jack

JMTT: Jack Morava Type Theory

Encompasses unstable homotopy, stable homotopy (e.g., π₀^S(S⁰) = ℤ), and chromatic phenomena (e.g., H^*(RP^2), spectral sequences), inspired by Morava’s chromatic vision.

To enable cohomology computations in Hopf Fibrations Type Theory (HFTT) using spectra like Hℤ or Hℚ, we need to refine and extend the spectrum-related rules. Cohomology in chromatic homotopy theory often involves spectra (e.g., Eilenberg-MacLane spectra Hℤ) and their stable homotopy groups, which represent cohomology groups when applied to other spectra or spaces. Our current HTT setup has spectra, stable homotopy groups (πₙ^S), and K(G, n) spaces trough n-Truncations and Groups, but lacks explicit rules for cohomology operations—pairings, cochain complexes, or spectrum maps—that make computations practical. Jack Morava Type Theory adds these rules, focusing on cohomology as H^n(X; G) = [X, K(G, n)] or, in the stable setting, π₋ₙ^S(HG ∧ X).

> H^*(RP^2; ℤ/2ℤ) = ℤ/2ℤ[α]/(α³)

Syntax

HFTT: Hopf Fibrations Type Theory

A Minimal Framework for Homotopy Groups of Spheres.

We introduce Hopf Fibrations Type Theory (HFTT), a novel type system designed to efficiently represent and compute homotopy groups of spheres, addressing the computational challenges of Cubical Homotopy Type Theory (CCHM). Built on a minimal set of primitives—Hopf fibrations (ℝ, ℂ, ℍ, 𝕆), suspension, and n-truncation — HFTT leverages fibrations to encode topological structure directly. Alongside standard types (Π, Σ, Id), HFTT includes natural numbers (ℕ) and an extended order type (ℕ∞) to access group properties. Key innovations include eliminators for suspensions and truncations, enabling streamlined reductions, and a derivable order function that extracts the order of elements in πₙ(Sᵐ), supporting both finite (e.g., π₄(S³) ≅ ℤ/2ℤ) and infinite (e.g., π₃(S²) ≅ ℤ) groups. Computational rules ensure efficient normalization, while the fibrations provide a basis for homotopy groups, potentially simplifying proofs of properties like π₄(S³). This article outlines HFTT’s syntax, rules, and its promise as a compact, expressive framework for homotopy type theory.

> π₃(S²) = ℤ

Syntax

Inference Rules

Formations

Γ ⊢ Sⁿ : U := ℕ-iter U 𝟐 Susp
Γ ⊢ HopfFibⁿ : U (n ∈ {1, 2, 4, 8})
Γ ⊢ Susp(A) : U
Γ ⊢ Truncⁿ(A) : U
Γ ⊢ ℕ : U
Γ ⊢ ℕ∞ : U
Γ ⊢ Π(x:A).B : U
Γ ⊢ Σ(x:A).B : U
Γ ⊢ Id_A(u, v) : U

Γ ⊢ Spec : U
Γ ⊢ πₙ^S(A) : U
Γ ⊢ A ∧ B : Spec
Γ ⊢ [A, B] : Spec
Γ ⊢ Hⁿ(X; G) : U
Γ ⊢ G ⊗ H : Group
Γ ⊢ SS(E, r) : U
Γ ⊢ Group : U
Γ ⊢ S⁰[p] : Spec

Introductions

ℕ: 𝟎 : ℕ, suc : ℕ → ℕ
ℕ∞: fin : ℕ → ℕ∞, inf : ℕ∞
Susp: susp(t) : Susp(A) if t : A
Truncⁿ: truncⁿ(t) : Truncⁿ(A) if t : A
Π: λx.t : Π(x:A).B if Γ, x:A ⊢ t : B
Σ: (t, u) : Σ(x:A).B if t : A, u : B[t/x]
Id: refl : Id_A(u, u)
Γ ⊢ t : HopfFibⁿ  ⇒  t ≡ hopfⁿ
fiber : HopfFibⁿ → U
fiber(HopfFib¹) = S⁰
fiber(HopfFib²) = S¹
fiber(HopfFib⁴) = S³
fiber(HopfFib⁸) = S⁷
total : HopfFibⁿ → U
total(HopfFib¹) = S¹
total(HopfFib²) = S³
total(HopfFib⁴) = S⁷
total(HopfFib⁸) = S¹⁵
fibration : Π(f:HopfFibⁿ).fiber(f) → total(f)
lift : Π(a:Sᵐ).Π(b:Sᵐ).Id_Fibⁿ(hopfⁿ, hopfⁿ) → Id_Sᵐ(a, b)
inv : Id_A(u, v) → Id_A(v, u)
proj : HopfFibⁿ → Sᵐ, (m = n/2, e.g., HopfFib² → S²)
Spec : U, Aₙ : ℕ → U, σₙ : Aₙ → Susp(Aₙ₊₁) ⊢ spec({Aₙ}, {σₙ}) : Spec
S⁰ : Spec, S⁰ := spec({Sⁿ}, {σₙ}) ⊢ σₙ : Sⁿ → Susp(Sⁿ₊₁) ≡ Sⁿ⁺¹
Γ ⊢ A : Spec, a : A₀ ⊢ πₙ^S(A) : U := colimₖ πₙ₊ₖ(Aₖ)
stable(t) : πₙ^S(A)
A : Spec, p : ℕ (prime), S⁰[p] : Spec ⊢ loc_p(t) : S⁰[p]
grp(G, e, op, inv) : Group
smash(t, u) : ∧
t(E,r,p,q) : SS
map(t) : [,]
tensor(g, h) : G ⊗ H

