Naturality (part 2)

Now that we’ve seen the definition, I’d like to give some motivation for what it means to be a natural transformation. In some sense, it’s a map between functors. Functors in some sense are a map between morphisms, so this is like one level more abstract, but that doesn’t seem to get the point across. I’m going to take the following example almost directly from Steve Awodey‘s book Category Theory. I think this is a great text for learning basic category theory.

In a category \mathcal C with products, and objects A, B, and C, the objects A\times(B\times C) and (A\times B)\times C are technically not the same object, but there’s an obvious isomorphism between them. But this is more than just your standard isomorphism. It’s not like we construct the isomorphism one way if we use a specific object A, and a different way for another object A'. It doesn’t really depend on the objects A, B, or C at all. It depends on the structure of the functor \times. How can we encode that?

We encode that by saying that the maps between objects A and A' can be “puffed up” to maps between (A\times B)\times C and (A'\times B)\times C, and to maps between A\times (B\times C) and A'\times (B\times C). Moreover, these puffed up maps are somehow not very different, in the sense that the way we “transform” from (A\times B)\times C to A\times (B\times C) is “the same” on the domain and codomain ends of the “puffed up” maps. Take some time to parse my attempt at writing a coherent sentence.

Take a look at the diagram again and see if you can understand what I was trying to get across. Here, F is the functor (-\times B)\times C and G is the functor -\times(B\times C). Can you write down what \eta_X should be? Does the diagram commute?

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