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:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)