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''monad, on a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943001.png" />''
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A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943002.png" />. In other words, a triple on a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943003.png" /> is a covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943004.png" /> endowed with natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943006.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943007.png" /> denotes the identity functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t0943008.png" />) such that the following diagrams are commutative:
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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/t/t094/t094300/t0943009.png" /></td> </tr></table>
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''monad, on a category  $  \mathfrak R $''
  
<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/t/t094/t094300/t09430010.png" /></td> </tr></table>
+
A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on  $  \mathfrak R $.
 +
In other words, a triple on a category  $  \mathfrak R $
 +
is a covariant functor  $  T: \mathfrak R \mathop \rightarrow \limits \mathfrak R $
 +
endowed with natural transformations  $  \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $
 +
and  $  \mu : T ^ {2} \mathop \rightarrow \limits T $
 +
(here  $  {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $
 +
denotes the identity functor on  $  \mathfrak R $)
 +
such that the following diagrams are commutative:
 +
 
 +
$$  \begin{array}{crclc} T (X)  & \mathop \rightarrow \limits ^ {T ( \eta _ {X} )}  &T ^ {2} (X)  & \mathop \leftarrow \limits ^ {\eta _ {T (X)}}  &T (X)  \\ {}  &{} _ {1 _ {T (X)}} \searrow  &\scriptsize {\mu _ {X}} \downarrow  &\swarrow _ {1 _ {T (X)}}  &{}  \\ {}  &{}  &T (X)  &{}  &{}  \\ \end{array}  $$
 +
 
 +
$$  \begin{array}{rcl} T ^ {3} (X)  & \mathop \rightarrow \limits ^ {T ( \mu _ {X} )}  &T ^ {2} (X)  \\ \scriptsize {\mu _ {T (X)}}  \downarrow  &{}  &\downarrow  \scriptsize {\mu _ {X}}  \\ T ^ {2} (X)  & \mathop \rightarrow \limits _ {\mu _ {X}}  &T (X)  \\ \end{array}  $$
  
 
A triple is sometimes called a standard construction, cf. [[#References|[2]]].
 
A triple is sometimes called a standard construction, cf. [[#References|[2]]].
  
For any pair of adjoint functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430012.png" /> (see [[Adjoint functor|Adjoint functor]]) with unit and co-unit of adjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430014.png" />, respectively, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430015.png" /> endowed with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430017.png" /> is a triple on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430018.png" />. Conversely, for any triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430019.png" /> there exist pairs of adjoint functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430022.png" />, and the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430024.png" /> are obtained from the unit and co-unit of the adjunction in the manner described above. The different such decompositions of a triple may form a proper class. In this class there is a smallest element (the Kleisli construction) and a largest element (the Eilenberg–Moore construction).
+
For any pair of adjoint functors $  F : \mathfrak R \mathop \rightarrow \limits \mathfrak L $
 +
and $  G: \mathfrak L \mathop \rightarrow \limits \mathfrak R $
 +
(see [[Adjoint functor|Adjoint functor]]) with unit and co-unit of adjunction $  \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits GF $
 +
and $  \epsilon : FG \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $,  
 +
respectively, the functor $  T = GF: \mathfrak R \mathop \rightarrow \limits \mathfrak R $
 +
endowed with $  \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $
 +
and $  \mu = G ( \epsilon _ {F} ): T ^ {2} \mathop \rightarrow \limits T $
 +
is a triple on $  \mathfrak R $.  
 +
Conversely, for any triple $  (T, \eta , \mu ) $
 +
there exist pairs of adjoint functors $  F $
 +
and $  G $
 +
such that $  T = GF $,  
 +
and the transformations $  \eta $
 +
and $  \mu $
 +
are obtained from the unit and co-unit of the adjunction in the manner described above. The different such decompositions of a triple may form a proper class. In this class there is a smallest element (the Kleisli construction) and a largest element (the Eilenberg–Moore construction).
  
 
===Examples.===
 
===Examples.===
  
 +
1) In the category of sets, the functor which sends an arbitrary set to the set of all its subsets has the structure of a triple. Each set  $  X $
 +
is naturally imbedded in the set of its subsets via singleton sets, and to each set of subsets of  $  X $
 +
one associates the union of these subsets.
  
1) In the category of sets, the functor which sends an arbitrary set to the set of all its subsets has the structure of a triple. Each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430025.png" /> is naturally imbedded in the set of its subsets via singleton sets, and to each set of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430026.png" /> one associates the union of these subsets.
+
2) In the category of sets, every representable functor $  H _ {A} (X) = H (A, X) $
 +
carries a triple: The mapping  $  \eta _ {X} :  X \mathop \rightarrow \limits H (A, X) $
 +
associates to each  $  x \in X $
 +
the constant function  $  f _ {x} :  A \mathop \rightarrow \limits X $
 +
with value  $  x $;
 +
the mapping  $  \mu _ {X} :  H (A, H (A, X)) \simeq H (A \times A, X) \mathop \rightarrow \limits H (A, X) $
 +
associates to each function of two variables its restriction to the diagonal.
  
