nLab (-2)-groupoid



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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

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


Compare the concepts of (1)(-1)-groupoid (a truth value) and 00-groupoid (a set). Compare also with (2)(-2)-category and (1)(-1)-poset, which mean the same thing for their own reasons.

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

As a category, (2)Grpd(-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)(-1)-groupoid, which is a truth value; and indeed, a groupoid enriched over (2)Grpd(-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)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere 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)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

Last revised on February 1, 2020 at 23:47:56. See the history of this page for a list of all contributions to it.