Eliminators

Γ ⊢ C : ℕ → U, z : C(𝟎), s : Π(k:ℕ).C(k) → C(suc(k)) ⊢ rec_ℕ(C, z, s, t) : C(t) (t : ℕ)
Γ ⊢ C : ℕ∞ → U, f : Π(k:ℕ).C(fin(k)), i : C(inf) ⊢ case_ℕ∞(C, f, i, t) : C(t) (t : ℕ∞)
Γ ⊢ A : U, t : Susp(A), C : Susp(A) → U, s : Π(a:A).C(susp(a)) ⊢ elim_Susp(C, s, t) : C(t)
Γ ⊢ A : U, t : Truncⁿ(A), C : Truncⁿ(A) → U, trunc : Π(a:A).C(truncⁿ(a)) ⊢ elim_Truncⁿ(C, trunc, t) : C(t)

Γ ⊢ t : A ∧ B, C : (A ∧ B) → U, s : Π(a:A).Π(b:B).C(smash(a, b)) Γ ⊢ elim_Smash(C, s, t) : C(t)
Γ ⊢ t : [A, B], C : [A, B] → U, m : Π(f:A→B).C(map(f)) Γ ⊢ elim_Map(C, m, t) : C(t)
Γ ⊢ E : Spec, C : Spec → U, : Π({Aₙ}:ℕ→U).Π({σₙ}:Π(n:ℕ).Aₙ→Susp(Aₙ₊₁)).C(spec({Aₙ},{σₙ})) ⊢ elim_Spec(C, s, E) : C(E)
Γ ⊢ A : Spec, t : πₙ^S(A), C : πₙ^S(A) → U, s : Π(k:ℕ).Π(a:πₙ₊ₖ(Aₖ)).C(stable(a)) ⊢ elim_πₙ^S(C, s, t) : C(t)

order : Π(n:ℕ).Π(m:ℕ).Π(x:πₙ(Sᵐ)).ℕ∞
order(n)(m)(x) = rec_ℕ(k ↦ ℕ∞, inf, λk.prev.case(test(k),
    λeq.fin(suc(k)), λ_.prev), suc(k_max))
    test(n)(m)(x)(k) = trunc⁰(pow(n)(m)(x)(k) = refl)

Computations

rec_ℕ(C, z, s, 𝟎) ≡ z
rec_ℕ(C, z, s, suc(k)) ≡ s(k, rec_ℕ(C, z, s, k))
case_ℕ∞(C, f, i, fin(k)) ≡ f(k)
case_ℕ∞(C, f, i, inf) ≡ i
elim_Susp(C, s, susp(a)) ≡ s(a)
Susp(Sⁿ) ≡ Sⁿ⁺¹
susp(HopfFibⁿ) ↦ HopfFibⁿ⁺¹, (n ∈ {1, 2, 4}, n+1 ≤ 8)
susp(truncⁿ(a)) ↦ truncⁿ⁺¹(susp(a))
Susp(HopfFib⁸) ≡ Susp(total(HopfFib⁸))    (fallback to S¹⁶)
Susp(Truncⁿ(A)) ↦ Truncⁿ⁺¹(Susp(A))    (term-level coherence)
elim_Truncⁿ(C, trunc, truncⁿ(a)) ≡ trunc(a)
πₖ(Truncⁿ(A)) ≡ 𝟎    (k > n)
HopfFib¹ ≡ S⁰ → S¹
HopfFib² ≡ S¹ → S³ → S²
HopfFib⁴ ≡ S³ → S⁷ → S⁴
HopfFib⁸ ≡ S⁷ → S¹⁵ → S⁸
fibration(HopfFibⁿ) : fiber(HopfFibⁿ) → total(HopfFibⁿ)
proj(fibration(HopfFibⁿ)(x)) ≡ baseᵐ
lift(baseᵐ, baseᵐ, refl) ≡ refl
lift(a, b, p) · q ≡ lift(a, c, p · q)    (path composition)
πₙ(Sᵐ) ≅ Id_total(HopfFibᵏ)(fibration(hopfᵏ)(x), fibration(hopfᵏ)(y))    (adjusted definition)
(λx.t) u ≡ t[u/x]
fst(t, u) ≡ t
snd(t, u) ≡ u
πₙ(Sᵐ) ≅ Id_Suspⁿ⁻ᵐ(HopfFibᵏ)(hopfᵏ, hopfᵏ)    (m ≤ k, k ∈ {1, 2, 4, 8})
pow(n)(m)(x)(k) = rec_ℕ(k’ ↦ πₙ(Sᵐ), refl, λk’.p.p · x, k)
(p · q) · r ≡ p · (q · r), p · refl ≡ p, refl · p ≡ p, p · inv(p) ≡ refl
proj(hopfⁿ) ≡ baseᵐ    (fixed point in Sᵐ)
lift(a, b, p) · q ≡ lift(a, c, p · q)    (path composition)

