nLab (-1)-category

Idea

As a degenerate case of the general notion of n-category, $(-1)$-categories may be understood as truth value. Compare the concept of 0-category (a set) and (−2)-category (which is trivial). The point of $(-1)$-categories (a kind of negative thinking) 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 $0$-category of $(-1)$-categories; this is the set of truth values, classically

$$(-1)Cat := \{true, false\} \,.$$

Similarly, $(-2)$-categories form a $(-1)$-category (specifically, the true one).

If we equip the category of $(-1)$-categories with the monoidal structure of conjunction (the logical AND operation), then a category enriched over this is a poset; an enriched groupoid is a set. Notice that this doesn't fit the proper patterns of the periodic table; we see that $(-1)$-categories work better as either $0$-posets or as $(-1)$-groupoids. Nevertheless, there is no better alternative for the term ‘$(-1)$-category’.

For an introduction to $(-1)$-categories and $(-2)$-categories see page 11 and page 34 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.