# Products (part 2)

Let’s step back for a second. We have a notion of products in sets, groups, vector spaces, etc. What would be great is if these are all really the same idea (they certianly look similar). So we want to make a product in an arbitrary category. We started out doing this by looking at projection maps. After all, that’s sort of the one thing that all products have in common.

Last time we mentioned that we need more that just the projection maps to define a product. That is, just because we have $A\leftarrow P\to B$ doesn’t mean that $P$ is the product of $A$ and $B$ as we want to define it. In fact, lots of things could map to both $A$ and $B$. We saw that in $\textsc{Set}$, the Cartesian products $A\times B\times C$ and $A\times B$ both have those maps. Shucks.

But in fact we’re not out of luck, because, if we have $f_A:A\times B\times C\to A$, we can actually “factor” this map into two pieces. The first will be a map $g:A\times B\times C\to A\times B$, and the second will be the map $\pi_A:A\times B\to A$. What I mean is, there is such a $g$ with $\pi_A\circ g=f$. With this in mind, here’s a definition for a product.

In a category $\mathcal C$, given two objects $A$ and $B$, consider the subcategory $\mathcal C'$ induced by all objects which have morphisms into both $A$ and $B$. Then the product is $A\times B$ is a terminal object in $\mathcal C'$.

Remember that terminal objects don’t necessarily exist in every category. Consequently products don’t necessarily exist in every category. However, as we already showed, if they do exist, they are unique. In the categories you know and love, products as we defined them are the same as products as we know them. The product in $\textsc{Set}$ is the Cartesian product. The product in vector spaces is the vector space product.