2) In the category of sets, every representable functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430027.png" /> carries a triple: The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430028.png" /> associates to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430029.png" /> the constant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430030.png" /> with value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430031.png" />; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430032.png" /> associates to each function of two variables its restriction to the diagonal.
+
3) In the category of topological spaces, each topological group  $  G $,
 +
with unit  $  e $,
 +
enables one to define a functor $  T _ {G} (X) = X \times G $
 +
that carries a triple: Each element  $  x \in X $
 +
is taken to the element  $  (x, e) $
 +
and the mapping $  \mu :  X \times G \times G \mathop \rightarrow \limits X \times G $
 +
is defined by  $  \mu _ {X} (x, g, g ^  \prime  ) = (x, gg ^  \prime  ) $.
  
3) In the category of topological spaces, each topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430033.png" />, with unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430034.png" />, enables one to define a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430035.png" /> that carries a triple: Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430036.png" /> is taken to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430037.png" /> and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430038.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430039.png" />.
+
4) In the category of modules over a commutative ring $  R $,  
 
+
each (associative, unital) $  R $-
4) In the category of modules over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430040.png" />, each (associative, unital) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430041.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430042.png" /> gives rise to a triple structure on the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430043.png" />, in a manner similar to Example 3).
+
algebra $  A $
 +
gives rise to a triple structure on the functor $  T _ {A} (X) = X \otimes _ {R} A $,  
 +
in a manner similar to Example 3).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Adams,  "Infinite loop spaces" , Princeton Univ. Press  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Categories"  ''J. Soviet Math.'' , '''7''' :  4  (1977)  pp. 532–586  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''13'''  (1975)  pp. 51–148</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.G. Manes,  "Algebraic theories" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Adams,  "Infinite loop spaces" , Princeton Univ. Press  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.Sh. Tsalenko,  E.G. Shul'geifer,  "Categories"  ''J. Soviet Math.'' , '''7''' :  4  (1977)  pp. 532–586  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''13'''  (1975)  pp. 51–148</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.G. Manes,  "Algebraic theories" , Springer  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The non-descriptive name  "triple"  for this concept has now largely been superseded by  "monad" , although there is an obstinate minority of category-theorists who continue to use it. A comonad (or cotriple) on a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430044.png" /> is a monad on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430045.png" />; in other words, it is a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430046.png" /> equipped with natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430048.png" /> satisfying the duals of the commutative diagrams above. Every adjoint pair of functors (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430049.png" />) gives rise to a comonad structure on the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430050.png" />, as well as a monad structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430051.png" />.
+
The non-descriptive name  "triple"  for this concept has now largely been superseded by  "monad" , although there is an obstinate minority of category-theorists who continue to use it. A comonad (or cotriple) on a category $  \mathfrak R $
 +
is a monad on $  \mathfrak R ^ {op} $;  
 +
in other words, it is a functor $  T: \mathfrak R \mathop \rightarrow \limits \mathfrak R $
 +
equipped with natural transformations $  \epsilon : T \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $
 +
and $  \delta : T \mathop \rightarrow \limits T ^ {2} $
 +
satisfying the duals of the commutative diagrams above. Every adjoint pair of functors ( $  F \dashv G $)  
 +
gives rise to a comonad structure on the composite $  FG $,  
 +
as well as a monad structure on $  GF $.
  
An important example of a functor which carries a comonad structure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430053.png" />, or, equivalently, the functor of big Witt vectors, cf. [[Lambda-ring|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430054.png" />-ring]]; [[Witt vector|Witt vector]]. A special case of the natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430055.png" /> occurs in algebraic number theory as the [[Artin–Hasse exponential]], [[#References|[a5]]].
+
An important example of a functor which carries a comonad structure is $  \Lambda : {\mathop{\rm Ring}\nolimits} \mathop \rightarrow \limits {\mathop{\rm Ring}\nolimits} $,  
 +
$  \Lambda (A)=1+tA[[t]] $,  
 +
or, equivalently, the functor of big Witt vectors, cf. [[Lambda-ring| $  \lambda $-
 +
ring]]; [[Witt vector|Witt vector]]. A special case of the natural transformation $  W(A) \mathop \rightarrow \limits \Lambda (W(A)) $
 +
occurs in algebraic number theory as the [[Artin–Hasse exponential]], [[#References|[a5]]].
  
