Subject: categories: (-2)-categories

From: [email-masked] (Robert Pare)
Date: Tue, 19 Dec 2000 11:24:17 -0400 (AST)

Here are some thoughts I had on (-2)-categories. When I read John Baez's
posting I thought I had missed the boat, but I guess the point is that
strict n-categories are not the same as weak n-categories (surprise!).

A *strong* n-category (henceforth called n-category) is a category enriched
in (n-1)-Cat, the cartesian closed *category* of small (n-1)-categories.
In the time honoured way, start with the empty category. It's cartesian closed
(it doesn't have a terminal object but let's glance over that). A category 
enriched in it can have no objects, for where would their homs land. So
there is only one, the empty category itself. So the category of small
categories enriched in the empty category is 1, the category with one
morphism, which is cartesian closed of course. A category enriched in 1
is just a class of objects, and a small one is just a set. So now the
category of small categories is just Set, and we are on our way. So 
what's the point? A set is a 0-category, there is only one (-1)-category,
the empty category, and there are no (-2)-categories. There *is* a notion
of (-2)-category, viz. an object of the empty category, but there aren't many
of those! I think it should be clear from this that there is not even a 
notion of (-3)-category so it doesn't even make sense to say there are none.

Bob Pare