarrow  \mathop{\rm Id} _ {\mathfrak C} $
 +
leads to an adjoint pair (or adjunction) if the following equations hold:
  
<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/a/a010/a010820/a01082044.png" /></td> </tr></table>
+
$$
 +
G ( \psi _ {Y} )
 +
\phi _ {G (Y) }  = \
 +
1 _ {G (Y) }  ,\ \
 +
\psi _ {F(X) }  F ( \phi _ {X} )  = \
 +
1 _ {F (X) }
 +
$$
  
for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082046.png" />. A natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082047.png" /> is the unit of some adjunction if and only if for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082049.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082052.png" />. This property expresses the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082053.png" /> is a free object over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082054.png" /> with respect to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082055.png" /> in the sense of the following definition. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082056.png" /> together with a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082057.png" /> is free over an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082058.png" /> if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082059.png" /> can be written uniquely in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082060.png" /> for some morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082061.png" />. A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082062.png" /> has a left adjoint if and only if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082063.png" /> there is an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082064.png" /> that is free over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082065.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082066.png" />.
+
for all objects $  X $
 +
and $  Y $.  
 +
A natural transformation $  \phi :   \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $
 +
is the unit of some adjunction if and only if for any morphism $  \alpha : X \rightarrow G (Y) $
 +
in $  \mathfrak K $
 +
there is a unique morphism $  \alpha  ^  \prime  : F (X) \rightarrow Y $
 +
in $  \mathfrak C $
 +
such that $  \alpha = G ( \alpha  ^  \prime  ) \epsilon _ {X} $.  
 +
This property expresses the fact that $  F (X) $
 +
is a free object over $  X $
 +
with respect to the functor $  G $
 +
in the sense of the following definition. An object $  Y \in  \mathop{\rm Ob}  \mathfrak C $
 +
together with a morphism $  \epsilon : X \rightarrow G (Y) $
 +
is free over an object $  X \in  \mathop{\rm Ob}  \mathfrak K $
 +
if every morphism $  \alpha : X \rightarrow G ( Y ^ { \prime } ) $
 +
can be written uniquely in the form $  \alpha = G ( \alpha  ^  \prime  ) \epsilon $
 +
for some morphism $  \alpha  ^  \prime  : Y \rightarrow Y ^ { \prime } $.  
 +
A functor $  G : \mathfrak C \rightarrow \mathfrak K $
 +
has a left adjoint if and only if for every $  X \in  \mathop{\rm Ob}  \mathfrak K $
 +
there is an object $  Y $
 +
that is free over $  X $
 +
with respect to $  G $.
  
 
===Examples of adjoint functors.===
 
===Examples of adjoint functors.===
  
 +
1) If  $  G :  \mathfrak C \rightarrow {\mathfrak S } $,
 +
where  $  \mathfrak S $
 +
is the category of sets, then  $  G $
 +
has a left adjoint only if it is representable. A representable functor  $  G \simeqH  ^ {A} = H _ {\mathfrak C} ( A , Y ) $
 +
has a left adjoint if and only if all co-products  $  \amalg _ {x \in X }  A _ {x} $
 +
exist in  $  \mathfrak C $,
 +
where  $  X \in  \mathop{\rm Ob}  \mathfrak S $
 +
and  $  A _ {x} = A $
 +
for all  $  x \in X $.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082068.png" /> is the category of sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082069.png" /> has a left adjoint only if it is representable. A representable functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082070.png" /> has a left adjoint if and only if all co-products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082071.png" /> exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082074.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082075.png" />.
+
2) In the category $  \mathfrak S $
 +
of sets, for any set  $  A $
 +
the basic functor $  H  ^ {A} (Y) = H ( A , Y ) $
 +
is the right adjoint of the functor  $  X \times A $.
  
2) In the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082076.png" /> of sets, for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082077.png" /> the basic functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082078.png" /> is the right adjoint of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082079.png" />.
+
3) In the category of Abelian groups, the functor $  \mathop{\rm Hom} ( A , Y ) $
 +
is the right adjoint of the functor $  X \otimes A $
 +
of tensor multiplication by  $  A $,
 +
and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.
  
