- Product category
In the mathematical field of

category theory , the**product**of two categories "C" and "D", denoted nowrap| "C" × "D" and called a**product category**, is a straightforward extension of the concept of theCartesian product of two sets.**Definition**The product category nowrap| "C" × "D" has:

*as objects:

*:pairs of objects nowrap| ("A", "B"), where "A" is an object of "C" and "B" of "D";

*as arrows from nowrap| ("A"_{1}, "B"_{1}) to nowrap| ("A"_{2}, "B"_{2}):

*:pairs of arrows nowrap| ("f", "g"), where nowrap| "f" : "A"_{1}→ "A"_{2}is an arrow of "C" and nowrap| "g" : "B"_{1}→ "B"_{2}is an arrow of "D";

*as composition, component-wise composition from the contributing categories:

*:nowrap|1= ("f"_{2}, "g"_{2}) o ("f"_{1}, "g"_{1}) = ("f"_{2}o "f"_{1}, "g"_{2}o "g"_{1});

*as identities, pairs of identities from the contributing categories:

*:1_{("A", "B")}= (1_{"A"}, 1_{"B"}).**Relation to other categorical concepts**For small categories, this is the same as the action on objects of the

categorical product in the category**Cat**. Afunctor whose domain is a product category is known as abifunctor . An important example is theHom functor , which has the product of the opposite of some category with the original category as domain::Hom : "C"^{op}× "C" →**Set**.**Generalization to several arguments**Just as the binary Cartesian product is readily generalized to an "n"-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of "n" categories. The product operation on categories is commutative and associative,

up to isomorphism , and so this generalization brings nothing new from a theoretical point of view.

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