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of a family of objects in a category

A concept describing the (categorical analogues of the) construction of a direct sum of modules or a discrete union (bouquet) of sets in the language of morphisms. Let , , be an indexed family of objects in a category . An object , together with morphisms , is called the coproduct of the family , , if for any family of morphisms , , there exists a unique morphism such that , . The morphisms are called the imbeddings of the coproduct; the coproduct is denoted by , , or in case . The morphism figuring in the definition of the coproduct is sometimes denoted by or . The coproduct of a family of objects is defined uniquely up to an isomorphism; it is associative and commutative. The coproduct is the dual concept of the product of a family of objects in a category.

The coproduct of the empty family of objects is the left zero (initial object) of the category. In an Abelian category, the coproduct is frequently called the direct sum of the family , , and is denoted by , or in case . In most categories of structured sets, the coproduct of a family of objects coincides with the free product of the family, and as a rule requires special description. Thus, in the category of groups, the coproduct is the free product of groups; in the category of modules it is the direct sum of modules; etc.

In a category with null morphisms, if is a coproduct, there exist uniquely defined morphisms such that , . In an Abelian category the coproduct and the product of a finite family of objects are one and the same.


[1] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)


Also in not necessarily Abelian categories the coproduct of a family of objects is frequently called the sum of a family of objects or the direct sum of a family of objects. Often used notations are , and .


[a1] N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)
[a2] J. Adámek, "Theory of mathematical structures" , Reidel (1983)
How to Cite This Entry:
Coproduct. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article