3) In the category of Abelian groups, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082080.png" /> is the right adjoint of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082081.png" /> of tensor multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082082.png" />, and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.
+
4) Let  $  P :  \mathfrak A \rightarrow \mathfrak S $
 +
be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor  $  P $
 +
has a left adjoint $  F :  \mathfrak S \rightarrow \mathfrak A $,
 +
which assigns to every set  $  X $
 +
the free algebra of the variety  $  \mathfrak A $
 +
with  $  X $
 +
as set of free generators.
  
4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082083.png" /> be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082084.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082085.png" />, which assigns to every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082086.png" /> the free algebra of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082088.png" /> as set of free generators.
+
5) The imbedding functor $  \mathop{\rm Id} _ {\mathfrak C , \mathfrak K }  : \mathfrak C \rightarrow \mathfrak K $
 
+
of an arbitrary reflective subcategory $  \mathfrak C $
5) The imbedding functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082089.png" /> of an arbitrary reflective subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082090.png" /> of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082091.png" /> is the right adjoint of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082092.png" />-reflector (cf. also [[Reflexive subcategory|Reflexive subcategor]]). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082093.png" /> its quotient group by the commutator subgroup.
+
of a category $  \mathfrak K $
 +
is the right adjoint of the $  \mathfrak C $-
 +
reflector (cf. also [[Reflexive subcategory|Reflexive subcategor]]). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group $  G $
 +
its quotient group by the commutator subgroup.
  
 
===Properties of adjoint functors.===
 
===Properties of adjoint functors.===
 
The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.
 
The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082095.png" /> be categories that are complete on the left and locally small on the left. A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082096.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082097.png" /> if and only if the following conditions hold: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082098.png" /> commutes with limits; b) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a01082099.png" />, at least one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820101.png" />, is non-empty; and c) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820102.png" />, there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820103.png" /> such that every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820104.png" /> is representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010820/a010820108.png" />.
+
Let $  \mathfrak K $
 +
and $  \mathfrak C $
 +
be categories that are complete on the left and locally small on the left. A functor $  G : \mathfrak G \rightarrow \mathfrak K $
 +
has a left adjoint $  F : \mathfrak K \rightarrow \mathfrak C $
 +
if and only if the following conditions hold: a) $  G $
 +
commutes with limits; b) for every $  X \in  \mathop{\rm Ob}  \mathfrak K $,  
 +
at least one of the sets $  H ( X , G (Y) ) $,  
 +
$  Y \in  \mathop{\rm Ob}  \mathfrak K $,  
 +
is non-empty; and c) for every $  X \in  \mathop{\rm Ob}  \mathfrak K $,  
 +
there is a set $  S \subset  \mathop{\rm Ob}  \mathfrak C $
 +
such that every morphism $  \alpha : X \rightarrow G (Y) $
 +
is representable in the form $  \alpha = G ( \alpha  ^  \prime  ) \phi $,  
 +
where $  \phi : X \rightarrow G (B) $,  
 +
$  B \in S $,  
 +
$  \alpha  ^  \prime  : B \rightarrow Y $.
  
 
By passing to dual categories, one may establish a duality between the concepts of a  "left adjoint functor"  and a  "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.
 
By passing to dual categories, one may establish a duality between the concepts of a  "left adjoint functor"  and a  "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Maclane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Maclane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.
 
A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.

Revision as of 16:09, 1 April 2020


A concept expressing the universality and naturalness of many important mathematical constructions, such as a free universal algebra, various completions, and direct and inverse limits.

