Teaching intuition for the universal property of the product (category theory)

Question

In category theory, there's the idea of the product as an object satisfying a particular universal property. Can you suggest ways to make the concept of the product intuitive? (So far, my attempt contains three examples: "product of finite sets" and two instances of "product in a poset category", and I'm teaching primarily by example.)

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Background:

I've recently been writing a series about the category-theoretic concept of the universal property, over on Arbital (an open collaborative maths-explanations site). The aim is to make an intuitive explanation for people who are comfortable enough with mathematical notation that they can read formulae if they have explanations attached, and who are completely comfortable with the notion of a "function", but who have no exposure to category theory. (For example, fairly advanced students of computer science or engineering, or second-year undergraduate mathematics students. I studied maths, so I know the level of mathematics study I want from the students; my guesses for compsci and engineering are less well-founded.)

I'm running into what I presume are two of the standard problems (certainly the nLab has both these problems in spades, because it's not really designed to teach, but to be a reference work):

• I don't really know whether my explanation would teach someone who didn't already know the content. The only person we can get to dog-food this content is someone who already kind of knows the shape of some basic category theory.
• Category theory turns out to be quite difficult to explain. (Who knew.)

We've already got Universal Property of the Empty Set, which I'm pretty pleased with; now I'm trying to make intuitive the notion of the product, on Universal Property of the Product. The latter is what this Question is about.

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The following is not necessary for an answer to this Question, but if you have any suggested specific improvements to the Product page, I'd be very happy to hear them. Similarly, anyone assisting in any way with the rest of the Universal Properties project would receive twenty Internet points from me.

Answer 1

I'm going to explain how I think of the category-theoretic definition of products. Unfortunately, this viewpoint won't lead directly to the usual universal property but rather to an equivalent formulation in terms of morphisms into products. I hope that it can still be of some use even if you'd rather go directly to the universal property.

In most of the familiar categories (e.g., the categories of sets, of topological spaces, of groups) one would like a description of the product to say (among other things) that the elements of $A\times B$ are ordered pairs $(a,b)$ consisting of an element $a\in A$ and an element $b\in B$. From a category-theoretic point of view, there are two problems with this idea.

First, we should not attempt to define an object individually but only up to isomorphism, since isomorphic objects have the same category-theoretic properties. So we should rather say that the elements of $A\times B$ are in (canonical) bijection with ordered pairs as above.

Second, we can't exactly talk about elements in the usual sense; the best we can do in category-theory is to talk about morphisms into an object. A morphism $X\to Y$ can usefully be regarded as an $X$-indexed family of elements of $Y$ (some people say "generalized element of $Y$" for this), and much of what we want to say about elements can be said instead about such families of elements.

In the case at hand, this means that we'd describe $A\times B$ by saying that an $X$-indexed family of elements of $A\times B$ corresponds to a pair consisting of an $X$-indexed family of elements of $A$ and an $X$-indexed family of elements of $B$. Formally, this means that the morphisms $X\to A\times B$ are in (canonical) bijection with ordered pairs of a morphism $X\to A$ and a morphism $X\to B$.

If, as is usual in category theory, we take "canonical" to mean (at least) natural, then this description is equivalent to the universal property. Specifically, taking $X$ to be $A\times B$ itself, we get that the identity morphism of $A\times B$ must correspond to a pair of morphisms $p_A:A\times B\to A$ and $p_B:A\times B\to B$, and it follows from naturality that these have the required universal property to serve as the projection morphisms of $A\times B$ to the factors $A$ and $B$.

Incidentally, in the second paragraph above, I said that the (non-categorical) description of $A\times B$ would include the specification of its elements "among other things". Of course, in the category of groups, those other things would include the group operation on $A\times B$, and in the category of topological spaces they would include the product topology of $A\times B$. One of the delightful aspects of category theory is that, by using morphisms into $A\times B$ (families of elements) instead of just elements, the definition automatically includes those other things, without having to describe them separately.

Answer 2

Here is my intuition:

First, each object can be used to encode some information which is accessible through morphisms. For example, take the $\mathbf{Set}$ category and two objects $A = \{0\}$, $B = \{1, 2\}$. Now you can encode one bit of information by picking one of the morphisms $f_1(0) = 1$ or $f_2(0) = 2$. Now, suppose that we could assign to each object some quantity that represents its potential for encoding information.

The product $A\times B$ of $A$ and $B$ is an object that holds exactly the amount of potential for encoding information as both $A$ and $B$ together, not more, not less. To make this more concrete:

Not less: If you have some information encoded as $f_A : Y \to A$ and $f_B : Y \to B$, then you can "store" that together in $f_{A \times B} : Y \to A \times B$ and then recover that information separately using projections $\pi_A$ and $\pi_B$ (separately meaning the information with regard to $A$ is independent from the information with regard to $B$).

Not more: There is no additional, hidden information you could recover from $A \times B$ besides what you get via $\pi_A$ and $\pi_B$. In other words, if both morphisms $f_{A \times B}$ and $g_{A \times B}$ get you the same information when used with $\pi_A$ and $\pi_B$, then they have to be the same morphism (otherwise we could use that difference to encode more information).

When the categories get more complicated the encoding/decoding gets more complicated too, but I could not find an example for which this intuition breaks (although, note that I'm not a category-theorist).

I hope this helps $\ddot\smile$

Answer 3

In my opinion, the intuition regarding the use of universal property comes most directly through its use in defining ordered pairs.

To any pair of morphisms $f : X \to A$ and $g : X \to B$, there is the corresponding ordered pair $(f,g) : X \to A \times B$. Conversely, any morphism $h : X \to A \times B$ is an ordered pair; specifically $h = (\pi_0 \circ h, \pi_1 \circ h)$.

Described at a higher level, $(\cdot,\cdot)$ is the canonical natural bijection $$\hom(-,A) \times \hom(-,B) \to \hom(-, A \times B)$$

Note that this is not entirely dissimilar from Andreas Blass's answer; one could even argue we're saying the same thing in different language.

I think the use of ordered pairs here is important, because it promotes the universal property from being a "description" to an "operation" — while you can do this in principle with any universal property, the product is especially amenable to being used in calculation.

Also, note that this emphasizes the morphisms between the objects involved, rather than the objects themselves.