Universal Properties I - Inspiring Examples

The general fact of the uniqueness of the universal arrows implies the uniqueness of the ... object, up to a unique isomorphism (who wants more?).

Many (perhaps even most!) constructions in abstract algebra involve an object (and morphism) that satisfies some defining "universal property." But often the mathematician unaccustomed with category theory is left to wonder:

What exactly is a universal property?
Does every object have a universal property?
Can an object have more than one universal property?
Why do we care about universal properties, anyway?

We'll soon see answers to all of these questions, but for now let's just remember the first motto of category theory: it's all about the arrows!

Rather than dive straight into the technical definitions, I think it's more illuminating to first review many specific situations in which a common construction is characterized by some property about maps.

Examples in $Set$

Eilenberg and Mac Lane

...whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these objects as an integral part of their definition.

You encounter many constructions on sets as soon as you start studying set theory. Here we list a few, along with a "universal property" they satisfy.

Cartesian product

Suppose $X$ and $Y$ are two sets. The Cartesian product $X \times Y$ is usually defined to be the set of all ordered pairs $(x, y)$ with $x \in X$ and $y \in Y$ . There are two (natural) projection maps from $X \times Y$ , namely $p_{1} : X \times Y \to X$ defined by $(x, y) \mapsto x$ and $p_{2} : X \times Y \to Y$ defined by $(x, y) \mapsto y$ .

One can prove that this collection of information (the set $X \times Y$ together with those two projection maps) is "universal among all such information." More precisely, for any set $Z$ equipped with two maps $q_{1} : Z \to X$ and $q_{2} : Z \to Y$ , there there is a unique set map $h : Z \to X \times Y$ such that $q_{1}$ factors through $p_{1}$ and $q_{2}$ factors through $p_{2}$ . This can be visualized below:

The map $h$ is easy to describe in terms of elements. It is simply given by $h (z) = (q_{1} (z), q_{2} (z))$ . (Convince yourself that if the diagram is to be commutative, there is actually no other choice!)

There is also a "naturality" to this construction, in that the construction of the Cartesian product $X \times Y$ is "the same" for every pair of sets $(X, Y)$ . This is an observation that will apply to many of the following examples and is something we will explore more later when we talk about adjoints.

Disjoint union

The disjoint union of two sets is exactly dual to the Cartesian product construction. Indeed, if $X$ and $Y$ are two sets then the disjoint union $X ⊔ Y$ is a set equipped with maps $i_{1} : X \to X ⊔ Y$ and $i_{2} : Y \to X ⊔ Y$ , "universal among all such information." Once again, what this means precisely is that whenever $Z$ is a set equipped with two maps $j_{1} : X \to Z$ and $j_{2} : Y \to Z$ , then there is a unique set map $h : X ⊔ Y \to Z$ such that $j_{1}$ factors through $i_{1}$ and $j_{2}$ factors through $i_{2}$ :

Quotient by equivalence relations

Suppose $E$ is an equivalence relation on a set $X$ and $X / E$ is the corresponding quotient set, consisting of the equivalence classes of elements of $X$ under $E$ . Let $p : X \to X / E$ be the "projection" map that sends each element to its $E$ -equivalence class. The "universal property" of $X / E$ (and really of the map $p : X \to X / E$ ) is that any function $f$ on $X$ that respects the equivalence relation descends to a function on $X / E$ . More precisely, any set map $f : X \to Y$ for which $f (s) = f (s^{'})$ whenever $s \sim_{E} s^{'}$ factors uniquely through $p$ :

(This example can be made even more "categorical" if one is willing to reframe the definition of "equivalence relation" without reference to elements.)

Equalizers

Have you ever wondered if there was an analogue in $S e t$ of the notion of a kernel? Since sets do not have an element identified as "zero", there is no direct analogue of a kernel. (Since $S e t$ has distinct initial and terminal objects, there also is no "zero set" that would allow us to formally define zero maps, either.)

There is something close, however, called the equalizer. Given a pair of set maps $f, g : X \to Y$ between the same two sets, the equalizer of $f$ and $g$ is the subset $E$ of $X$ consisting of all elements at which $f$ and $g$ agree. In other words, it is the set $E = {x \in X ∣ f (x) = g (x)}$ . Let $i : E \to X$ be the inclusion map. Then we have a set $E$ together with a map $i : E \to X$ such that $f \circ i = g \circ i$ . The equalizer is "universal for such information." That is, if $Z$ is any set together with a map $j : Z \to X$ such that $f \circ j = g \circ j$ , then $j$ factors uniquely through $i$ :

At the level of elements, the map $h$ is given by $h (z) = j (z)$ ; note that since $f (j (z)) = g (j (z))$ the element $j (z)$ is indeed in the set $E$ . (Once again, the commutativity of the diagram forces this as the only possible definition of $h$ .)

