nLab
(-2)-groupoid

Definition

A $(-2)$-groupoid 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.