Monads in the category of sets can be equivalently described by sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430057.png" />-ary operations for each cardinal number (or set) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430058.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430059.png" /> gives the projection operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430061.png" /> gives the rules for composing operations. See [[#References|[5]]] or [[#References|[a1]]]. This approach extends to monads in arbitrary categories, but it has not proved useful in general, as it has in or near sets.
+
Monads in the category of sets can be equivalently described by sets $  T(n) $
 +
of $  n $-
 +
ary operations for each cardinal number (or set) $  n $;  
 +
$  \eta _ {n} : n \mathop \rightarrow \limits T(n) $
 +
gives the projection operations $  (x _ {1} , x _ {2} ,\dots) \mapsto x _ {i} $,  
 +
and $  \mu $
 +
gives the rules for composing operations. See [[#References|[5]]] or [[#References|[a1]]]. This approach extends to monads in arbitrary categories, but it has not proved useful in general, as it has in or near sets.
  
Of the two canonical ways of constructing an adjunction from a given monad, mentioned in the main article above, the Eilenberg–Moore construction (or category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430063.png" />-algebras) is by far the more important. Given a monad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430064.png" /> on a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430065.png" />, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430067.png" />-algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430068.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430069.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430070.png" /> is a morphism such that
+
Of the two canonical ways of constructing an adjunction from a given monad, mentioned in the main article above, the Eilenberg–Moore construction (or category of $  T $-
 +
algebras) is by far the more important. Given a monad $  (T, \eta , \mu ) $
 +
on a category $  \mathfrak R $,  
 +
a $  T $-
 +
algebra in $  \mathfrak R $
 +
is a pair $  (A, \alpha ) $
 +
where $  \alpha : TA \mathop \rightarrow \limits A $
 +
is a morphism 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/t/t094/t094300/t09430071.png" /></td> </tr></table>
+
$$  \begin{array}{lcr} A  \mathop \rightarrow \limits ^ {\eta _ {A}}  &TA  & \mathop \leftarrow \limits ^ {\mu _ {A}}  T ^ {2} A  \\ {} _ {1 _ {A}} \nwarrow  &\scriptsize \alpha  \downarrow  &\downarrow  \scriptsize {T _ {A}}  \\ {}  & A  & \mathop \leftarrow \limits _  \alpha  TA  \\ \end{array}  $$
  
commutes. A homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430073.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430074.png" /> is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430076.png" /> such that
+
commutes. A homomorphism of $  T $-
 +
algebras $  (A, \alpha ) \mathop \rightarrow \limits (B, \beta ) $
 +
is a morphism $  f: A \mathop \rightarrow \limits B $
 +
in $  \mathfrak R $
 +
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/t/t094/t094300/t09430077.png" /></td> </tr></table>
+
$$  \begin{array}{rcl} TA  & \mathop \rightarrow \limits ^ {Tf}  &TB  \\ \scriptsize \alpha  \downarrow  &{}  &\downarrow  \scriptsize \beta  \\  A  &\mathop \rightarrow \limits _ {f}  & B  \\ \end{array}  $$
  
commutes; thus, one has a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430079.png" />-algebras, with an evident forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430080.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430081.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430082.png" />, which sends an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430084.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430085.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430086.png" />, and the monad induced by the adjunction (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430087.png" />) is the one originally given.
+
commutes; thus, one has a category $  \mathfrak R ^ {T} $
 +
of $  T $-
 +
algebras, with an evident forgetful functor $  G ^ {T} : \mathfrak R ^ {T} \mathop \rightarrow \limits \mathfrak R $.  
 +
The functor $  G ^ {T} $
 +
has a left adjoint $  F ^ { T} $,  
 +
which sends an object $  A $
 +
of $  \mathfrak R $
 +
to the $  T $-
 +
algebra $  (TA, \mu _ {A} ) $,  
 +
and the monad induced by the adjunction ( $  F ^ { T} \dashv G ^ {T} $)  
 +
is the one originally given.
  
Now the Kleisli category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430088.png" /> is just the full subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430089.png" /> on the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430090.png" />: the category of free algebras (cf. also [[Category|Category]]).
+
Now the Kleisli category of $  (T, \eta , \mu ) $
 +
is just the full subcategory of $  \mathfrak R ^ {T} $
 +
on the objects $  F ^ { T} (A) $:  
 +
the category of free algebras (cf. also [[Category|Category]]).
  
