Triple

monad, on a category

A monoid in the category of all endomorphism functors on . In other words, a triple on a category is a covariant functor endowed with natural transformations and (here denotes the identity functor on ) such that the following diagrams are commutative:

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

For any pair of adjoint functors and (see Adjoint functor) with unit and co-unit of adjunction and , respectively, the functor endowed with and is a triple on . Conversely, for any triple there exist pairs of adjoint functors and such that , and the transformations and 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 is naturally imbedded in the set of its subsets via singleton sets, and to each set of subsets of one associates the union of these subsets.

2) In the category of sets, every representable functor carries a triple: The mapping associates to each the constant function with value ; the mapping associates to each function of two variables its restriction to the diagonal.

3) In the category of topological spaces, each topological group , with unit , enables one to define a functor that carries a triple: Each element is taken to the element and the mapping is defined by .

4) In the category of modules over a commutative ring , each (associative, unital) -algebra gives rise to a triple structure on the functor , 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The non-descriptive name "triple" for this concept has now largely been superseded by "monad" , although there is an obstinate minority of category-theorists who continue to use it.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 is a monad on ; in other words, it is a functor equipped with natural transformations and satisfying the duals of the commutative diagrams above. Every adjoint pair of functors () gives rise to a comonad structure on the composite , as well as a monad structure on .

An important example of a functor which carries a comonad structure is , , or, equivalently, the functor of big Witt vectors, cf. -ring; Witt vector. A special case of the natural transformation occurs in algebraic number theory as the Artin–Hasse exponential, [a5].

Monads in the category of sets can be equivalently described by sets of -ary operations for each cardinal number (or set) ; gives the projection operations , and 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 -algebras) is by far the more important. Given a monad on a category , a -algebra in is a pair where is a morphism such that

commutes. A homomorphism of -algebras is a morphism in such that

commutes; thus, one has a category of -algebras, with an evident forgetful functor . The functor has a left adjoint , which sends an object of to the -algebra , and the monad induced by the adjunction () is the one originally given.

Now the Kleisli category of is just the full subcategory of on the objects : the category of free algebras (cf. also Category).

For a monad on , in the Kleisli construction the category has as objects the objects of , and as hom-sets the sets

The composition rule for assigns to and the -composite:

as identity mapping one uses the -morphism .

An adjoint pair , is obtained by setting for ,

for , for , and for .

Then will serve as unit for the adjunction, while the co-unit is given by

Co-algebras are defined in the same manner. In practice, co-algebras very often occur superposed on algebras; a comonad will be constructed on a category of algebras of some sort, , leading to the category of bi-algebras. An important class of cases involves a monad and a cotriple on the same category . There is a standard lifting of to a cotriple on . A "TG-bi-algebraTG-bi-algebra" means an object of ; 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 is said to be idempotent if is an isomorphism. In this case it can be shown that any -algebra structure on an object is necessarily a two-sided inverse for , and hence that is isomorphic to the full subcategory consisting of all objects such that is an isomorphism. is a reflective subcategory of , the left adjoint to the inclusion being given by itself. Conversely, for any reflective subcategory of , the monad on 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)