nLab suspension type



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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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




The suspension type is an axiomatization of the suspension object in the context of homotopy type theory.


As a higher inductive type, the suspension ΣA\Sigma A of a type AA is given by

  • A point N:ΣA\mathrm{N} : \Sigma A
  • A point S:ΣA\mathrm{S} : \Sigma A
  • A function merid:A(N= ΣAS)merid : A \to (\mathrm{N} =_{\Sigma A} \mathrm{S})

In Coq pseudocode it becomes

Inductive Suspension (A : Type) : Type
  | north : Suspension A
  | south : Suspension A
  | meridian : A -> Id Suspension A north south

This says that the type is dependent on the type A and inductive constructed from two terms in the suspension, whose interpretation is as the north and south poles of the suspension, together with a term in the function type from A to the identity type of paths between these two terms, representing the meridians from the north to the south pole.


  • The two-valued type 2\mathbf{2} is the suspension type of the empty type 0\mathbf{0}.

  • The interval type II is the suspension type of the unit type 1\mathbf{1}.

  • The circle type S 1S^1 is the suspension type of 2\mathbf{2}.

  • The homotopical disk type G 2G_2 is the suspension type of II.

  • Homotopical nn-sphere types of dimension n:n:\mathbb{N}, S nS^n, are suspension types of S n1S^{n-1}.

  • Homotopical nn-globe types of dimension n:n:\mathbb{N}, G nG_n, are suspension types of G n1G_{n-1}.

  • In general, the suspension type ΣA\Sigma A of a type AA is the homotopy pushout of the span 1A1\mathbf{1} \leftarrow A \rightarrow \mathbf{1}.


Last revised on June 7, 2022 at 22:13:16. See the history of this page for a list of all contributions to it.