Products (part 1)
January 25, 2012 1 Comment
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 be a category with two objects and . We want to make a definition of the product that, in the spirit of category theory, doesn’t use specific elements in or . So what do we know about products?
We normally think of a product as consisting of ordered pairs with and . Then if we have any other rules to satisfy (a group operation, for instance), each component takes the rules from it’s component. In , there’s nothing else to be satisfied. In , if we have two groups and , then the product has .
We know that products come equipped with certain maps known as “projections.” Consider given by . There’s a similar projection which projects the product onto the second coordinate.
So given a category and two objects and , their product should be some object that has morphisms and . That is, something that looks like this:
But that’s not good enough. There could be lots of objects that map into both and . For instance, in , if , , and are sets, the Cartesian product also has maps to both and . So how do we distinguish between what we want to be the product and something we don’t want, such as . Next time we’ll finish this characterization.