Here is a rather pleasant introduction to the philosophical point of view underlying Category Theory.
When is one thing equal to some other thing-
One of the templates of modern mathematics, category theory, offers its own formulation of equivalence as opposed to equality; the spirit of category theory allows us to be content to determine a mathematical object, as one says in the language of that theory, up to canonical isomorphism. The categorical viewpoint is, however, more than merely “content” with the inevitability that any particular mathematical object tends to come to us along with the contingent scaffolding of the specific way in which it is presented to us, but has this inevitability built in to its very vocabulary,and in an elegant way, makes profound use of this. It will allow itself the further flexibility of viewing any mathematical object “as” a representation of the theory in which the object is contained to the proto-theory of modern mathematics, namely,to set theory.
I have been spending some amount of time looking into Category theory and it is truly something elegant. A small selection of topics from Category theory have made their way into my reading list for my Oral Qualifiers. Hence, interested reinforced by need.
One of the strange things about category theory, and probably the most elegant thing about it as well is that category theory really has nothing much to do with objects in the way that Set theory does. Set theory and I think most things deal with objects, collections of objects and such. Category theory on the other hand is a theory about relationships, rather than the objects they relate. This shift, to me, is reminiscent of the Leibniz-Clarke viewpoints on the notion of space.