For a monad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430091.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430092.png" />, in the Kleisli construction the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430093.png" /> has as objects the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430094.png" />, and as hom-sets the sets
+
For a monad $  (T, \eta , \mu ) $
 +
on $  \mathfrak R $,  
 +
in the Kleisli construction the category $  \mathfrak L $
 +
has as objects the objects of $  \mathfrak R $,  
 +
and as hom-sets the sets
  
<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/t/t094/t094300/t09430095.png" /></td> </tr></table>
+
$$  \mathfrak L (A, B)  = \mathfrak R (A, TB). $$
  
The composition rule for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430096.png" /> assigns to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430098.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t09430099.png" />-composite:
+
The composition rule for $  \mathfrak L $
 +
assigns to $  f \in \mathfrak L (A, B) $
 +
and $  g \in \mathfrak L (B, C) $
 +
the $  \mathfrak R $-
 +
composite:
  
<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/t/t094/t094300/t094300100.png" /></td> </tr></table>
+
$$  [A  \mathop \rightarrow \limits ^ {T}  TB  \mathop \rightarrow \limits ^ {T(g)}  TTC  \mathop \rightarrow \limits ^ {\mu _ {C}}  TC ]  \in  \mathfrak L (A, C); $$
  
as identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300101.png" /> one uses the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300102.png" />-morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300103.png" />.
+
as identity mapping $  1 _ {A} \in \mathfrak L (A, A) = \mathfrak R (T, TA) $
 +
one uses the $  \mathfrak R $-
 +
morphism $  \eta _ {A} : A \mathop \rightarrow \limits TA $.
  
An adjoint pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300105.png" /> is obtained by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300106.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300107.png" />,
+
An adjoint pair $  F: \mathfrak R \mathop \rightarrow \limits \mathfrak L $,  
 +
$  U: \mathfrak L \mathop \rightarrow \limits \mathfrak R $
 +
is obtained by setting $  F(A)=A $
 +
for $  A \in \mathfrak R $,
  
<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/t/t094/t094300/t094300108.png" /></td> </tr></table>
+
$$  F(f)  = \eta _ {B} \circ f : A  \mathop \rightarrow \limits  B  \mathop \rightarrow \limits  TB  \in  \mathfrak R (A, TB)  = \mathfrak L (A, B)  $$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300110.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300111.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300112.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300113.png" />.
+
for $  f \in \mathfrak R (A, B) $,  
 +
$  U(B)=TB $
 +
for $  B \in \mathfrak L $,  
 +
and $  U(g ) = \mu _ {G} \circ T(g) $
 +
for $  g \in \mathfrak L (B, C)= \mathfrak R (B, TC) $.
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300114.png" /> will serve as unit for the adjunction, while the co-unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300115.png" /> is given by
+
Then $  \eta $
 +
will serve as unit for the adjunction, while the co-unit $  \epsilon : FU \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak L} $
 +
is given by
  
<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/t/t094/t094300/t094300116.png" /></td> </tr></table>
+
$$  \epsilon _ {B}  = \mathop{\rm Id} _ {T(B)}  \in  \mathfrak R (TB, TB)  = \mathfrak L (FUB, B). $$
  
Co-algebras are defined in the same manner. In practice, co-algebras very often occur superposed on algebras; a comonad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300117.png" /> will be constructed on a category of algebras of some sort, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300118.png" />, leading to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300119.png" /> of bi-algebras. An important class of cases involves a monad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300120.png" /> and a cotriple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300121.png" /> on the same category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300122.png" />. There is a standard lifting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300123.png" /> to a cotriple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300124.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300125.png" />. A  "TG-bi-algebraTG-bi-algebra"  means an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300126.png" />; the reverse order is also possible, but rarely occurs, and the objects would not be called bi-algebras.
+
Co-algebras are defined in the same manner. In practice, co-algebras very often occur superposed on algebras; a comonad $  G $
 +
will be constructed on a category of algebras of some sort, $  \mathfrak R $,  
 +
leading to the category $  {} ^ {G} \mathfrak R $
 +
of bi-algebras. An important class of cases involves a monad $  T $
 +
and a cotriple $  G $
 +
on the same category $  \mathfrak R $.  
 +
There is a standard lifting of $  G $
 +
to a cotriple $  G ^ {*} $
 +
on $  \mathfrak R ^ {T} $.  
 +
A  "TG-bi-algebraTG-bi-algebra"  means an object of $  {} ^ {G ^ {*}} ( \mathfrak R ^ {T} ) $;  
 +
the reverse order is also possible, but rarely occurs, and the objects would not be called bi-algebras.
  
 
For the role of comonads in (algebraic) cohomology theories see [[Cohomology of algebras|Cohomology of algebras]] and [[#References|[a2]]], [[#References|[a3]]]; particularly [[#References|[a2]]] for explicit interpretation.
 
