nLab
1-groupoid

Context

Homotopy theory

Category theory

Contents

Idea

By 1-groupoid one means – for emphasis – a groupoid regarded as an ∞-groupoid.

A quick way to make this precise is to says that a 1-groupoid is a Kan complex which is equivalent (homotopy equivalent) to the nerve of a groupoid: a 2-coskeletal Kan complex. More abstractly this is a 1-truncated ∞-groupoid.

More generally and more vaguely: Fix a meaning of \infty-groupoid, however weak or strict you wish. Then a 11-groupoid is an \infty-groupoid such that all parallel pairs of jj-morphisms are equivalent for j2j \geq 2. Thus, up to equivalence, there is no point in mentioning anything beyond 11-morphisms, except whether two given parallel 11-morphisms are equivalent. If you rephrase equivalence of 11-morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a groupoid. Thus one may also say that a 11-groupoid is simply a groupoid.

The point of all this is simply to fill in the general concept of nn-groupoid; nobody thinks of 11-groupoids as a concept in their own right except simply as groupoids. Compare 11-category and 11-poset, which are defined on the same basis.

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 valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

Revised on April 29, 2013 21:03:23 by Urs Schreiber (89.204.138.79)