Universal Property - Formal Definition

Formal Definition

Suppose that U: DC is a functor from a category D to a category C, and let X be an object of C. Consider the following dual (opposite) notions:

An initial morphism from X to U is an initial object in the category of morphisms from X to U. In other words, it consists of a pair (A, φ) where A is an object of D and φ: XU(A) is a morphism in C, such that the following initial property is satisfied:

  • Whenever Y is an object of D and f: XU(Y) is a morphism in C, then there exists a unique morphism g: AY such that the following diagram commutes:

A terminal morphism from U to X is a terminal object in the comma category of morphisms from U to X. In other words, it consists of a pair (A, φ) where A is an object of D and φ: U(A) → X is a morphism in C, such that the following terminal property is satisfied:

  • Whenever Y is an object of D and f: U(Y) → X is a morphism in C, then there exists a unique morphism g: YA such that the following diagram commutes:

The term universal morphism refers either to an initial morphism or a terminal morphism, and the term universal property refers either to an initial property or a terminal property. In each definition, the existence of the morphism g intuitively expresses the fact that (A, φ) is "general enough", while the uniqueness of the morphism ensures that (A, φ) is "not too general".

Read more about this topic:  Universal Property

Other articles related to "formal definition":

Big O Notation - Example
... In typical usage, the formal definition of O notation is not used directly rather, the O notation for a function f(x) is derived by the following simplification rules If f(x) is a sum ... One may confirm this calculation using the formal definition let f(x) = 6x4 − 2x3 + 5 and g(x) = x4 ... Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, for some suitable choice of x0 and M and for all x > x0 ...
Atomic SP-DEVS - Formal Definition
... The above controller for crosswalk lights can be modeled by an atomic SP-DEVS model ... Formally, an atomic SP-DEVS is a 7-tuple where is a finite set of input events is a finite set of output events is a finite set of states is the initial state is the time advanced function which defines the lifespan of a state where is the set of non-negative rational numbers plus infinity ...

Famous quotes containing the words definition and/or formal:

    The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.
    Ralph Waldo Emerson (1803–1882)

    Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?”
    Horace Walpole (1717–1797)