Triple

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, monads, and theories" , Springer (1985)
[a2]	J.W. Duskin, "-torsors and the interpretation of "monad" cohomology" Proc. Nat. Acad. Sci. USA , 71 (1974) pp. 2554–2557
[a3]	J.W. Duskin, "Simplicial methods and the interpretation of "monad" cohomology" Mem. Amer. Math. Soc. , 3 (1975)
[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 monads and categorical homology theory ETH 1966/7 , Lect. notes in math. , 80 , Springer (1969)
[a7]	S. Eilenberg, J.C. Moore, "Adjoint functors and monads" Ill. J. Math. , 9 (1965) pp. 381–398
[a8]	S. Eilenberg (ed.) et al. (ed.) , Proc. conf. categorical algebra (La Jolla, 1965) , Springer (1966)