(-1)-category

- 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
- 2-category, (2,1)-category
- 1-category
- 0-category
- (?1)-category?
- (?2)-category?

- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- Kan complex
- quasi-category
- simplicial model for weak ∞-categories?

- algebraic definition of higher category
- stable homotopy 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.

Last revised on November 16, 2016 at 08:03:38. See the history of this page for a list of all contributions to it.