nLab
0-groupoid

Contents

Idea

A 0-groupoid or 0-type is a set. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, groupoids, 2-groupoids, 3-groupoids, etc. Doing so reveals patterns such as the periodic table. (It also sheds light on the theory of homotopy groups and n-stuff.)

For example, there should be a 11-groupoid of 00-groupoids; this is the underlying groupoid of the category of sets. Then a groupoid enriched over this is a 11-groupoid (more precisely, a locally small groupoid). Furthermore, an enriched category is a category (or 11-category), so a 00-groupoid is the same as a 0-category.

One can continue to define a (−1)-groupoid to be a truth value and a (−2)-groupoid to be a triviality (that is, there is exactly one).

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 July 6, 2015 20:50:01 by Mateo Carmona? (200.116.21.208)