nLab
homotopy n-types - table

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		(-1)-groupoid/truth value		h-proposition
h-level 2	0-truncated	discrete space	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		h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid		h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid		h-$n$-groupoid
h-level $\mathrm{\infty }$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/∞-stack	h-$\mathrm{\infty }$-groupoid