negative thinking

**higher category theory**
* category theory
* homotopy theory
## Basic concepts
* k-morphism, coherence
* looping and delooping
* looping and suspension
## Basic theorems
* homotopy hypothesis-theorem
* delooping hypothesis-theorem
* periodic table
* stabilization hypothesis-theorem
* exactness hypothesis
* holographic principle
## Applications
* applications of (higher) category theory
* higher category theory and physics
## Models
* (n,r)-category
* Theta-space
* ∞-category/ω-category
* (∞,n)-category
* n-fold complete Segal space
* (∞,2)-category
* (∞,1)-category
* quasi-category
* algebraic quasi-category
* simplicially enriched category
* complete Segal space
* model category
* (∞,0)-category/∞-groupoid
* Kan complex
* algebraic Kan complex
* simplicial T-complex
* n-category = (n,n)-category
* 2-category, (2,1)-category
* 1-category
* 0-category
* (−1)-category
* (−2)-category
* n-poset = (n-1,n)-category
* poset = (0,1)-category
* 2-poset = (1,2)-category
* n-groupoid = (n,0)-category
* 2-groupoid, 3-groupoid
* categorification/decategorification
* geometric definition of higher category
* Kan complex
* quasi-category
* simplicial model for weak ω-categories
* complicial set
* weak complicial set
* algebraic definition of higher category
* bicategory
* bigroupoid
* tricategory
* tetracategory
* strict ω-category
* Batanin ω-category
* Trimble ω-category
* Grothendieck-Maltsiniotis ∞-categories
* stable homotopy theory
* symmetric monoidal category
* symmetric monoidal (∞,1)-category
* stable (∞,1)-category
* dg-category
* A-∞ category
* triangulated category
## Morphisms
* k-morphism
* 2-morphism
* transfor
* natural transformation
* modification
## Functors
* functor
* 2-functor
* pseudofunctor
* lax functor
* (∞,1)-functor
## Universal constructions
* 2-limit
* (∞,1)-adjunction
* (∞,1)-Kan extension
* (∞,1)-limit
* (∞,1)-Grothendieck construction
## Extra properties and structure
* cosmic cube
* k-tuply monoidal n-category
* strict ∞-category, strict ∞-groupoid
* stable (∞,1)-category
* (∞,1)-topos
## 1-categorical presentations
* homotopical category
* model category theory
* enriched category 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

- (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

Revised on March 19, 2011 04:27:54
by Toby Bartels
(98.23.132.28)