nLab Hopf construction in homotopy type 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


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




In classical algebraic topology we have four Hopf fibrations (of spheres):

  1. S 0S 1S 1S^0 \hookrightarrow S^1 \to S^1 The real Hopf fibration
  2. S 1S 3S 2S^1 \hookrightarrow S^3 \to S^2 The usual complex Hopf fibration
  3. S 3S 7S 4S^3 \hookrightarrow S^7 \to S^4 The quaternionic Hopf fibration
  4. S 7S 15S 8S^7 \hookrightarrow S^15 \to S^8 The octonionic Hopf fibration

These can be constructed in homotopy type theory as part of a more general construction:

An H-space structure on a pointed type AA gives a fibration over ΣA\Sigma A via the Hopf construction. This fibration can be written classically as: AA*AΣAA \to A\ast A \to \Sigma A where A*AA\ast A is the join of AA and AA. This is all done in the HoTT book. Note that ΣA\Sigma A can be written as a homotopy pushout ΣA1 A1\Sigma A \coloneqq \mathbf 1 \sqcup^A \mathbf 1 , and there is a lemma in the HoTT book allowing you to construct a fibration on a pushout (the equivalence AAA \to A needed is simply the multiplication from the H-space μ(a,)\mu(a,-)).

Thus the problem of constructing a Hopf fibration reduces to finding a H-space structure on the spheres: the S 1S^1, S 3S^3 and S 7S^7.

  • The space S 0=2(Bool)S^0=\mathbf 2(\equiv Bool) is not connected so we cannot perform the construction from the book on it. However it is very easy to construct a family S 1𝒰S^1 \to \mathcal{U} with fiber BoolBool by induction on S 1S^1. (Note: loop maps to ua(neg)ua(neg) where negneg is the equivalence of negation and uaua is the univalence axiom.

  • For S 1S^1 Peter Lumsdaine gave the construction in 2012 and Guillaume Brunerie proved it was correct in 2013. By induction on the circle we can define the multiplication: μ(base)id S 1\mu(base)\equiv id_{S^1}, and ap μ(loop)funext(h)ap_\mu(loop)\equiv funext(h) where h:(x:S 1)(x=x)h : (x : S^1) \to (x = x) is also defined by circle induction: h(base)=looph(base) = loop and ap h(loop)=reflap_h(loop) = refl. funextfunext denotes functional extensionality.

  • For S 3S^3 Buchholtz-Rijke 16 solved this through a homotopy theoretic version of the Cayley-Dickson construction. This has been formalised in Lean.

  • For S 7 S^7 this is still an open problem.

It is still an open problem to show that these are the only spheres to have a H-space structure. This would be done by showing these are the only spheres with Hopf invariant 11 which has been defined in Brunerie 2016


category: homotopy theory

Last revised on June 15, 2022 at 12:55:16. See the history of this page for a list of all contributions to it.