Learning Category Theory, cont

A very important aspect of category theory is that it lets us talk very generally about structures and "structure preserving maps" between those structures. The structures are the objects of the category and the morphisms the maps between the structures. In the category Set, objects as sets morphisms as function, there is no extra structure to preserve in some sense, as we often refer to many mathematical objects as "sets with additional structure". Morphisms will then often be whatever notion of homomorphism is appropriate in the theory we are designing the category for. Notice, however, that categories are not always about theories- they are just points with lines between them, and they can describe all sorts of interesting things.

An object with additional structure to preserve would be groups. The category called Grp has groups as its objects, and group homomorphisms as its morphisms. The addition structure here is the identity, inverses, and the action of the operator. The cool thing about CT is that it can study very general situations that occur when we talk about structures and how to map between them. For example, it is the case that a lot of structures have some basic, trivial example that has only the minimal content to satisfy the axioms of whatever structure we are talking about. In the theory of groups this would be the one element group, containing in its underlying set only the identity of its operator. CT satisfies our desire to generalize this situation to all sorts of different structures, and shows us some very high level similarities between structures of different types when their categories turn out to have similar properties. The reason groups have a trivial example is that they are "pointed sets", sets with a distinguished element. Other algebraic structures also have a "point" at their identity (assuming one operator they will have one identity), and the category of their structures as objects with their structure preserving maps will have a similar feature as Grp. This object is called the initial object and it can be used to show that the objects in the category have some sort of underlying structure that is exemplified by this initial object in the simplest way possible.

That explanation may have been confusing, but the point is that CT lets us talk about the similarities between seemingly different objects, and gives a way to talk rigorously about notions like the "smallest" or "simplest" or "best" of something.

One way to think of category theory is as being about morphisms more than being about objects. We can talk about sets without looking inside of them, or talk about groups without ever noticing that they are constructed from sets with additional structure. The properties of objects are revealed by arrows. This is like considering an object to be defined by how it can be used, or more like by its relationship with other objects. It may seem circular- how do we define objects as their relation to other objects if we only define those other objects by their relation with still other objects? It turns out not to be a problem, as we can examine arrows and find objects that must have certain properties in the category and those objects will tell us a lot about the category. In a (bounded) lattice we can find a top and bottom element, and even not knowing anything about the element itself, we know the top the "largest" object in the lattice and the bottom element the "smallest" in the sense of those words most appropriate to the lattice. Incidentally, a lattice forms a category, and there is a category of lattices with morphisms as functions that preserve the top and bottom element and the structure of the meets and joins.

I hope to describe a series of categories in the next post to give us some practice thinking in this manner. Hopefully I can then show some similarities between them and talk about how we characterize these similarities. Things will then get more interesting when we talk about an interesting category called Cat, the category of categories. Can you see from the definition of a category what additional structure must be preserved by the morphisms in Cat?