Category Product

The product of a family {X_i}_(i in I) of objects of a category is an object P=product_(i in I)X_i, together with a family of morphisms {p_i:P->X_i}_(i in I) such that for every object Q and every family of morphisms {q_i:Q->X_i} there is a unique morphism q:Q->P such that

 p_i degreesq=q_i

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

In the category of sets, the product is the Cartesian product, and in the category of groups it is the group direct product. In both cases, P=product_(i in I)X_i, and p_i:P->X_i is the projection onto the ith factor.

See also

Cartesian Product, Coproduct, Direct Product, Group Direct Product

This entry contributed by Margherita Barile

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Joshi, K. D. "Products and Coproducts." 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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Category Product

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

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