Products & Coproducts

A cartesian product, a disjoint union and a greatest common divisor, all pinned down by the same demand about arrows.

The idea

The product and the coproduct are the categorical forms of the cartesian product and the disjoint union. The set-level recipes describe what the new object contains, and the objects of a general category have no contents to describe, so both constructions are pinned down another way: by how other objects map into them, or out of them.

Definition (Product).

Let $X$ and $Y$ be objects of a category $\mathcal{A}$. A product of $X$ and $Y$ is an object $P$ together with two arrows $p \colon P \to X$ and $q \colon P \to Y$ such that for every object $T$ and every pair of arrows $f \colon T \to X$ and $g \colon T \to Y$ there is exactly one arrow $h \colon T \to P$ satisfying $p \circ h = f, \qquad q \circ h = g.$

An arrow into $P$ is the same thing as a pair of arrows, one to each factor. When such a $P$ exists it is written $X \times Y$.

In $\mathbf{Set}$ the cartesian product of ordered pairs, with $p$ and $q$ the two coordinate projections, meets the demand. Given $f$ and $g$, the two equations dictate both coordinates of $h(t)$, so the only candidate is $h(t) = (f(t), g(t))$ — and that formula does satisfy them. Existence and uniqueness, which is exactly what the definition asks.

Nothing in the definition prefers one direction of arrow, so reversing every arrow gives the coproduct $X + Y$: an object equipped with arrows from $X$ and from $Y$, out of which an arrow is the same thing as a pair of arrows. In $\mathbf{Set}$ the coproduct is the disjoint union.

Ways to work on it

• Walkthrough. Define products and coproducts by their universal property and meet them in familiar categories.
• Proof. A product is a terminal object in the category of cones, so uniqueness up to a unique isomorphism is the theorem you already have.
• Practice. Identify products and coproducts in familiar categories and compute with the universal property.
• Hardest. Explore categories where products and coproducts differ — or fail to exist.

Not sure where to start? Take the ten-question placement test.