Product (category theory)

In the category of the product of a theory is indexed by the set I family of objects, a pair, wherein

• P is an object
• A morphism ( called projection) of P to be ( for each i of I),
• And for each object C and any family of morphisms from C to there is a unique morphism f from C to P are with.

This synthetic product is defined via a so-called universal property and is only up to natural isomorphism clearly.

This very general definition includes many other definitions occurring in the mathematics of the concept of product, and beyond more:

The dual notion is that of Koprodukts.

Examples

For abelian groups, modules, vector spaces and Banach spaces, the finite products comply with the finite coproducts. This is called a bi-product. Their existence is required in the definition of abelian categories (especially form Abelian groups, modules over a ring, or vector spaces over a field abelian categories).