Michael Barr wrote:
And here is a question: are categories more abstract or less abstract than sets?
A higher-dimensional category theorist's answer: "Neither - a set is merely a 0-category, and a category a 1-category." There's a more serious thought behind this. Sometimes I've wondered, in a vague way, whether the much-discussed hierarchy 0-categories (sets) form a (1-)category, (1-)categories form a 2-category, ... has a role to play in foundations. After all, set-theorists seek to found mathematics on the theory of 0-categories; category-theorists sometimes talk about founding mathematics on the theory of 1-categories and providing a (Lawverian) axiomatization of the 1-category of 0-categories; you might ask "what next"? Axiomatize the 2-category of (1-)categories? Or the (n+1)-category of n-categories? Could it even be, I ask with tongue in cheek and head in clouds, that general n-categories provide a more natural foundation than either 0-categories or 1-categories? Tom