nLab
(-2)-category

There is just one (2)(-2)-category, namely the truth value True. Compare the concepts of (−1)-category (a truth value in general) and 0-category (a set). The point of (2)(-2)-categories is that they complete some patterns in the periodic table of nn-categories. (They also shed light on the theory of homotopy groups and n-stuff.)

For example, there should be a (1)(-1)-category of (2)(-2)-categories; this is the true truth value. The category of (2)(-2)-categories is a monoidal category in a unique way; then a category enriched over this is a (1)(-1)-category; such is necessarily an enriched groupoid. If you think of a (1)(-1)-category as a 0-poset, then this makes a (2)(-2)-category a (−1)-poset. If you think of a (1)(-1)-category as a (−1)-groupoid, then this makes a (2)(-2)-category a (−2)-groupoid.

For an introduction to (1)(-1)-categories and (2)(-2)-categories see page 11 of

(1)(-1)-categories and (2)(-2)-categories were discovered (or invented) by James Dolan and Toby Bartels. To witness these concepts in the process of being discovered, read the discussion here:

Revised on June 30, 2010 22:07:26 by Toby Bartels (75.88.78.90)