nLab
(-2)-groupoid

Definition

A $(-2)$-groupoid or (-2)-type is a (−2)-truncated object in ∞Grpd.

There is, up to equivalence, just one $(-2)$-groupoid, namely the point.

Remarks

Compare the concepts of $(-1)$-groupoid (a truth value) and $0$-groupoid (a set). Compare also with $(-2)$-category and $(-1)$-poset, which mean the same thing for their own reasons.

The point of $(-2)$-groupoids is that they complete some patterns in the periodic tables and complete the general concept of $n$-groupoid. For example, there should be a $(-1)$-groupoid $(-2)\Grpd$ of $(-2)$-groupoids; a $(-1)$-groupoid is simply a truth value, and $(-2)\Grpd$ is the true truth value.

As a category, $(-2)\Grpd$ is a monoidal category in a unique way, and a groupoid enriched over this should be (at least up to equivalence) a $(-1)$-groupoid, which is a truth value; and indeed, a groupoid enriched over $(-2)\Grpd$ is a groupoid in which any two objects are isomorphic in a unique way, which is equivalent to a truth value.

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

Related concepts

• true

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