universal construction

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Universal properties are commonly used in mathematics, often without mentioning the term “universal property”.

For example, if one were asked to give a map $\mathbb{R} \to \mathbb{R} \times \mathbb{C}$, they might write down something like $x \mapsto (x^2, x + i x)$. In effect, what is done is that a pair of maps $\mathbb{R} \to \mathbb{R}$ and $\mathbb{R} \to \mathbb{C}$ is given, namely $(x \mapsto x^2, x \mapsto x + i x)$. The **universal property** of the product says that giving a map to $A \times B$ is the same as giving a map to $A$ and a map to $B$, and moreover this correspondence is natural in some precise sense.

Similarly, given rings $R$ and $S$, if we want to extend a ring homomorphism $R \to S$ to a homomorphism from the polynomial ring $R[x] \to S$, all we have to do is to specify an element of $S$ that we send $x$ to. In other words, a homomorphism $R[x] \to S$ is the same as a homomorphism $R \to S$ and an element of $S$.

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 $f: X \to A \times B$ biject naturally with pairs of maps $(f_1: X \to A, f_2: X \to B)$.

The naturality condition is that if $f: X \to A \times B$ corresponds to $f_1: X \to A, f_2: X \to B$, and $g: Y \to X$ is a map, then $f \circ g$ corresponds to $f_1 \circ g$ and $f_2 \circ g$, so that the bijection respects composition.

In particular, the identity map $id: A \times B \to A \times B$ corresponds to a pair of projection maps $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$. Then if $f: X \to A \times B$ is a map, then by naturality, it corresponds to $\pi_1 \circ f: X \to A$ and $\pi_2 \circ f: X \to B$.

Suppose we are not given a bijection, but just an object $P$ with maps $\pi_1: P \to A$ and $\pi_2: P \to B$. Then as above, we obtain a function from maps $X \to P$ to pairs of maps $X \to A, X \to B$ by composition. This makes $P$ into the product of $A$ and $B$ exactly when this function is a bijection, ie. for any pair of maps $f_1 : X \to A, f_2: X \to B$, there is a unique map $f: X \to P$ whose compositions with $\pi_1, \pi_2$ are $f_1, f_2$ 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: $C$ is a product of $A$ and $B$ if there exists maps $\pi_1: C \to A$ and $\pi_2: C \to B$ such that given any pair of maps $f_1: X \to A$ and $f_2: X \to B$, there is a unique map $f: X \to C$ such that the following diagram commutes:

$\array{
& & X & & \\
& ^\mathllap{f_1}\swarrow & \downarrow^\mathrlap{f} & \searrow^\mathrlap{f_2}\\
A & \underset{\pi_1}{\leftarrow} & C & \underset{\pi_2}{\rightarrow} & B
}$

In this case, we tend to write $A \times B$ for $C$.

Note that if we are talking about sets, then an element of a set $X$ is equivalent to a map $1 \to X$ from the singleton set $1$. Thus in particular, the above definition says an element of $A \times B$ is the same as a pair of elements $(a, b)$, where $a \in A$ and $b \in B$.

The free group on $n$ generators is a group $F_n$ such that group homomorphisms $F_n \to G$ bijects (naturally) with $n$ elements of $G$ (not necessarily distinct).

Similar to the above, the naturality condition says if $f: F_n \to G$ corresponds to $g_1, ..., g_n \in G$, and $h: G \to H$ is a map, then $h \circ f$ corresponds to the elements $h(g_1), ..., h(g_n)$. In particular, suppose the identity map $id: F_n \to F_n$ corresponds to $n$ elements $x_1, ..., x_n\in F_n$. Then any homomorphism $f: F_n \to G$ corresponds to the elements $f(x_1), ..., f(x_n)$ of $G$.

Thus, given the specified elements $x_1, ..., x_n \in F_n$, the universal property says given any $n$ elements of $G$, we can find a unique homomorphism $f: F_n \to G$ that sends $x_1, ..., x_n$ to the $n$ elements.

Diagrammatically, picking $n$ elements out of a set $X$ is the same as a function (of sets) $n \to X$. If we write $U(G)$ for the underlying set of the group $G$ (ie. $U$ is the forgetful functor to $Set$), the universal property of the free group says that there is a specified function $\phi: n \to U(F_n)$, such that for every function $f: n \to U(G)$, we can find a unique group homomorphism $\tilde{f}: F_n \to G$ such that the following diagram commutes:

$\array{
U(F_n) & \overset{U(\tilde{f})}{\to} & U(G)\\
^\mathllap{\phi}\uparrow & \nearrow_{\mathrlap{f}}\\
n
}$

In other words, every map $f: n \to U(G)$ factors through the universal map $\phi: n \to U(F_n)$ uniquely.

The tensor product of vector spaces has the universal property that a bilinear map $V \times W \to U$ bijects naturally with linear maps $V \otimes W \to U$. The naturality condition is given by the existence of a universal bilinear map $\phi: V \times W \to V \otimes W$ such that every bilinear map $V \times W \to U$ factors through $\phi$ uniquely.

We have more degenerate examples such as the terminal object:

In the category of sets, the singleton $1$ satisfies the property that there is always a unique map from any object to $1$. So we can say that the maps $A \to 1$ biject (necessarily naturally) with the set $1$. More generally, in any category, if an object $X$ is such that there is always a unique map from any object to $X$, then $X$ is called the terminal object.

Dually, an initial object is an object $0$ such that there is a unique map from $0$ to any object $X$.

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.

Some or all of these have analogs in higher category theory, notably in 2-category theory and (∞,1)-category theory:

An introductory but thorough treatment of universal properties and their relationship with the Yoneda lemma is given in

- Emily Riehl,
*Category Theory in context*, Chapter 2. (pdf)

A more informal and intuitive account is given in

- Paolo Perrone,
*Notes on Category Theory with examples from basic mathematics*, Chapter 2. (arXiv)