Universal properties
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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