For the role of comonads in (algebraic) cohomology theories see [[Cohomology of algebras|Cohomology of algebras]] and [[#References|[a2]]], [[#References|[a3]]]; particularly [[#References|[a2]]] for explicit interpretation.
Line 72: Line 190:
 
An adjunction is said to be monadic (or monadable) if the Eilenberg–Moore construction applied to the monad it induces yields an adjunction equivalent to the original one. Many important examples of adjunctions are monadic; for example, for any [[Variety of universal algebras|variety of universal algebras]], the forgetful functor from the variety to the category of sets and its left adjoint (the free algebra functor) form a monadic adjunction.
 
An adjunction is said to be monadic (or monadable) if the Eilenberg–Moore construction applied to the monad it induces yields an adjunction equivalent to the original one. Many important examples of adjunctions are monadic; for example, for any [[Variety of universal algebras|variety of universal algebras]], the forgetful functor from the variety to the category of sets and its left adjoint (the free algebra functor) form a monadic adjunction.
  
A monad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300127.png" /> is said to be idempotent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300128.png" /> is an isomorphism. In this case it can be shown that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300129.png" />-algebra structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300130.png" /> on an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300131.png" /> is necessarily a two-sided inverse for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300132.png" />, and hence that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300133.png" /> is isomorphic to the full subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300134.png" /> consisting of all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300135.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300136.png" /> is an isomorphism. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300137.png" /> is a [[Reflective subcategory|reflective subcategory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300138.png" />, the left adjoint to the inclusion being given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300139.png" /> itself. Conversely, for any reflective subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300140.png" />, the monad on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300141.png" /> induced by the inclusion and its left adjoint is idempotent; thus, the adjunctions corresponding to reflective subcategories are always monadic.
+
A monad $  (T, \eta , \mu ) $
 +
is said to be idempotent if $  \mu $
 +
is an isomorphism. In this case it can be shown that any $  T $-
 +
algebra structure $  \alpha $
 +
on an object $  A $
 +
is necessarily a two-sided inverse for $  \eta _ {A} $,  
 +
and hence that $  \mathfrak R ^ {T} $
 +
is isomorphic to the full subcategory $  {\mathop{\rm Fix}\nolimits} (T) \subset  \mathfrak R $
 +
consisting of all objects $  A $
 +
such that $  \eta _ {A} $
 +
is an isomorphism. $  {\mathop{\rm Fix}\nolimits} (T) $
 +
is a [[Reflective subcategory|reflective subcategory]] of $  \mathfrak R $,  
 +
the left adjoint to the inclusion being given by $  T $
 +
itself. Conversely, for any reflective subcategory of $  \mathfrak R $,  
 +
the monad on $  \mathfrak R $
 +
induced by the inclusion and its left adjoint is idempotent; thus, the adjunctions corresponding to reflective subcategories are always monadic.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Toposes, monads, and theories" , Springer (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Duskin,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094300/t094300142.png" />-torsors and the interpretation of  "monad"  cohomology"  ''Proc. Nat. Acad. Sci. USA'' , '''71'''  (1974)  pp. 2554–2557</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W. Duskin,  "Simplicial methods and the interpretation of  "monad"  cohomology"  ''Mem. Amer. Math. Soc.'' , '''3'''  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Adamek,  H. Herrlich,  G.E. Strecker,  "Abstract and concrete categories" , Wiley (Interscience)  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Hazewinkel,  "Formal groups" , Acad. Press  (1978)  pp. Sects. 14.5; 14.6, E2</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Appelgate (ed.)  et al. (ed.) , ''Seminar on monads and categorical homology theory ETH 1966/7'' , ''Lect. notes in math.'' , '''80''' , Springer  (1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Eilenberg,  J.C. Moore,  "Adjoint functors and monads"  ''Ill. J. Math.'' , '''9'''  (1965)  pp. 381–398</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Eilenberg (ed.)  et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer  (1966)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Toposes, triples, and theories" , Springer (1985). {{ZBL|0567.18001}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Duskin,  "$K(\pi,n)$-torsors and the interpretation of  "triple"  cohomology"  ''Proc. Nat. Acad. Sci. USA'' , '''71'''  (1974)  pp. 2554–2557. {{ZBL|0288.18013}}</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W. Duskin,  "Simplicial methods and the interpretation of  "triple"  cohomology"  ''Mem. Amer. Math. Soc.'' , '''3'''  (1975). {{ZBL|0376.18011}}</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Adamek,  H. Herrlich,  G.E. Strecker,  "Abstract and concrete categories" , Wiley (Interscience)  (1990)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Hazewinkel,  "Formal groups" , Acad. Press  (1978)  pp. Sects. 14.5; 14.6, E2</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Appelgate (ed.)  et al. (ed.) , ''Seminar on triples and categorical homology theory ETH 1966/7'' , Lect. notes in math., '''80''' , Springer  (1969). {{ZBL| 1134.18001}}</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Eilenberg,  J.C. Moore,  "Adjoint functors and triples"  ''Ill. J. Math.'' , '''9'''  (1965)  pp. 381–398. {{ZBL|0135.02103}}</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Eilenberg (ed.)  et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer  (1966)</TD></TR>
 +
</table>

