Subject: categories: Re: Categories ridiculously abstract

From: Tom Leinster <[email-masked]>
Date: Thu, 30 Nov 2000 17:30:30 +0000 (GMT)

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