nLab stable homotopy type



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


Stable Homotopy theory

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




A pointed homotopy type X *X_\ast could be called linear (following the terminology of linear model category) if the canonical morphism

ΣΩXX \Sigma \Omega X \longrightarrow X

from the suspension object of its loop space object is an equivalence.

By Goodwillie calculus the tangent (∞,1)-topos THT\mathbf{H} of some (∞,1)-topos H\mathbf{H} is the localization of the classifying (∞,1)-topos H[X *]\mathbf{H}[X_\ast] (of pointed objects), that regard each pointed finite powering X * X_\ast^\bullet of the generic pointed object X *X_\ast as a linear homotopy type.

See at excisive functor – Characterization via the generic pointed object.

Therefore a pointed homotopy type X *X_\ast such that all its finite pointed powers are linear could be called a stable homotopy type.

For H\mathbf{H} an (∞,1)-topos and THT \mathbf{H} its tangent (∞,1)-topos, then all homotopy types in T *HTHT_\ast \mathbf{H} \hookrightarrow T \mathbf{H} are stable in this sense, these are equivalently the spectrum objects in H\mathbf{H}.


An account using modal homotopy type theory is in

Last revised on June 8, 2022 at 02:21:11. See the history of this page for a list of all contributions to it.