nLab sequential spectrum 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


Stable Homotopy theory



This entry is about the formulation of sequential spectra in homotopy type theory, specifically of sequential spectra that need not be genuine stable homotopy types in the sense of Omega-spectra and thus may be regarded as “pre-spectra” (they become genuine spectra under spectrification).


In homotopy type theory, a prespectrum or sequential spectrum is a dependent bi-infinite sequence of types n:E ntypen:\mathbb{Z} \vdash E_n\;\mathrm{type} with a dependent bi-infinite sequence of types n:* n:E nn \colon \mathbb{Z} \vdash *_n \colon E_n and a dependent bi-infinite sequence of functions n:e n:E nΩE n+1n \colon \mathbb{Z} \vdash e_n \colon E_n \to \Omega E_{n+1} which preserves the point, where ΩA\Omega A is the loop space type of AA.

In Coq pseudocode, this becomes

Definition prespectrum :=
  {X \colon int -> Type & 
   { pt \colon forall n, X n &
    { glue \colon forall n, X n -> pt (S n) == pt (S n) &
      forall n, glue n (pt n) == idpath (pt (S n)) }}}.

See also


Last revised on June 8, 2022 at 23:37:04. See the history of this page for a list of all contributions to it.