nLab (-1)-groupoid

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 symmetric proset, 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.

Related concepts

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid