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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One Response to Products (part 1)

  1. Pingback: Products (part 2) « Andy Soffer

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