The categorical notion which is dual to product. The coproduct of a family {X_i}_(i in I) of objects of a category is an object C=coproduct_(i in I)X_i, together with a family of morphisms {c_i:X_i->C}_(i in I) such that for every object D and every family of morphisms {d_i:X_i->D} there is an unique morphism d:C->D such that

 d degreesc_i=d_i

for all i in I. The coproduct is unique up to isomorphisms.

In the category of sets, the coproduct is the disjoint union C= union ^._(i in I)X_i, and c_i:X_i->C is the inclusion. In the category of Abelian groups, the coproduct is the group direct sum C= direct sum _(i in I)X_i, and c_i:X_i->C is the injection of the ith summand. In the category of groups, the coproduct is the free product of groups.

See also

Category Product, Direct Sum, Free Product, Group Direct Sum

This entry contributed by Margherita Barile

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Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.Joshi, K. D. "Coproducts and Products." Ch. 8 in Introduction to General Topology. New Delhi, India: Wiley, pp. 189-216, 1983.Kasch, F. "Construction of Products and Coproducts." §4.80 in Modules and Rings. New York: Academic Press, pp. 80-84, 1982.Rowen, L. "Products and Coproducts." In Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 73-76, 1988.Strooker, J. R. "Products and Sums." §1.5 in Introduction to Categories, Homological Algebra and Sheaf Cohomology Cambridge, England: Cambridge University Press, pp. 14-21, 1978.

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Barile, Margherita. "Coproduct." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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