Definition
Let be a category with some objects and . An object is the product of and, denoted, iff it satisfies this universal property:
- there exist morphisms, called the canonical projections or projection morphisms, such that for every object and pair of morphisms there exists a unique morphism such that the following diagram commutes:
The unique morphism is called the product of morphisms and and is denoted .
Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set . Then we obtain the definition of a product.
An object is the product of a family of objects iff there exist morphisms, such that for every object and a -indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all :
The product is denoted ; if, then denoted and the product of morphisms is denoted .
Read more about this topic: Product (category Theory)
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