πₙ^S(S⁰) ≡ colimₖ πₙ₊ₖ(Sᵏ)
stable(aₖ) ↦ colimₖ(aₖ)    (aₖ : πₙ₊ₖ(Aₖ))
elim_Spec(C, s, spec({Aₙ},{σₙ})) ≡ s({Aₙ},{σₙ})
cup(map(f), map(g)) ↦ map(λx.kgn(tensor(f(x), g(x)), n+m))
cup(t, u) associative, graded-commutative
diffᵣ(diffᵣ(t)) ≡ 0    (d² = 0)
SS(E, r+1) ≅ ker(diffᵣ) / im(diffᵣ)    (next page)
diffᵣ(diffᵣ(t)) ≡ 0, SS(E, r+1) ≅ ker(diffᵣ) / im(diffᵣ)
Hⁿ(X; G) ≅ π₀^S([X, K(G, n)]), Hⁿ(X; G) ≅ π₋ₙ^S(HG ∧ X)
Hⁿ(X; G) ≅ π₀^S([X, K(G, n)]), Hⁿ(X; G) ≅ π₋ₙ^S(HG ∧ X)    (stable equivalence)
HG ∧ S⁰ ≡ HG
elim_Map(C, m, map(f)) ≡ m(f), πₙ^S([X, Y]) ≅ [Suspⁿ(X), Y]₀   (adjointness)
[spec({Aₙ},{σₙ}), spec({Bₙ},{τₙ})]ₖ ≡ [Aₖ, Bₖ]    (component-wise)
elim_Smash(C, s, smash(a, b)) ≡ s(a, b), S⁰ ∧ X ≡ X
(spec({Aₙ},{σₙ})) ∧ (spec({Bₙ},{τₙ})) ≡ spec({Aₙ ∧ Bₙ},{σₙ ∧ τₙ})
πₙ(K(G, n)) ≅ G, susp(kgn(g, n)) ↦ kgn(g, n+1)
loc_p(spec({Aₙ},{σₙ})) ↦ spec({Aₙ[p]},{σₙ[p]})
πₙ^S(A) ≡ colimₖ πₙ₊ₖ(Aₖ), stable(aₖ) ↦ colimₖ(aₖ)
πₙ^S(S⁰[p]) ≡ πₙ^S(S⁰) ⊗ ℤ_{(p)}    (p-local integers)
elim_Spec(C, s, spec({Aₙ},{σₙ})) ≡ s({Aₙ},{σₙ})
πₙ^S(A ∧ B) ≅ colimₖ πₙ₊ₖ(Aₖ ∧ Bₖ) - stable Homotopy Refinement
Γ ⊢ finite : Π(n:ℕ).Π(m:ℕ).Trunc⁰(πₙ(Sᵐ)) → U, finite(n)(m)(trunc⁰(x)) = Σ(k:ℕ).Id_πₙ(Sᵐ)(pow(n)(m)(x)(k), refl)

Coherences

(p · q) · r ≡ p · (q · r)
p · refl ≡ p    refl · p ≡ p
p · inv(p) ≡ refl    (inv(p) : Id_A(v, u) if p : Id_A(u, v))
Γ ⊢ t : Trunc⁰(A), u : Trunc⁰(A) ⊢ t ≡ u : Trunc⁰(A) or  Γ ⊢ t ≠ u : Trunc⁰(A)
Γ ⊢ t : HopfFibⁿ  ⇒  t ≡ hopfⁿ
Γ ⊢ t : Truncⁿ(A)    Γ ⊢ u : Truncⁿ(A)    πₖ(t) ≡ πₖ(u) (k ≤ n)  ⇒  t ≡ u

Publications

Namdak Tonpa