Let $ F : \mathfrak K \rightarrow \mathfrak C $ be a covariant functor in one argument from a category $ \mathfrak K $ into a category $ \mathfrak C $. $ F $ induces a functor

$$ H ^ {F} ( X , Y ) = \ H _ {\mathfrak C } ( F (X) , Y ) : \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S , $$

where $ \mathfrak K ^ {*} $ is the category dual to $ \mathfrak K $, $ \mathfrak S $ is the category of sets, and $ H _ {\mathfrak C} ( X , Y ) : \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S $ is the basic set-valued functor. The functor $ H ^ {F} $ is contravariant in the first argument and covariant in the second. Similarly, any covariant functor $ G : \mathfrak C \rightarrow \mathfrak K $ induces a functor

$$ H _ {G} ( X , Y ) = \ H _ {\mathfrak K } ( X , G (Y) ) : \ \mathfrak K ^ {*} \times \mathfrak C \rightarrow \mathfrak S , $$

which is also contravariant in the first argument and covariant in the second. The functors $ F $ and $ G $ are adjoint, or form an adjoint pair, if $ H ^ {F} $ and $ H _ {G} $ are isomorphic, that is, if there is a natural transformation $ \theta : H ^ {F} \rightarrow H _ {G} $ that establishes a one-to-one correspondence between the sets of morphisms $ H _ {\mathfrak C} ( F (X) , Y ) $ and $ H _ {\mathfrak K} ( X , G (Y) ) $ for all objects $ X \in \mathop{\rm Ob} \mathfrak K $ and $ Y \in \mathop{\rm Ob} \mathfrak C $. The transformation $ \theta $ is called the adjunction of $ F $ with $ G $, $ F $ is called the left adjoint of $ G $ and $ G $ the right adjoint of $ F $( this is written $ \theta : F \lTL G $, or simply $ F \lTL G $). The transformation $ \theta ^ {-1} : H _ {G} \rightarrow H ^ {F} $ is called the co-adjunction.

Let $ \theta : F \lTL G $. For all $ X \in \mathop{\rm Ob} \mathfrak K $ and $ Y \in \mathop{\rm Ob} \mathfrak C $, let

$$ \epsilon _ {X} = \theta ( 1 _ {F (X) } ) ,\ \ \eta _ {Y} = \theta ^ {-1} ( 1 _ {G (Y) } ) . $$

The families of morphisms $ \{ \epsilon _ {X} \} $ and $ \{ \eta _ {Y} \} $ define natural transformations $ \epsilon : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ and $ \eta : F G \rightarrow \mathop{\rm Id} _ {\mathfrak C} $, called the unit and co-unit of the adjunction $ \theta $. They satisfy the following equations:

$$ G ( \eta _ {Y} ) \epsilon _ {G (Y) } = \ 1 _ {G (Y) } ,\ \ \eta _ {F (X) } F ( \epsilon _ {X} ) = \ 1 _ {F (X) } . $$

In general, a pair of natural transformations $ \phi : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ and $ \psi : F G \rightarrow \mathop{\rm Id} _ {\mathfrak C} $ leads to an adjoint pair (or adjunction) if the following equations hold:

$$ G ( \psi _ {Y} ) \phi _ {G (Y) } = \ 1 _ {G (Y) } ,\ \ \psi _ {F(X) } F ( \phi _ {X} ) = \ 1 _ {F (X) } $$

for all objects $ X $ and $ Y $. A natural transformation $ \phi : \mathop{\rm Id} _ {\mathfrak K} \rightarrow G F $ is the unit of some adjunction if and only if for any morphism $ \alpha : X \rightarrow G (Y) $ in $ \mathfrak K $ there is a unique morphism $ \alpha ^ \prime : F (X) \rightarrow Y $ in $ \mathfrak C $ such that $ \alpha = G ( \alpha ^ \prime ) \epsilon _ {X} $. This property expresses the fact that $ F (X) $ is a free object over $ X $ with respect to the functor $ G $ in the sense of the following definition. An object $ Y \in \mathop{\rm Ob} \mathfrak C $ together with a morphism $ \epsilon : X \rightarrow G (Y) $ is free over an object $ X \in \mathop{\rm Ob} \mathfrak K $ if every morphism $ \alpha : X \rightarrow G ( Y ^ { \prime } ) $ can be written uniquely in the form $ \alpha = G ( \alpha ^ \prime ) \epsilon $ for some morphism $ \alpha ^ \prime : Y \rightarrow Y ^ { \prime } $. A functor $ G : \mathfrak C \rightarrow \mathfrak K $ has a left adjoint if and only if for every $ X \in \mathop{\rm Ob} \mathfrak K $ there is an object $ Y $ that is free over $ X $ with respect to $ G $.

