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)
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)