Products (part 2)
January 26, 2012 1 Comment
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 doesn’t mean that is the product of and as we want to define it. In fact, lots of things could map to both and . We saw that in , the Cartesian products and both have those maps. Shucks.
But in fact we’re not out of luck, because, if we have , we can actually “factor” this map into two pieces. The first will be a map , and the second will be the map . What I mean is, there is such a with . With this in mind, here’s a definition for a product.
In a category , given two objects and , consider the subcategory induced by all objects which have morphisms into both and . Then the product is is a terminal object in .
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 is the Cartesian product. The product in vector spaces is the vector space product.