# nLab (-1)-groupoid

### Context

#### Higher category theory

higher category theory

# Contents

## Definition

A $(-1)$-groupoid or (-1)-type is a truth value, or equivalently an (−1)-truncated object in ∞Grpd. By excluded middle, this is either the empty groupoid (false) or the terminal groupoid (true, the point).

## Remarks

Compare the concept of 0-groupoid (a set) and (−2)-groupoid (which is trivial). The point of $(-1)$-groupoids 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 $(-1)$-groupoids; a $0$-category is also a set, and this set is the set of truth values: classically

$(-1)Grpd := \{\bot, \top\}$

Actually, since for other values of $n$, n-groupoids form not just an $(n+1)$-category but an $(n+1,1)$-category, we should expect the $0$-category of $(-1)$-groupoids to be a $(0,1)$-category, or $1$-poset. This simply means a poset, and indeed truth values do always form a poset, classically ($\bot \leq \top$).

If we equip the category of $(-1)$-groupoids with the monoidal structure of conjunction (the logical AND operation), then a groupoid enriched over this is a setoid, and a category enriched over it is a proset. Up to equivalence of categories, these are the same as a set (a $0$-groupoid) and a poset (a (0,1)-category); this fits the patterns of the periodic table.

See (−1)-category for more on this sort of negative thinking.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth value(0,1)-sheafmere proposition, h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheafh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheafh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheafh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on September 10, 2012 20:22:24 by Urs Schreiber (131.174.188.17)