Soon we will see how to extend this idea to general categories, i.e., outside of $S e t$ . In an abelian category we will also see that we can replace the pair of maps $(f, g)$ with their difference $f - g$ , in which the equalizer of the pair $(f, g)$ can be replaced by $\ker (f - g)$ .

Coequalizers

As will always be the case, there is a notion dual to equalizers, named ... (drumroll) ... coequalizers. I will simply leave a diagram here with no explanation. Can you fill in the details?

Pushouts/Pushforwards

Suppose $f : Z \to X$ and $g : Z \to Y$ are set maps. Let $X ⊔_{Z} Y$ be the quotient of the disjoint union $X ⊔ Y$ by the equivalence relation that identifies the elements $f (z)$ and $g (z)$ for every $z \in Z$ . Visually, one imagines gluing together the sets $X$ and $Y$ along the image in each of $Z$ . This set comes equipped with two set maps $i : X \to X ⊔_{Z} Y$ and $j : Y \to X ⊔_{Z} Y$ such that the diagram below is commutative:

We call the diagram above a pushout (or pushforward) diagram. This information is universal among all such pairs of set maps from $X$ and $Y$ with the analogous commutativity conditions.

Pullbacks

Exactly dual to the notion of a pushout is that of a pullback. The relevant diagram is shown below. Can you fill in the details?

The pullback object is sometimes also called a fibered product or a product of $X$ and $Y$ over $Z$ .

Examples in $Grp$

Many of the constructions in $S e t$ described above can also be carried out analogously in $G r p$ , where they frequently (and unfortunately) have a different name. For example, in the pushout example above, that same diagram in $G r p$ defines the so-called free product with amalgamation. Rather than repeat the diagrams, let's just focus on one heavy-hitter: the quotient group construction.

Quotient groups

The quotient group construction is probably the first time most of us officially encounter the phrase "universal property." Let's recall the details. Suppose $G$ is a group and $N$ is a normal subgroup. Then the set of left cosets of $N$ in $G$ has the structure of a group and there is a group morphism $π : G \to G / N$ with $N \subseteq \ker (π)$ , universally so. That is, every other morphism $f : G \to H$ with $N \subseteq \ker (f)$ factors uniquely through $π$ :

(You might recall that we actually have $\ker (π_{N}) = N$ . Can you prove this must be the case regardless, simply from the universal property?)

You might notice something a little different about this example compared to all the others listed here. In this example, the universal property involves information not encoded in the diagram, namely that the arrows have been restricted (to only morphisms with a certain property). We could try to fix this, by adding additional information to the diagram in such a way to encode this condition, but I think a better idea is to instead look for a characterization of "universal property" that goes behind diagrams. That's exactly what Yoneda's Lemma is going to give us.

Saunders Mac Lane

Now the quotient group is usually described as a group whose elements are cosets. However, once the cosets are used to prove this one "universal" property of $π : G \to G / N$ , all other properties of quotient groups - for example, the isomorphism theorems - can be proved with no further mention of cosets ... All that is needed is the existence of a universal element ... For that matter, even this existence could be proved without using cosets (see the adjoint functor theorem ...)

Examples in $Ab$

Direct sum

Suppose $A$ and $B$ are two abelian groups. What is the smallest abelian group that contains $A$ and $B$ ? Spoiler: it's not their (disjoint) union, at least not in general. One reason is that the disjoint union of their sets of elements is likely not even a group. However, there is a construction that produces from $A$ and $B$ a new group with the universal property in $Ab$ directly analogous to that of the disjoint union in $Set$ . This construction is the so-called direct sum of $A$ and $B$ . Usually denoted $A \oplus B$ , as a set this group consists of all "formal sums"^[1] of the form $a + b$ with $a \in A$ and $b \in B$ . The operation is as expected, with $(a_{1} + b_{1}) + (a_{2} + b_{2}) = (a_{1} + a_{2}) + (b_{1} + b_{2})$ . One can verify that $A \oplus B$ really is an abelian group, and that there are evident morphisms $i_{1} : A \to A \oplus B$ and $i_{2} : B \to A \oplus B$ defined by $i_{1} (a) = a + 0_{B}$ and $i_{2} (b) = 0_{A} + b$ , respectively. As with the disjoint union, this information is universal. In other words, if $C$ is any other abelian group together with group morphisms $j_{1} : A \to C$ and $j_{2} : B \to C$ , then these morphisms factor uniquely through $i_{1}$ and $i_{2}$ , respectively:

Look at this picture and compare with the picture (in $Set$ ) for disjoint union:

At least as far as category theory is concerned ...

Suggested next notes

Universal Properties II - Commutative diagrams, cones and limits
Direct products of modules
Direct sums of modules
Sums of submodules
Direct products vs. direct sums vs. sums

Does the informal nature of our use of the phrase "formal sums" bother you? It probably should. We'll try to do better later on, when we look at direct sums of modules. ↩︎