In a category, a product (also called a cartesian product) of two objects and is an object equipped with morphisms and , called projections, such that for any object equipped with maps and there exists a unique (called the pairing of and ) such that and .
Remarks
When they exist, products are unique up to unique canonical isomorphism, so we often say “the product.”
One can define in a similar way a product of any family of objects. A product of the empty family is a terminal object.
A product is a special case of a limit in which the domain category is discrete.
This interactive demonstration in the category of finite sets lets you type two sets and see a product, showing the unique map to it from a randomly chosen equipped with and . It also generates a second product, showing the unique canonical isomorphism between this and the first product.
Revised on October 8, 2009 23:45:22
by Toby Bartels
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