Latest revision as of 09:29, 3 July 2021


monad, on a category $ \mathfrak R $

A monoid in the category of all endomorphism functors on $ \mathfrak R $. In other words, a triple on a category $ \mathfrak R $ is a covariant functor $ T: \mathfrak R \mathop \rightarrow \limits \mathfrak R $ endowed with natural transformations $ \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $ and $ \mu : T ^ {2} \mathop \rightarrow \limits T $ (here $ {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $ denotes the identity functor on $ \mathfrak R $) such that the following diagrams are commutative:

$$ \begin{array}{crclc} T (X) & \mathop \rightarrow \limits ^ {T ( \eta _ {X} )} &T ^ {2} (X) & \mathop \leftarrow \limits ^ {\eta _ {T (X)}} &T (X) \\ {} &{} _ {1 _ {T (X)}} \searrow &\scriptsize {\mu _ {X}} \downarrow &\swarrow _ {1 _ {T (X)}} &{} \\ {} &{} &T (X) &{} &{} \\ \end{array} $$

$$ \begin{array}{rcl} T ^ {3} (X) & \mathop \rightarrow \limits ^ {T ( \mu _ {X} )} &T ^ {2} (X) \\ \scriptsize {\mu _ {T (X)}} \downarrow &{} &\downarrow \scriptsize {\mu _ {X}} \\ T ^ {2} (X) & \mathop \rightarrow \limits _ {\mu _ {X}} &T (X) \\ \end{array} $$

A triple is sometimes called a standard construction, cf. [2].

For any pair of adjoint functors $ F : \mathfrak R \mathop \rightarrow \limits \mathfrak L $ and $ G: \mathfrak L \mathop \rightarrow \limits \mathfrak R $ (see Adjoint functor) with unit and co-unit of adjunction $ \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits GF $ and $ \epsilon : FG \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $, respectively, the functor $ T = GF: \mathfrak R \mathop \rightarrow \limits \mathfrak R $ endowed with $ \eta : {\mathop{\rm Id}\nolimits} _ {\mathfrak R} \mathop \rightarrow \limits T $ and $ \mu = G ( \epsilon _ {F} ): T ^ {2} \mathop \rightarrow \limits T $ is a triple on $ \mathfrak R $. Conversely, for any triple $ (T, \eta , \mu ) $ there exist pairs of adjoint functors $ F $ and $ G $ such that $ T = GF $, and the transformations $ \eta $ and $ \mu $ are obtained from the unit and co-unit of the adjunction in the manner described above. The different such decompositions of a triple may form a proper class. In this class there is a smallest element (the Kleisli construction) and a largest element (the Eilenberg–Moore construction).

Examples.

1) In the category of sets, the functor which sends an arbitrary set to the set of all its subsets has the structure of a triple. Each set $ X $ is naturally imbedded in the set of its subsets via singleton sets, and to each set of subsets of $ X $ one associates the union of these subsets.

2) In the category of sets, every representable functor $ H _ {A} (X) = H (A, X) $ carries a triple: The mapping $ \eta _ {X} : X \mathop \rightarrow \limits H (A, X) $ associates to each $ x \in X $ the constant function $ f _ {x} : A \mathop \rightarrow \limits X $ with value $ x $; the mapping $ \mu _ {X} : H (A, H (A, X)) \simeq H (A \times A, X) \mathop \rightarrow \limits H (A, X) $ associates to each function of two variables its restriction to the diagonal.

3) In the category of topological spaces, each topological group $ G $, with unit $ e $, enables one to define a functor $ T _ {G} (X) = X \times G $ that carries a triple: Each element $ x \in X $ is taken to the element $ (x, e) $ and the mapping $ \mu : X \times G \times G \mathop \rightarrow \limits X \times G $ is defined by $ \mu _ {X} (x, g, g ^ \prime ) = (x, gg ^ \prime ) $.

4) In the category of modules over a commutative ring $ R $, each (associative, unital) $ R $- algebra $ A $ gives rise to a triple structure on the functor $ T _ {A} (X) = X \otimes _ {R} A $, in a manner similar to Example 3).

