homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
Negative thinking is a way of thinking about categorification by considering what the original concept is a categorification of. That is, to better understand how foos are categorified to become -foos, -foos, and so on, you think about how foos are themselves a categorification of -foos, -foos, and so on. Generally, the concept of -foo stops making sense for small values of after a few steps, but it does make sense surprisingly often for at least some non-positive values. Experienced negative thinkers can compete to see ‘how low can you go’.
More generally, negative thinking can apply whenever you have a sequence of mathematical objects and ask yourself what came before the beginning? Examples outside category theory include the -sphere and the -simplex (which are both empty), although maybe it means something that these are both from homotopy theory. Tim Gowers has called this ’generalizing backwards’.
For low values of -category, see Section 2 of Lectures on n-Categories and Cohomology. Related issues appear at category theory vs order theory. See also nearly any page here with ‘0’ or ‘(-1)’ in the title, such as