nLab homotopy level

Idea

The term homotopy level (or h-level), originating in homotopy type theory, is another name for the notion of truncation (particularly in (∞,1)-categories and their internal language of homotopy type theory) in which the numbering is offset by 2:

a homotopy n-type is a type of homotopy level $n+2$.

This offset in counting enables it to “start” at 0 rather than (-2), which is convenient when defining it by induction over the natural numbers in type theory. Thus, the correspondence between the various terminologies is indicated in the following 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	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

Definition

A type $A$ has a homotopy level or h-level of $n$ if the type $hasHLevel(n, A)$ is inhabited, for natural number $n:\mathbb{N}$. $hasHLevel(n, A)$ is inductively defined as

$$hasHLevel(0, A) \coloneqq isContr(A)$$

$$hasHLevel(s(n), A) \coloneqq \prod_{a:A} \prod_{b:A} hasHLevel(n, a =_A b)$$

Examples

• Every proposition has a homotopy level of $1$.

• Every set has a homotopy level of $2$.

Referencees

• Ayberk Tosun, Formal Topology in Univalent Foundations, (pdf)