In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Formal Definition). 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.
Let be a functor between categories and . In what follows, let be an object of , while and are objects of .
Thus, the functor maps , and in to , and in .
A universal morphism from to is a unique pair in which has the following property, commonly referred to as a universal property. For any morphism of the form in , there exists a unique morphism in such that the following diagram commutes:
We can dualize this categorical concept. A universal morphism from to is a unique pair that satisfies the following universal property. For any morphism of the form in , there exists a unique morphism in such that the following diagram commutes:
Note that in each definition, the arrows are reversed. Both definitions are necessary to describe universal constructions which appear in mathematics; but they also arise due to the inherent duality present in category theory. In either case, we say that the pair which behaves as above satisfies a universal property.
Universal morphisms can be described more concisely as initial and terminal objects in a comma category.
Let be a functor and an object of . Then recall that the comma category is the category where
Now suppose that the object in is initial. Then for every object , there exists a unique morphism such that the following diagram commutes.
Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a universal morphism from to . Therefore, we see that a universal morphism from to is equivalent to an initial object in the comma category .
Conversely, recall that the comma category is the category where
Suppose is a terminal object in . Then for every object , there exists a unique morphism such that the following diagrams commute.
The diagram on the right side of the equality is the same diagram pictured when defining a universal morphism from to . Hence, a universal morphism from to corresponds with a terminal object in the comma category .
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.
Let be the category of vector spaces -Vect over a field and let be the category of algebras -Alg over (assumed to be unital and associative). Let
be the forgetful functor which assigns to each algebra its underlying vector space.
Given any vector space over we can construct the tensor algebra . The tensor algebra is characterized by the fact:
This statement is an initial property of the tensor algebra since it expresses the fact that the pair , where is the inclusion map, is a universal morphism from the vector space to the functor .
Since this construction works for any vector space , we conclude that is a functor from -Vect to -Alg. This means that is left adjoint to the forgetful functor (see the section below on relation to adjoint functors).
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.
Let and be objects of a category with finite products. The product of and is an object × together with two morphisms
such that for any other object of and morphisms and there exists a unique morphism such that and .
To understand this characterization as a universal property, take the category to be the product category and define the diagonal functor
by and . Then is a universal morphism from to the object of : if is any morphism from to , then it must equal a morphism from to followed by .
Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.
Let and be categories with a small index category and let be the corresponding functor category. The diagonal functor
is the functor that maps each object in to the constant functor to (i.e. for each in ).
Given a functor (thought of as an object in ), the limit of , if it exists, is nothing but a universal morphism from to . Dually, the colimit of is a universal morphism from to .
Defining a quantity does not guarantee its existence. Given a functor and an object of , there may or may not exist a universal morphism from to . If, however, a universal morphism does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism: if is another pair, then there exists a unique isomorphism such that . This is easily seen by substituting in the definition of a universal morphism.
It is the pair which is essentially unique in this fashion. The object itself is only unique up to isomorphism. Indeed, if is a universal morphism and is any isomorphism then the pair , where is also a universal morphism.
The definition of a universal morphism can be rephrased in a variety of ways. Let be a functor and let be an object of . Then the following statements are equivalent:
The dual statements are also equivalent:
Suppose is a universal morphism from to and is a universal morphism from to . By the universal property of universal morphisms, given any morphism there exists a unique morphism such that the following diagram commutes:
If every object of admits a universal morphism to , then the assignment and defines a functor . The maps then define a natural transformation from (the identity functor on ) to . The functors are then a pair of adjoint functors, with left-adjoint to and right-adjoint to .
Similar statements apply to the dual situation of terminal morphisms from . If such morphisms exist for every in one obtains a functor which is right-adjoint to (so is left-adjoint to ).
Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let and be a pair of adjoint functors with unit and co-unit (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in and :
Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of (equivalently, every object of ).
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.
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