- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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

Last revised on July 22, 2014 at 21:11:01. See the history of this page for a list of all contributions to it.