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.

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

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

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

Rules for hasHLevel

Suppose the dependent type theory has identity types, contraction types, and a natural numbers type, but no dependent product types or dependent sum types. Then one could still define the type family $\mathrm{hasHLevel}(n, A)$ for all types $A$ in the context of the natural numbers variable $n:\mathrm{N}$, with the following rules:

Formation rules for hasHLevel types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash \mathrm{Contr}_A(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{hasHLevel}(0, A) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N}, x:A, y:A \vdash \mathrm{hasHLevel}(n, x =_A y) \; \mathrm{type}}{\Gamma, n:\mathbb{N} \vdash \mathrm{hasHLevel}(s(n), A) \; \mathrm{type}}$$

Introduction rules for hasHLevel types:

$$\frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:\mathrm{Contr}_A[a/x]}{\Gamma, n:\mathbb{N} \vdash (a, b):\mathrm{hasHLevel}(0, A)}$$

$$\frac{\Gamma, n:\mathbb{N}, x:A, y:A \vdash p(n, x, y):\mathrm{hasHLevel}(n, x =_A y)}{\Gamma, n:\mathbb{N} \vdash \lambda(x).\lambda(y).p(n, x, y):\mathrm{hasHLevel}(s(n), A)}$$

Elimination rules for hasHLevel types:

$$\frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \pi_2(z):\mathrm{Contr}_A(\pi_1(z))}$$

$$\frac{\Gamma, n:\mathbb{N} \vdash p:\mathrm{hasHLevel}(s(n), A) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma, n:\mathbb{N} \vdash p(n, a, b):\mathrm{hasHLevel}(n, a =_A b)}$$

Computation rules for hasHLevel types:

$$\frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{\mathrm{hasHLevel} 1}^0:\pi_1(a, b) =_A a} \qquad \frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{\mathrm{hasHLevel} 2}^0:\pi_2(a, b) =_{\mathrm{Contr}_A(a)} b}$$

$$\frac{\Gamma, n:\mathrm{N}, x:A, y:A \vdash p(n, x, y):\mathrm{hasHLevel}(n, x =_A y) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma, n:\mathrm{N} \vdash \beta_{\mathrm{hasHLevel}}^s:(\lambda(x).\lambda(y).p(n, x, y))(a, b) =_{\mathrm{hasHLevel}(n, a =_A b)} p(n, a, b)}$$

Uniqueness rules for hasHLevel types:

$$\frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \eta_{\mathrm{hasHLevel}}^0:z =_{\mathrm{hasHLevel}(0, A)} (\pi_1(z), \pi_2(z))}$$

$$\frac{\Gamma, n:\mathbb{N} \vdash p:\mathrm{hasHLevel}(s(n), A)}{\Gamma, n:\mathbb{N} \vdash \eta_{\mathrm{hasHLevel}}^s:p(n) =_{\mathrm{hasHLevel}(s(n), A)} \lambda(x).\lambda(y).p(n, x, y)}$$

Examples

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

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

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

Referencees

The notion of “homotopy levels” (as a notion in type theory) originates with:

• Vladimir Voevodsky, Univalent Foundations Project (2010) [pdf, pdf]

Further discussion:

• Guillaume Brunerie, Truncations and truncated higher inductive types (2012) [blog post]

Discussion with Agda:

• 1lab, h-Levels

Synonymous discussion, but under the name $n$-types, is in

• Univalent Foundations Project, §7 of: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

• Nicolai Kraus, Truncation levels in Homotopy Type Theory, Nottingham (2015) [pdf, eprints:28986]

See also:

• Thierry Coquand, Ayberk Tosun, Formal Topology in Univalent Foundations, Ch. 6 in: Proof and Computation II – From Proof Theory and Univalent Mathematics to Program Extraction and Verification, Wold Scientific (2021) [doi:10.1142/12263]