# Universal Property

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.

This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

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

Apply - Universal Property
... 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 ...
Universal Property - History
... Universal properties of various topological constructions were presented by Pierre Samuel in 1948 ...
Multiplier Algebra - Definition
... Its multiplier algebra M(A) is 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 ... 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 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 ... elements 〈〉, itself satisfies this property.) The existence and uniqueness of the morphism is proved the same way as for H0, except that one must modify the ... quotient of the free Heyting 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 〈〉 ...
Tensor Product Of Hilbert Spaces - Definition - Universal Property
... 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 ...

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

No man acquires property without acquiring with it a little arithmetic, also.
Ralph Waldo Emerson (1803–1882)

And universal Nature, through her vast
And crowded whole, an infinite paroquet,
Repeats one note.
Ralph Waldo Emerson (1803–1882)