Examples of adjoint functors.

1) If $ G : \mathfrak C \rightarrow {\mathfrak S } $, where $ \mathfrak S $ is the category of sets, then $ G $ has a left adjoint only if it is representable. A representable functor $ G \simeqH ^ {A} = H _ {\mathfrak C} ( A , Y ) $ has a left adjoint if and only if all co-products $ \amalg _ {x \in X } A _ {x} $ exist in $ \mathfrak C $, where $ X \in \mathop{\rm Ob} \mathfrak S $ and $ A _ {x} = A $ for all $ x \in X $.

2) In the category $ \mathfrak S $ of sets, for any set $ A $ the basic functor $ H ^ {A} (Y) = H ( A , Y ) $ is the right adjoint of the functor $ X \times A $.

3) In the category of Abelian groups, the functor $ \mathop{\rm Hom} ( A , Y ) $ is the right adjoint of the functor $ X \otimes A $ of tensor multiplication by $ A $, and the imbedding functor of the full subcategory of torsion groups is the left adjoint of the functor of taking the torsion part of any Abelian group.

4) Let $ P : \mathfrak A \rightarrow \mathfrak S $ be the forgetful functor from an arbitrary variety of universal algebras into the category of sets. The functor $ P $ has a left adjoint $ F : \mathfrak S \rightarrow \mathfrak A $, which assigns to every set $ X $ the free algebra of the variety $ \mathfrak A $ with $ X $ as set of free generators.

5) The imbedding functor $ \mathop{\rm Id} _ {\mathfrak C , \mathfrak K } : \mathfrak C \rightarrow \mathfrak K $ of an arbitrary reflective subcategory $ \mathfrak C $ of a category $ \mathfrak K $ is the right adjoint of the $ \mathfrak C $- reflector (cf. also Reflexive subcategor). In particular, the imbedding functor of the category of Abelian groups in the category of groups has a left adjoint, which assigns to every group $ G $ its quotient group by the commutator subgroup.

Properties of adjoint functors.

The left adjoint functor of a given functor is uniquely determined up to isomorphism of functors. Left adjoints commute with co-limits (e.g. co-products) and send null objects and null morphism into null objects and null morphisms, respectively.

Let $ \mathfrak K $ and $ \mathfrak C $ be categories that are complete on the left and locally small on the left. A functor $ G : \mathfrak G \rightarrow \mathfrak K $ has a left adjoint $ F : \mathfrak K \rightarrow \mathfrak C $ if and only if the following conditions hold: a) $ G $ commutes with limits; b) for every $ X \in \mathop{\rm Ob} \mathfrak K $, at least one of the sets $ H ( X , G (Y) ) $, $ Y \in \mathop{\rm Ob} \mathfrak K $, is non-empty; and c) for every $ X \in \mathop{\rm Ob} \mathfrak K $, there is a set $ S \subset \mathop{\rm Ob} \mathfrak C $ such that every morphism $ \alpha : X \rightarrow G (Y) $ is representable in the form $ \alpha = G ( \alpha ^ \prime ) \phi $, where $ \phi : X \rightarrow G (B) $, $ B \in S $, $ \alpha ^ \prime : B \rightarrow Y $.

By passing to dual categories, one may establish a duality between the concepts of a "left adjoint functor" and a "right adjoint functor" ; this enables one to deduce the properties of right adjoints from those of left adjoints.

The concept of an adjoint functor is directly connected with the concept of a triple (or monad) in a category.

References

[1] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
[2] S. Maclane, "Categories for the working mathematician" , Springer (1971)

Comments

A category is called complete on the left if small diagrams have limits. A category is called locally small on the left if it has small hom-sets. The statement that a functor has a left adjoint if and only if a), b) and c) above holds, is called the Freyd adjoint functor theorem.

How to Cite This Entry:
Adjoint functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_functor&oldid=16692
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article