Coproducts

Hooray for duality! We get a notion of coproducts for free.

We said that a product of A and B was a terminal object in the subcategory induced by the collection of objects that have morphisms to both A and B. 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 A and B.

In other words, the coproduct A\coprod B has injection maps i_A:A\to A\coprod B and i_B:B\to A\coprod B, and if there is any other C and f_A:A\to C and f_B:B\to C, then there is a unique morphism g which makes

commute.

Let me juxtapose the diagrams for products and coproducts so you can really see the duality.

(a) is a diagram commonly used to describe products. (b) is a diagram commonly used to explain coproducts.

Notice that even the symbol for coproducts is the symbol \prod (commonly used for products) upside down. For whatever reason, some people still use \times for products sometimes. Now that I think about it, I should go back and change all of the \times to \prod. 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.

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