negative thinking

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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 22-foos, 33-foos, and so on, you think about how foos are themselves a categorification of 00-foos, (1)(-1)-foos, and so on. Generally, the concept of nn-foo stops making sense for small values of nn 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)(-1)-sphere and the (1)(-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 nn-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

Revised on July 22, 2014 21:11:01 by Anonymous Coward (