nLab homotopy level



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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+2n+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.


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

hasHLevel(0,A)isContr(A)hasHLevel(0, A) \coloneqq isContr(A)
hasHLevel(s(n),A) a:A b:AhasHLevel(n,a= Ab)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 hasHLevel(n,A)\mathrm{hasHLevel}(n, A) for all types AA in the context of the natural numbers variable n:Nn:\mathrm{N}, with the following rules:

Formation rules for hasHLevel types:

ΓAtypeΓ,x:AContr A(x)typeΓhasHLevel(0,A)type\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}}
ΓAtypeΓ,n:,x:A,y:AhasHLevel(n,x= Ay)typeΓ,n:hasHLevel(s(n),A)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:

Γ,x:Ab(x):Contr A(x)Γa:AΓb:Contr A[a/x]Γ,n:(a,b):hasHLevel(0,A)\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)}
Γ,n:,x:A,y:Ap(n,x,y):hasHLevel(n,x= Ay)Γ,n:λ(x).λ(y).p(n,x,y):hasHLevel(s(n),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:

Γz:hasHLevel(0,A)Γπ 1(z):AΓz:hasHLevel(0,A)Γπ 2(z):Contr A(π 1(z))\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))}
Γ,n:p:hasHLevel(s(n),A)Γa:AΓb:AΓ,n:p(n,a,b):hasHLevel(n,a= Ab)\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:

Γ,x:Ab(x):Contr A(x)Γa:AΓβ hasHLevel1 0:π 1(a,b)= AaΓ,x:Ab(x):Contr A(x)Γa:AΓβ hasHLevel2 0:π 2(a,b)= Contr A(a)b\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}
Γ,n:N,x:A,y:Ap(n,x,y):hasHLevel(n,x= Ay)Γa:AΓb:AΓ,n:Nβ hasHLevel s:(λ(x).λ(y).p(n,x,y))(a,b)= hasHLevel(n,a= Ab)p(n,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:

Γz:hasHLevel(0,A)Γη hasHLevel 0:z= hasHLevel(0,A)(π 1(z),π 2(z))\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))}
Γ,n:p:hasHLevel(s(n),A)Γ,n:η hasHLevel s:p(n)= hasHLevel(s(n),A)λ(x).λ(y).p(n,x,y)\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)}


  • Every proposition has a homotopy level of 11.

  • Every set has a homotopy level of 22.

The correspondence between the various terminologies is indicated in the following table:

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+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid


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

Further discussion:

Discussion with Agda:

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

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]

Last revised on February 18, 2023 at 11:12:14. See the history of this page for a list of all contributions to it.