(-1)-category

**category theory**
## Concepts
* category
* functor
* natural transformation
* Cat
## Universal constructions
* universal construction
* representable functor
* adjoint functor
* limit/colimit
* weighted limit
* end/coend
* Kan extension
## Theorems
* Yoneda lemma
* Isbell duality
* Grothendieck construction
* adjoint functor theorem
* monadicity theorem
* adjoint lifting theorem
* Tannaka duality
* Gabriel-Ulmer duality
* small object argument
* Freyd-Mitchell embedding theorem
* relation between type theory and category theory
## Extensions
* sheaf and topos theory
* enriched category theory
* higher category theory
## Applications
* applications of (higher) category theory
***
**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

A *$(-1)$-category* is a truth value. Compare the concept of 0-category (a set) and (−2)-category (which is trivial). The point of $(-1)$-categories (a kind of negative thinking) is that they complete some patterns in the periodic table of $n$-categories. (They also shed light on the theory of homotopy groups and n-stuff.)

For example, there should be a $0$-category of $(-1)$-categories; this is the set of truth values, classically

$(-1)Cat := \{true, false\}
\,.$

Similarly, $(-2)$-categories form a $(-1)$-category (specifically, the true one).

If we equip the category of $(-1)$-categories with the monoidal structure of conjunction (the logical AND operation), then a category enriched over this is a poset; an enriched groupoid is a set. Notice that this doesn't fit the proper patterns of the periodic table; we see that $(-1)$-categories work better as either $0$-posets or as $(-1)$-groupoids. Nevertheless, there is no better alternative for the term ‘$(-1)$-category’.

For an introduction to $(-1)$-categories and $(-2)$-categories see page 11 and page 34 of

- John C. Baez, Michael Shulman,
*Lectures on n-Categories and Cohomology*(arXiv).

$(-1)$-categories and $(-2)$-categories were discovered (or invented) by James Dolan and Toby Bartels. To witness these concepts in the process of being discovered, read the discussion here:

- John Baez, Toby Bartels, David Corfield and James Dolan, Property, structure and stuff. See also stuff, structure, property for more on that material.

Revised on June 30, 2010 22:04:53
by Toby Bartels
(75.88.78.90)