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 $1$-groupoid is an $\infty$-groupoid such that all parallel pairs of $j$-morphisms are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent. If you rephrase equivalence of $1$-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 $1$-groupoid is simply a groupoid.

The point of all this is simply to fill in the general concept of $n$-groupoid; nobody thinks of $1$-groupoids as a concept in their own right except simply as groupoids. Compare $1$-category and $1$-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)-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+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid

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