Tom Leinster wrote:
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?
Surely we should start with the set of (-1)-categories? <gd&r> -Robert Dawson