nLab 1-groupoid

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.

Related concepts

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid