Universal properties are commonly used in mathematics, often without mentioning the term “universal property”.
For example, if one were asked to give a map , they might write down something like . In effect, what is done is that a pair of maps and is given, namely . The universal property of the product says that giving a map to is the same as giving a map to and a map to , and moreover this correspondence is natural in some precise sense.
Similarly, given rings and , if we want to extend a ring homomorphism to a homomorphism from the polynomial ring , all we have to do is to specify an element of that we send to. In other words, a homomorphism is the same as a homomorphism and an element of .
For it to be a universal property, just the existence of such a bijection is not sufficient. We will need some conditions to make sure the bijection is “natural”. Abstractly, this says that the bijection is given by a natural isomorphism of certain functors. More concretely, by the Yoneda lemma, this is equivalent to saying the bijection is “mediated” by some “universal maps”, which is how universal properties are usually formulated. See Concrete examples for more details.
Recall that by the Yoneda lemma, specifying how we can map in or out of an object uniquely determines the object up to isomorphism. So we can use these universal properties as definitions of the constructions! These are known as universal constructions. Of course, these definitions are not actually “constructions”. We still have to do the concrete constructions the good, old way to show that there are objects satisfying the universal property (or apply general theorems such as the adjoint functor theorem).
We first look at a few concrete examples of universal properties. These are all special cases of the ones described below.
The product of two objects (eg. sets, groups, rings etc.) is specified by the property that maps biject naturally with pairs of maps .
The naturality condition is that if corresponds to , and is a map, then corresponds to and , so that the bijection respects composition.
Suppose we are not given a bijection, but just an object with maps and . Then as above, we obtain a function from maps to pairs of maps by composition. This makes into the product of and exactly when this function is a bijection, ie. for any pair of maps , there is a unique map whose compositions with are respectively (naturality is easy to check).
(The experienced reader will notice that this is just a special case of the Yoneda lemma)
Thus, the universal property can be stated as follows: is a product of and if there exists maps and such that given any pair of maps and , there is a unique map such that the following diagram commutes:
In this case, we tend to write for .
Note that if we are talking about sets, then an element of a set is equivalent to a map from the singleton set . Thus in particular, the above definition says an element of is the same as a pair of elements , where and .
Similar to the above, the naturality condition says if corresponds to , and is a map, then corresponds to the elements . In particular, suppose the identity map corresponds to elements . Then any homomorphism corresponds to the elements of .
Thus, given the specified elements , the universal property says given any elements of , we can find a unique homomorphism that sends to the elements.
Diagrammatically, picking elements out of a set is the same as a function (of sets) . If we write for the underlying set of the group (ie. is the forgetful functor to ), the universal property of the free group says that there is a specified function , such that for every function , we can find a unique group homomorphism such that the following diagram commutes:
In other words, every map factors through the universal map uniquely.
The tensor product of vector spaces has the universal property that a bilinear map bijects naturally with linear maps . The naturality condition is given by the existence of a universal bilinear map such that every bilinear map factors through uniquely.
We have more degenerate examples such as the terminal object:
In the category of sets, the singleton satisfies the property that there is always a unique from any object to . So we can say that the maps biject (necessarily naturally) with the set . More generally, in any category, if an object is such that there is always a unique map from any object to , then is called the terminal object.
Dually, an intial object is an object such that there is a unique map from to any object .
In general, the universal constructions in category theory include
Each of these may be defined by requiring it to satisfy a universal property. A universal property is a property of some construction which boils down to (is manifestly equivalent to) the property that an associated object is a universal initial object of some (auxiliary) category.
In good cases, every single one of these is a special case of every other, so somehow one single concept here comes to us with many different faces.