Good Math, Bad Math

The thing that I think is most interesting about category theory is that what it's really fundamentally about is structure. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate structures in a simple visual way. Morphisms express the structure of a category; functors are higher level morphisms that express the structure of relationships between categories.

In my last category theory post, one of the things I mentioned was how category theory lets you explain the idea of symmetry and group actions - which are a kind of structural immunity to transformation, and a definition of transformation - in a much simpler way than it could be talked about without categories.

It turns out that symmetry transformations are just the tip of the iceberg of the kinds of structural things we can talk about using categories. In fact, as I alluded to in my last post, if we create a category of categories, we end up with functors as arrows between categories.

What happens if we take the same kind of thing that we did to get group actions, and we pull out a level, so that instead of looking at the category of categories, focusing on arrows from the specific category of a group to the category of sets, we do it with arrows between members of the category of functors?

We get the general concept of a natural transformation. A natural transformation is a morphism from functor to functor, which preserves the full structure of morphism composition within the categories mapped by the functors.

Suppose we have two categories, C and D. And suppose we also have two functors, F, G : C → D. A natural transformation from F to G, which we'll call η maps every object x in C to an arrow η_x : F(x) → G(x). η_x has the property that for every arrow a : x → y in C, η_y º F(a) = G(a) º η_x. If this is true, we call η_x the component of η for (or at) x.

That paragraph is a bit of a whopper to interpret. Fortunately, we can draw a diagram to help illustrate what that means. The following diagram commutes if η has the property described in that paragraph.

I think this is one of the places where the diagrams really help. We're talking about a relatively straightforward property here, but it's very confusing to write about in equational form. But given the commutative diagram, you can see that it's not so hard: the path η_y º F(a) and the path G(a) º η compose to the same thing: that is, the transformation η hasn't changed the structure expressed by the morphisms.

And that's precisely the point of the natural transformation: it's a way of showing the relationships between different descriptions of structures - just the next step up the ladder. The basic morphisms of a category express the structure of the category; functors express the structure of relationships between categories; and natural transformations express the structure of relationships between relationships.

Coming soon: a few examples of natural transformation, and then... Yoneda's lemma. Yoneda's lemma takes the idea we mentioned before of a group being representable by a category with one object, and generalizes all the way up from the level of a single category to the level of natural transformations.