In "Towards a categorical foundation of mathematics" (Logic Colloquium '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic no.11, 1998; pp.153-190) and in subsequent work, I am proposing an approach to a foundation whose universe consists of the weak n-categories and whatever things are needed to connect them. This is done on the basis of a general point of view concerning the role of identity of mathematical objects. Readers of said paper who have followed developments on weak higher dimensional categories will note that much has been done since towards fleshing out the program. Michael Makkai On Thu, 30 Nov 2000, 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? 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