Product (part 3)

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

such that if there is any other object P and maps f_A:P\to A and f_B:P\to B, then there is a unique map g which makes the following diagram commute:

The diagram commuting is simply saying that \pi_A\circ g=f_A and \pi_B\circ g=f_B. 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 A and B. This is a wonderful exercise to test your understanding of these definitions. I highly suggest working through it.

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3 Responses to Product (part 3)

  1. Pingback: Coproducts « Andy Soffer

  2. Pingback: Coproducts in the category of Sets « Andy Soffer

  3. Pingback: Back to functors « Andy Soffer

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