Uniqueness of product in the category of Sets

Question

I understand why the Cartesian product $A \times B$ is the product of $A$ and $B$ in the category of sets. But is it the only product from the point of view of the category theory? Following the categorical definition, product is defined up to an isomorphism and there can be more than one product.

I think $B \times A$ is another product (and there is a bijection between $B\times A$ and $A \times B$). Am I right? Are there any other products?

Answer 1

In category theory, a product of $A$ and $B$ is not merely an object, but rather it is an object $P$ together with two morphism $\pi_A\colon P\to A$ and $\pi_B\colon P\to B$, called the "projections", that satisfy the universal property associated to the product.

The universal property ensures that if $(P,\pi_A,\pi_B)$ is a product, and $(R,\rho_A,\rho_B)$ is also a product, then there exists a unique isomorphism $\phi\colon P\to R$ such that $\pi_A = \rho_A\circ \phi$ and $\pi_B=\rho_B\circ\phi$. So we say the product is "unique up to unique isomorphism", but that uniqueness requires you to have the full information about the product: the object and the projection morphisms.

Conversely, if $Q$ is any object and $\psi\colon P\to Q$ is an isomorphism, then we can make $Q$ into a product of $A$ and $B$ by defining $q_A\colon Q\to A$ to be $q_A=\pi_A\circ\psi^{-1}$ and $q_B=\pi_B\circ\psi^{-1}$.

So, in your case, yes: $A\times B$ with $\pi_A(a,b)=a$ and $\pi_B(a,b)=b$ is a product of $A$ and $B$. Also, $B\times A$ with $\pi_A(b,a)=a$ and $\pi_B(b,a)=b$, is another product. And any set that can be bijected with $A\times B$ can be given the structure of a product by defining suitable "projection functions".

(The above holds for any family of objects, not just pairs)

Answer 2

This was briefly mentioned in Arturo's answer, but let me spell it out more fully.

Let $A$, $B$ and $P$ be sets and suppose that $f : P \cong A \times B$. Define $p_1: P \to A$ by $\pi_1 \circ f$ and $p_2 : P \to B$ by $\pi_2 \circ f$. I claim that $(P, p_1, p_2)$ is a categorical product of $A$ and $B$.

First, $p_1$ and $p_2$ give the necessary projections. For the universal property, let $h : S \to A$ and $k : S \to B$ be any functions (for any set $S$). We want to prove that there is a unique function $u: S \to P$ such that $p_1 \circ u = h$ and $p_2 \circ u = k$. By the universal property for the ordinary product, (h, k) is the unique function $S \to A \times B$ such that $\pi_1 \circ (h, k) = h$ and $\pi_2 \circ (h, k) = k$. Since $f$ is an isomorphism, this implies that $u = f^{-1} \circ (h, k)$ is the unique function such that $\pi_1 \circ f \circ u = h$ and $\pi_2 \circ f \circ u = k$. But that means that $u$ is the unique map we wanted, since $\pi_i \circ f = p_i$.

So in a sense, $A \times B$ is maximally non-unique: any set with the right cardinality can be made to work. But on the other hand, any such set is of the form above, meaning that the projection functions for one product are equal to the projection functions for the other product composed with an isomorphism between the two objects. This follows from the universal property. Phrased differently, any two products are isomorphic via a unique isomorphism that commutes with the projections.