Apply - Universal Property

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.

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

Universal Property - History
... Universal properties of various topological constructions were presented by Pierre Samuel in 1948 ...
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... 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) ...
Tensor Product Of Hilbert Spaces - Definition - 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 ...
Universal Constructions - Heyting Algebra of Formulas Equivalent With Respect To A Theory 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 〈〉 ...

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