# Triple

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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. .

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).

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### 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).

How to Cite This Entry:
Triple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triple&oldid=16681
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article