January 30, 2012 Leave a comment
Hooray for duality! We get a notion of coproducts for free.
We said that a product of and was a terminal object in the subcategory induced by the collection of objects that have morphisms to both and . Then the dual notion would be a called a coproduct, and would be an initial object in the subcategory induced by the collection of objects that have morphisms from both and .
In other words, the coproduct has injection maps and , and if there is any other and and , then there is a unique morphism which makes
Let me juxtapose the diagrams for products and coproducts so you can really see the duality.
Notice that even the symbol for coproducts is the symbol (commonly used for products) upside down. For whatever reason, some people still use for products sometimes. Now that I think about it, I should go back and change all of the to . Oh well. Laziness prevails.
I should also note that, since we showed initial and terminal objects are unique up to isomorphism, so are coproducts and products.