February 10, 2012 Leave a comment
Here’s the most common example of a natural transformation that I know of, and probably the most enlightening. Take a vector space and let denote it’s dual. If is finite dimensional, then , but not in any “natural” way. Moreover, even if is infinite dimensional, embeds in , but again, not in a “natural” way. That is, we have to pick a basis to show how embeds in . If we pick a different basis, we get a different embedding. We don’t have any method that is inherent to the structure. And it’s not just that we haven’t found one. They don’t exist.
However, does embed inside naturally (i.e., in a way independent of the structure of . How are we to describe this? We say that there is a natural transformation between the functors and . I’m suppressing the field over which we’re working in the notation. Oh well. It’s not too important.
So here goes. Let by . By what I mean is the map that takes in a functional from and evaluates it at . That is . Since , and is a map from to the underlying field, is an element of . Does this make
If we go across the top and then down, we take and send it to
If we go down and then across, we take and send it to
Woohoo! We just proved naturality. In fact, we did two things. First, we showed that we could embed in , and we didn’t have to make any choices about elements to do so. This embedding is canonical. Second, we showed that linear transformations between and correspond to linear transformations between and in a nice way (such that the diagram commutes).