What is "universal property"

n. (context mathematics English) A definition of a mathematical object, up to isomorphism, in terms of abstract maps between it and other objects of the same category. More specifically, it is either the ''initial property'' of an initial object in a coslice category-like comma category or the ''terminal property'' of a terminal object in a slice category-like comma category; these are roughly analogous to the minimum and the maximum (respectively) of a certain lattice, and are used to define the object uniqueness up to isomorphism.

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.