# Product (part 3)

January 27, 2012 3 Comments

Here’s another way to think about products. In fact, this is how it’s more commonly described. Let be a category, and and objects in . The product (if it exists) is some object with maps and in the diagram

such that if there is any other object and maps and , then there is a unique map which makes the following diagram commute:

The diagram commuting is simply saying that and . Notice that this exactly the same property that I said last time. I said “it’s the terminal object in the subcategory of things that map into both and . This is a wonderful exercise to test your understanding of these definitions. I highly suggest working through it.

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