# Products (part 1)

I want to put functors on the back burner for a while. I first want to talk about products. In fact, what I really want to talk about are universal properties, but it’s hard to grasp without seeing examples first. The definition of a product is one that uses a universal property.

Let $\mathcal C$ be a category with two objects $A$ and $B$. We want to make a definition of the product $A\times B$ that, in the spirit of category theory, doesn’t use specific elements in $A$ or $B$. So what do we know about products?

We normally think of a product as consisting of ordered pairs $(a,b)$ with $a\in A$ and $b\in B$. Then if we have any other rules to satisfy (a group operation, for instance), each component takes the rules from it’s component. In $\textsc{Set}$, there’s nothing else to be satisfied. In $\textsc{Grp}$, if we have two groups $(A,\cdot)$ and $(B,*)$, then the product has $(a_1,b_1)(a_2,b_2)=(a_1\cdot a_2,b_1*b_2)$.

We know that products come equipped with certain maps known as “projections.” Consider $\pi_A:A\times B\to A$ given by $\pi_A:(a,b)\mapsto a$. There’s a similar projection $\pi_B$ which projects the product onto the second coordinate.

So given a category $\mathcal C$ and two objects $A$ and $B$, their product should be some object $A\times B$ that has morphisms $\pi_A:A\times B\to A$ and $\pi_B:A\times B\to B$. That is, something that looks like this:

But that’s not good enough. There could be lots of objects that map into both $A$ and $B$. For instance, in $\textsc{Set}$, if $A$, $B$, and $C$ are sets, the Cartesian product $A\times B\times C$ also has maps to both $A$ and $B$. So how do we distinguish between what we want to be the product $A\times B$ and something we don’t want, such as $A\times B\times C$. Next time we’ll finish this characterization.

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