nLab negative thinking

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 $2$-foos, $3$-foos, and so on, you think about how foos are themselves a categorification of $0$-foos, $(-1)$-foos, and so on. Generally, the concept of $n$-foo stops making sense for small values of $n$ 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 $(-1)$-sphere and the $(-1)$-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 $n$-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

• (0,1)-category
• 0-category
• (-1)-category
• (-2)-category
• 0-poset
• (-1)-poset
• 0-groupoid
• (-1)-groupoid
• (-2)-groupoid
• 0-functor
• (-1)-functor
• (0,1)-topos
• (0,1)-site