nLab (-2)-category

There is just one $(-2)$-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 $(-2)$-categories is that they complete some patterns in the periodic table of $n$-categories. (They also shed light on the theory of homotopy groups and n-stuff.)

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

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

• John C. Baez, Michael Shulman, Lectures on n-Categories and Cohomology (arXiv).

$(-1)$-categories and $(-2)$-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:

• John Baez, Toby Bartels, David Corfield and James Dolan, Property, structure and stuff. See also stuff, structure, property for more on that material.