**Universal Property**

Consider a function, that is, where the bracket notation denotes the space of functions from *A* to *B*. By means of currying, there is a unique function . Then **Apply** provides the universal morphism

- ,

so that

or, equivalently one has the commuting diagram

The notation for the space of functions from *A* to *B* occurs more commonly in computer science. In category theory, however, is known as the exponential object, and is written as . There are other common notational differences as well; for example *Apply* is often called *Eval*, even though in computer science, these are not the same thing, with eval distinguished from *Apply*, as being the evaluation of the quoted string form of a function with its arguments, rather than the application of a function to some arguments.

Also, in category theory, *curry* is commonly denoted by, so that is written for *curry*(*g*). This notation is in conflict with the use of in lambda calculus, where lambda is used to denote free variables. With all of these notational changes accounted for, the adjointness of *Apply* and *curry* is then expressed in the commuting diagram

The articles on exponential object and Cartesian closed category provide a more precise discussion of the category-theoretic formulation of this idea. Thus use of lambda here is not accidental; Cartesian-closed categories provide the general, natural setting for lambda calculus.

Read more about this topic: Apply

### Other articles related to "universal property, universal, property":

**Universal Property**- History

...

**Universal**properties of various topological constructions were presented by Pierre Samuel in 1948 ...

... the C*-algebra satisfying the following

**universal property**for all C*-algebra D containing A as an ideal, there exists a unique *-homomorphism φ D → M(A) such that φ extends the identity homomorphism on A ... Uniqueness up to isomorphism is specified by the

**universal property**... The above lemma, together with the

**universal property**of the multiplier algebra, yields that M(A) is isomorphic to the idealizer of π(A) in B(H) ...

**Universal Property**

... The Hilbert tensor product is characterized by the following

**universal property**(Kadison Ringrose 1983, Theorem 2.6.4) There is a weakly Hilbert-Schmidt mapping p H1 × H2 → H ... As with any

**universal property**, this characterizes the tensor product H up to isomorphism ... The same

**universal property**, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces ...

*T*

... Then HT satisfies the same

**universal property**as H0 above, but with respect to Heyting algebras H and families of elements 〈ai〉 satisfying the

**property**that J(〈ai〉)=1 for any axiom J(〈Ai〉) in T ... of its elements 〈〉, itself satisfies this

**property**.) The existence and uniqueness of the morphism is proved the same way as for H0, except that ... algebra H0 on the same set of variables, by applying the

**universal property**of H0 with respect to HT, and the family of its elements 〈〉 ...

### Famous quotes containing the words property and/or universal:

“I have no concern with any economic criticisms of the communist system; I cannot enquire into whether the abolition of private *property* is expedient or advantageous. But I am able to recognize that the psychological premises on which the system is based are an untenable illusion. In abolishing private *property* we deprive the human love of aggression of one of its instruments ... but we have in no way altered the differences in power and influence which are misused by aggressiveness.”

—Sigmund Freud (1856–1939)

“We are often made to feel that there is another youth and age than that which is measured from the year of our natural birth. Some thoughts always find us young, and keep us so. Such a thought is the love of the *universal* and eternal beauty.”

—Ralph Waldo Emerson (1803–1882)