Universal property

In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Formal Definition). Informally, it represents an intuition that two mathematical theories work in the same way in some general sense, regardless of their specific differences; if those two theories have the same universal property, there is a formally defined transformation that represents this similarity in a precise way.

Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category (see Connection with comma categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory.

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.

To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see Examples below). Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples.

Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right).

Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

be the forgetful functor which assigns to each algebra its underlying vector space.

A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist.

Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.

Universal properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.