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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)-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+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheafh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

Revised on September 10, 2012 20:17:09 by Urs Schreiber (