# nLab (-2)-groupoid

### Context

#### Higher category theory

higher category theory

# Contents

## 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.

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 20:17:09 by Urs Schreiber (131.174.188.17)