References

[1] J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
[2] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[3] M.Sh. Tsalenko, E.G. Shul'geifer, "Categories" J. Soviet Math. , 7 : 4 (1977) pp. 532–586 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 13 (1975) pp. 51–148
[4] S. MacLane, "Categories for the working mathematician" , Springer (1971)
[5] E.G. Manes, "Algebraic theories" , Springer (1976)

Comments

The non-descriptive name "triple" for this concept has now largely been superseded by "monad" , although there is an obstinate minority of category-theorists who continue to use it. A comonad (or cotriple) on a category $ \mathfrak R $ is a monad on $ \mathfrak R ^ {op} $; in other words, it is a functor $ T: \mathfrak R \mathop \rightarrow \limits \mathfrak R $ equipped with natural transformations $ \epsilon : T \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak R} $ and $ \delta : T \mathop \rightarrow \limits T ^ {2} $ satisfying the duals of the commutative diagrams above. Every adjoint pair of functors ( $ F \dashv G $) gives rise to a comonad structure on the composite $ FG $, as well as a monad structure on $ GF $.

An important example of a functor which carries a comonad structure is $ \Lambda : {\mathop{\rm Ring}\nolimits} \mathop \rightarrow \limits {\mathop{\rm Ring}\nolimits} $, $ \Lambda (A)=1+tA[[t]] $, or, equivalently, the functor of big Witt vectors, cf. $ \lambda $- ring; Witt vector. A special case of the natural transformation $ W(A) \mathop \rightarrow \limits \Lambda (W(A)) $ occurs in algebraic number theory as the Artin–Hasse exponential, [a5].

Monads in the category of sets can be equivalently described by sets $ T(n) $ of $ n $- ary operations for each cardinal number (or set) $ n $; $ \eta _ {n} : n \mathop \rightarrow \limits T(n) $ gives the projection operations $ (x _ {1} , x _ {2} ,\dots) \mapsto x _ {i} $, and $ \mu $ gives the rules for composing operations. See [5] or [a1]. This approach extends to monads in arbitrary categories, but it has not proved useful in general, as it has in or near sets.

Of the two canonical ways of constructing an adjunction from a given monad, mentioned in the main article above, the Eilenberg–Moore construction (or category of $ T $- algebras) is by far the more important. Given a monad $ (T, \eta , \mu ) $ on a category $ \mathfrak R $, a $ T $- algebra in $ \mathfrak R $ is a pair $ (A, \alpha ) $ where $ \alpha : TA \mathop \rightarrow \limits A $ is a morphism such that

$$ \begin{array}{lcr} A \mathop \rightarrow \limits ^ {\eta _ {A}} &TA & \mathop \leftarrow \limits ^ {\mu _ {A}} T ^ {2} A \\ {} _ {1 _ {A}} \nwarrow &\scriptsize \alpha \downarrow &\downarrow \scriptsize {T _ {A}} \\ {} & A & \mathop \leftarrow \limits _ \alpha TA \\ \end{array} $$

commutes. A homomorphism of $ T $- algebras $ (A, \alpha ) \mathop \rightarrow \limits (B, \beta ) $ is a morphism $ f: A \mathop \rightarrow \limits B $ in $ \mathfrak R $ such that

$$ \begin{array}{rcl} TA & \mathop \rightarrow \limits ^ {Tf} &TB \\ \scriptsize \alpha \downarrow &{} &\downarrow \scriptsize \beta \\ A &\mathop \rightarrow \limits _ {f} & B \\ \end{array} $$

commutes; thus, one has a category $ \mathfrak R ^ {T} $ of $ T $- algebras, with an evident forgetful functor $ G ^ {T} : \mathfrak R ^ {T} \mathop \rightarrow \limits \mathfrak R $. The functor $ G ^ {T} $ has a left adjoint $ F ^ { T} $, which sends an object $ A $ of $ \mathfrak R $ to the $ T $- algebra $ (TA, \mu _ {A} ) $, and the monad induced by the adjunction ( $ F ^ { T} \dashv G ^ {T} $) is the one originally given.

Now the Kleisli category of $ (T, \eta , \mu ) $ is just the full subcategory of $ \mathfrak R ^ {T} $ on the objects $ F ^ { T} (A) $: the category of free algebras (cf. also Category).

For a monad $ (T, \eta , \mu ) $ on $ \mathfrak R $, in the Kleisli construction the category $ \mathfrak L $ has as objects the objects of $ \mathfrak R $, and as hom-sets the sets

$$ \mathfrak L (A, B) = \mathfrak R (A, TB). $$

The composition rule for $ \mathfrak L $ assigns to $ f \in \mathfrak L (A, B) $ and $ g \in \mathfrak L (B, C) $ the $ \mathfrak R $- composite:

$$ [A \mathop \rightarrow \limits ^ {T} TB \mathop \rightarrow \limits ^ {T(g)} TTC \mathop \rightarrow \limits ^ {\mu _ {C}} TC ] \in \mathfrak L (A, C); $$

as identity mapping $ 1 _ {A} \in \mathfrak L (A, A) = \mathfrak R (T, TA) $ one uses the $ \mathfrak R $- morphism $ \eta _ {A} : A \mathop \rightarrow \limits TA $.

