Learning Category Theory, natural transformations

One of the nice things about category theory is that is abstracts a very common pattern in math- we have some objects in our theory and some mappings between these objects. The morphisms in category theory are, if we are talking about the mappings between objects in our theory (categories are very general objects and can be used for all sorts of stuff), the appropriate notion of homomorphism in whatever theory we are concerned with.

Last time we saw functors as the notion of mapping between categories, but of course we can go much (much) further than that. If we take functors as themselves objects of a category (which is actually a nice thing to do sometimes) than we need some notion of morphism between functors. This is whats called a natural transformation.

So- what data specifies a natural transformation? Recall the definition of a functor- a functor F:C->D gives, for each object c in C, an object d in D, and for each morphism f in C a morphism g in D satisfying the diagrams

 c----->Fc
 |       |
f|     Ff|
 |       |
 v       v
 c`----->Fc`

A natural transformation T must be between two functors (which the same source and target) F, G:C -> D, and gives for each object c in C a morphism f:Fc -> Gc in D such that the following diagram commutes:

     Tc
  Fc--->Gc
  |      |
Ff|      |Gf
  |      |
  v  Td  v
  Fd---->Gd

The idea may be difficult at first- especially if we are not comfortable with the commutativity of diagrams, but the idea is so pervasive in category theory that it is in some sense obvious (at least conceptually)- a natural transformation must "preserve additional structure" of its objects. Since its objects are functors it must preserve the structure of the action of the functors (which in turn preserve the structure of a category (composition and identity)).

Natural transformations are one of the basic parts of category theory. Some categories turn out to be isomorphic to categories of functors, so the morphism of such categories (direct graphs for example) can be understood as natural transformations.

That's all I'm going to say about that for now. Next time: products and coproducts?