nLab n-truncation modality



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


Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



For all n{2,1,0,1,2,3,}n \in \{-2, -1, 0,1,2,3, \cdots\}, n n -truncation is a modality (in homotopy type theory).


The following applies to dependent type theory with suspension types and localizations?, such as to standard homotopy type theory.



( n n -sphere type S nS^n)

As nullification of the n+1n+1-sphere

The nn-truncation modality may also be defined to be the localization at the unique function from the (n+1)(n + 1)-dimensional sphere type (Def. ) to the unit type S n+1𝟙S^{n + 1} \to \mathbb{1}

[A] nL S n+1(A)\left[A\right]_n \coloneqq L_{S^{n + 1}}(A)

By definition, the type of functions (𝟙[A] n)(S n+1[A] n)(\mathbb{1} \to \left[A\right]_n) \to (S^{n + 1} \to \left[A\right]_n) is an equivalence of types.

Explicit inference rules

The following are the inference rules for the nn-truncation [X] n[X]_n of a given X:TypeX \,\colon\, Type regarded as a higher inductive type according to UFP13, §7.3, p. 223 (the diagram indicates the categorical semantics, for orientation):

type formation rule:

X:Type| || |[X] n:Type \frac{ X \,\colon\, Type \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} [X]_n \,\colon\, Type }

term introduction rule:

x:X| || |η X(x):[X] nϕ:S n+1[X] n| || |hub X ϕ:[X] nϕ:S n+1[X] n;s:S n+1| || |hmt X ϕ(s):Id [X] n(ϕ(s),hub X ϕ) \frac{ x \,\colon\, X \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} \eta_X(x) \,\colon\, [X]_n } \;\;\;\;\;\;\;\;\; \frac{ \phi \,\colon\, S^{n+1} \to [X]_n \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} hub_X^{\phi} \,\colon\, [X]_n } \;\;\;\;\;\;\;\;\; \frac{ \phi \,\colon\, S^{n+1} \to [X]_n \;; \;\;\; s \,\colon\, S^{n+1} \mathclap{\phantom{\vert_{\vert}}} }{ \mathclap{\phantom{\vert^{\vert}}} hmt_X^{\phi}(s) \,\colon\, Id_{[X]_n}\big( \phi(s) ,\, hub_X^{\phi} \big) }

term elimination rules:

x¯:[X] nD(x¯):Type; x:Xη D(x):D(η X(x)); ϕ:S n+1[X] n,ϕ D:s:S n+1D(ϕ(s))hub D ϕ D:D(hub X ϕ); ϕ:S n+1[X] n,ϕ D:s:S n+1D(ϕ(s)),s:S n+1hmt D ϕ D(s):Id D(hub X ϕ)((hmt X ϕ(s)) *ϕ D(s),hub D ϕ D)| | || | |ind (D,η D,hub D,hmt D):x¯:[X] nD(x¯) \frac{ \begin{array}{l} \overline{x} \,\colon\, [X]_n \;\vdash\; D(\overline{x}) \,\colon\, Type \;; \\ x \,\colon\, X \;\vdash\; \eta_D(x) \,\colon\, D\big( \eta_X(x) \big) \;; \\ \phi \,\colon\, S^{n+1} \to [X]_n \,, \; \phi_D \,\colon\, \underset{s \,\colon\,S^{n+1}}{\prod} \, D\big( \phi(s) \big) \;\;\vdash\;\; hub^{\phi_D}_D \,\colon\, D\big( hub_X^\phi \big) \;; \\ \phi \,\colon\, S^{n+1} \to [X]_n \,, \; \phi_D \,\colon\, \underset{s \,\colon\,S^{n+1}}{\prod} \, D\big( \phi(s) \big) \,, \; s \,\colon\, S^{n+1} \;\;\vdash\;\; hmt_D^{\phi_D}(s) \,\colon\, Id_{D\big(hub_X^{\phi}\big)}\Big( \big(hmt^{\phi}_X(s)\big)_\ast \phi_D(s) ,\, hub_D^{\phi_D} \Big) \mathclap{\phantom{\vert^{\vert^{\vert}}}} \end{array} }{ \mathclap{\phantom{\vert^{\vert_{\vert}}}} ind_{\big(D,\,\eta_D,\, hub_D,\, hmt_D \big)} \,\colon\, \underset{\overline{x}\colon [X]_n}{\prod} \, D(\overline{x}) }

computation rule:

ind (D,η D,hub D,hmt D)(η(x))=η D(x) ind_{\big(D,\,\eta_D,\, hub_D,\, hmt_D \big)} \big( \eta(x) \big) \;=\; \eta_D(x)


In low degree

(2)(-2)-truncation is the unit type modality (constant on the unit type).

(1)(-1)-truncation is given by forming bracket types, turning types into genuine propositions.

Classically, this is the same as the double negation modality; in general, the bracket type A 1{\|A\|_{-1}} only entails the double negation ¬(¬A)\neg(\neg{A}):

there is a canonical function

A 1¬(¬A) {\|A\|_{-1}} \longrightarrow \neg(\neg{A})

and this is a 1-epimorphism precisely if the law of excluded middle holds.

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


Construction of nn-truncation as a one-step higher inductive type in homotopy type theory:

Alternative construction of nn-truncation as an iterated pushout type is (somewhat implicit) in:

Discussion of nn-truncation as a modality:

and in addition via lifting properties against n-spheres:

Earlier discussion (and in view of homotopy levels):

Precursor discussion of the material that became UFP (2013, §7.3):

and precursur discussion of the material that became RSS (2020):

Considering the combination of nn-truncation modality and shape modality:

Last revised on February 20, 2024 at 12:44:33. See the history of this page for a list of all contributions to it.