An adjoint pair $ F: \mathfrak R \mathop \rightarrow \limits \mathfrak L $, $ U: \mathfrak L \mathop \rightarrow \limits \mathfrak R $ is obtained by setting $ F(A)=A $ for $ A \in \mathfrak R $,

$$ F(f) = \eta _ {B} \circ f : A \mathop \rightarrow \limits B \mathop \rightarrow \limits TB \in \mathfrak R (A, TB) = \mathfrak L (A, B) $$

for $ f \in \mathfrak R (A, B) $, $ U(B)=TB $ for $ B \in \mathfrak L $, and $ U(g ) = \mu _ {G} \circ T(g) $ for $ g \in \mathfrak L (B, C)= \mathfrak R (B, TC) $.

Then $ \eta $ will serve as unit for the adjunction, while the co-unit $ \epsilon : FU \mathop \rightarrow \limits {\mathop{\rm Id}\nolimits} _ {\mathfrak L} $ is given by

$$ \epsilon _ {B} = \mathop{\rm Id} _ {T(B)} \in \mathfrak R (TB, TB) = \mathfrak L (FUB, B). $$

Co-algebras are defined in the same manner. In practice, co-algebras very often occur superposed on algebras; a comonad $ G $ will be constructed on a category of algebras of some sort, $ \mathfrak R $, leading to the category $ {} ^ {G} \mathfrak R $ of bi-algebras. An important class of cases involves a monad $ T $ and a cotriple $ G $ on the same category $ \mathfrak R $. There is a standard lifting of $ G $ to a cotriple $ G ^ {*} $ on $ \mathfrak R ^ {T} $. A "TG-bi-algebraTG-bi-algebra" means an object of $ {} ^ {G ^ {*}} ( \mathfrak R ^ {T} ) $; the reverse order is also possible, but rarely occurs, and the objects would not be called bi-algebras.

For the role of comonads in (algebraic) cohomology theories see Cohomology of algebras and [a2], [a3]; particularly [a2] for explicit interpretation.

An adjunction is said to be monadic (or monadable) if the Eilenberg–Moore construction applied to the monad it induces yields an adjunction equivalent to the original one. Many important examples of adjunctions are monadic; for example, for any variety of universal algebras, the forgetful functor from the variety to the category of sets and its left adjoint (the free algebra functor) form a monadic adjunction.

A monad $ (T, \eta , \mu ) $ is said to be idempotent if $ \mu $ is an isomorphism. In this case it can be shown that any $ T $- algebra structure $ \alpha $ on an object $ A $ is necessarily a two-sided inverse for $ \eta _ {A} $, and hence that $ \mathfrak R ^ {T} $ is isomorphic to the full subcategory $ {\mathop{\rm Fix}\nolimits} (T) \subset \mathfrak R $ consisting of all objects $ A $ such that $ \eta _ {A} $ is an isomorphism. $ {\mathop{\rm Fix}\nolimits} (T) $ is a reflective subcategory of $ \mathfrak R $, the left adjoint to the inclusion being given by $ T $ itself. Conversely, for any reflective subcategory of $ \mathfrak R $, the monad on $ \mathfrak R $ induced by the inclusion and its left adjoint is idempotent; thus, the adjunctions corresponding to reflective subcategories are always monadic.

References

[a1] M. Barr, C. Wells, "Toposes, triples, and theories" , Springer (1985). Zbl 0567.18001
[a2] J.W. Duskin, "$K(\pi,n)$-torsors and the interpretation of "triple" cohomology" Proc. Nat. Acad. Sci. USA , 71 (1974) pp. 2554–2557. Zbl 0288.18013
[a3] J.W. Duskin, "Simplicial methods and the interpretation of "triple" cohomology" Mem. Amer. Math. Soc. , 3 (1975). Zbl 0376.18011
[a4] J. Adamek, H. Herrlich, G.E. Strecker, "Abstract and concrete categories" , Wiley (Interscience) (1990)
[a5] M. Hazewinkel, "Formal groups" , Acad. Press (1978) pp. Sects. 14.5; 14.6, E2
[a6] H. Appelgate (ed.) et al. (ed.) , Seminar on triples and categorical homology theory ETH 1966/7 , Lect. notes in math., 80 , Springer (1969). 1134.18001 Zbl 1134.18001
[a7] S. Eilenberg, J.C. Moore, "Adjoint functors and triples" Ill. J. Math. , 9 (1965) pp. 381–398. Zbl 0135.02103
[a8] S. Eilenberg (ed.) et al. (ed.) , Proc. conf. categorical algebra (La Jolla, 1965) , Springer (1966)
How to Cite This Entry:
Triple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triple&oldid=49482
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article