Examples of natural transformations (part 2)

Here’s the most common example of a natural transformation that I know of, and probably the most enlightening. Take a vector space V and let V^* denote it’s dual. If V is finite dimensional, then V\cong V^*, but not in any “natural” way. Moreover, even if V is infinite dimensional, V embeds in V^*, but again, not in a “natural” way. That is, we have to pick a basis to show how V embeds in V^*. 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, V does embed inside V^{**} naturally (i.e., in a way independent of the structure of V. How are we to describe this? We say that there is a natural transformation between the functors \mbox{id}_{\textsc{Vec}}:\textsc{Vec}\to\textsc{Vec} and -^{**}:\textsc{Vec}\to\textsc{Vec}. I’m suppressing the field over which we’re working in the notation. Oh well. It’s not too important.

So here goes. Let \eta_V:V\to V^{**} by \eta_V(v)=\mbox{ev}_v. By \mbox{ev}_v what I mean is the map that takes in a functional \phi from V^* and evaluates it at v. That is \mbox{ev}_v:\phi\mapsto \phi(v). Since \phi\in V^*, and \mbox{ev}_v is a map from V^* to the underlying field, \mbox{ev}_v is an element of V^{**}. Does this make


If we go across the top and then down, we take v\in V and send it to

T^{**}\circ\eta_V(v)=T^{**}\circ \mbox{ev}_v=T**(\phi\mapsto \phi(v))

=(\phi\mapsto (T^*\circ\phi)(v))=(\phi\mapsto\phi(Tv))=\mbox{ev}_{Tv}.

If we go down and then across, we take v\in V and send it to

(\eta_W\circ T)v=\mbox{ev}_{Tv}

Woohoo! We just proved naturality. In fact, we did two things. First, we showed that we could embed V in V^{**}, 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 V and W correspond to linear transformations between V^{**} and W^{**} in a nice way (such that the diagram commutes).


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