In , we attempt to avoid thinking about what an object is, and look only at how it interacts with other objects. It turns out that even if we're not allowed to talk about the "internal" structure of an object, we can still pin down some objects just by talking about their interactions. For example, if we are not allowed to define the as "the set with no elements", we by means of a "universal property", talking instead about the functions from the empty set rather than about the elements of the empty set.

This page is not designed to teach you any particular universal properties, but rather to convey a sense of what the idea of "universal property" is about. You are supposed to let it wash over you gently, without worrying particularly if you don't understand words or even entire sentences.

Examples

• The can be . Specifically, it is an instance of the idea of an , in the category of sets. The same idea captures the trivial , the $Z$ of , and the natural number $0$.
• The has a universal property, generalising the , the product of integers, the in a , the products of many different , and many other things besides.
• The has a universal property (we refer to this property by the unwieldy phrase "the free-group is left- to the "). The same property can be used to create free rings, the on a set, and the free on a set. This idea of the left adjoint can also be used to define initial objects (which is the generalised version of the universal property of the empty set). ^[1]

The above examples show that the ideas of category theory are very general. For instance, the third example captures the idea of a "free" object, which turns up all over .

Definition "up to isomorphism"

Universal properties might not define objects

Universal properties are often good ways to define things, but just like with any definition, we always need to check in each individual case that we've actually defined something coherent. There is no silver bullet for this: universal properties don't just magically work all the time.

For example, consider a very similar universal property to that of the (detailed ), but instead of working with sets, we'll work with , and instead of between sets, we'll work with field homomorphisms.

The corresponding universal property will turn out not to be coherent:

The initial field ^[2] is the unique field $F$ such that for every field $A$, there is a unique field homomorphism from $F$ to $A$.

1. ^{^︎}

   Indeed, an initial object of category $C$ is exactly a left adjoint to the unique functor from $C$ to $1$ the one-arrow category.

2. ^{^︎}

   Analogously with the empty set, but fields can't be empty so we'll call it "initial" for reasons which aren't important right now.

3. ^{^︎}

   $F_2$ has elements $0$ and $1$, and the relation $1+1=0$.

4. ^{^︎}

   That is, the group whose is $F$ without $0$, with the group operation being "multiplication in $F$".