Universal mapping 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, Dedekind–MacNeille completion, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.
Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.
Suppose that U: D → C 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: X → U(A) is a morphism in C, such that the following initial property is satisfied:
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