A^1-Homotopy Fabien Morel Type Theory (FMTT) is built on top of: 1) Cubical Type Theory (CTT), 2) Base field k, 3) A^1-indexed affine line, 4) Localization modality to enforce A^1-contractibility, 5) Suspension, 6) Nisnevich Cover, 7) K(Z,n), 8) N,Z,9) MGL (Motivic cobordism spectrum), 10) BGL (Classifying space of the general linear group). It bridges type theory and motivic homotopy, offering a lightweight, extensible platform for formalizing algebraic geometry’s homotopical landscape. Applications to motivic cohomology and stable homotopy categories are discussed.
Targets:
𝐻¹¹(ℙ,ℤ(1))=𝑘ˣ.𝜋₁ᴬ(S¹¹)=𝑘ˣ.𝑀𝐺𝐿²¹(pt)≅𝑘ˣ.Type Formers:
-- Base field with operations
def k : Type strict -- Abstract field type
def 0_k : k := sorry
def 1_k : k := sorry
def add_k : k → k → k := sorry
def mul_k : k → k → k := sorry
def neg_k : k → k := sorry
def inv_k : k → k := sorry
-- Field axioms as propositions
def add_comm : ∀ (x y : k), add_k x y = add_k y x := sorry
def add_assoc : ∀ (x y z : k), add_k (add_k x y) z = add_k x (add_k y z) := sorry
def add_zero : ∀ (x : k), add_k x 0_k = x := sorry
def add_neg : ∀ (x : k), add_k x (neg_k x) = 0_k := sorry
def mul_comm : ∀ (x y : k), mul_k x y = mul_k y x := sorry
def mul_assoc : ∀ (x y z : k), mul_k (mul_k x y) z = mul_k x (mul_k y z) := sorry
def mul_one : ∀ (x : k), mul_k x 1_k = x := sorry
def distrib : ∀ (x y z : k), mul_k x (add_k y z) = add_k (mul_k x y) (mul_k x z) := sorry
def one_neq_zero : 1_k ≠ 0_k := sorry
def mul_inv : ∀ (x : k), x ≠ 0_k → mul_k x (inv_k x) = 1_k := sorry
def k_mult : Type strict := { x : k // x ≠ 0_k }
def rec_k {A : k → Type strict} (z : A 0_k) (o : A 1_k)
(a : ∀ (x y : k), A x → A y → A (add_k x y))
(m : ∀ (x y : k), A x → A y → A (mul_k x y))
(n : ∀ (x : k), A x → A (neg_k x))
(i : ∀ (x : k) (h : x ≠ 0_k), A x → A (inv_k x))
(x : k) : A x := sorry -- Implementation dependent on k
-- Integers
inductive Z : Type strict where
| zero : Z
| pos : Nat → Z
| neg : Nat → Z
-- Affine line
inductive A1 : Type fibrant where
| point : k → A1
def 0_A1 : A1 := A1.point 0_k
-- Projective line
inductive P1 : Type fibrant where
| zero : P1
| inf : P1
| point : A1 → Path P1 zero inf
| eq_zero : Path P1 (point 0_A1) (refl zero)
-- S^{1,1}
inductive S11 : Type fibrant where
| base : S11
| loop : A1 → Path S11 base base
| zero : Path S11 (loop 0_A1) (refl base)
-- EM spaces
inductive K (n : Nat) : Type fibrant where
| base : K n
| loop_n : Z → Path (Omega_n (K n) (K.base) n) → K n
| trunc_n : isTrunc (n - 1) (K n)
-- BGL
inductive BGL : Type fibrant where
| base : BGL
| cell : Nat → BGL → BGL
| path : Nat → k_mult → Path BGL (cell n base) base
-- MGL
inductive MGL : Type fibrant where
| base : MGL
| thom : S11 → MGL
| orient : k_mult → Path MGL (thom S11.base) base
-- Localization (simplified for brevity)
def L_A1 (X : Type fibrant) : Type fibrant := (i : I) → X
def eta_A1 {X : Type fibrant} (x : X) : L_A1 X := <i> x
-- GW
inductive GW : Type strict where
| zero : GW
| form : (k → k → k) → (∀ (x y : k), f x y = f y x) → GW
| add : GW → GW → GW
| tensor : GW → GW → GW
| neg : GW → GW
-- KH
inductive KH : Type fibrant where
| base : KH
| loop : Nat → k_mult → Path KH base base
def pi_0_0_S : Type fibrant := Trunc 0 (L_A1 Unit)
def to_GW : pi_0_0_S → GW :=
λ t =>
match t with
| Trunc.tm γ => GW.form (λ x y => mul_k x y) (λ x y => mul_comm x y)
def from_GW : GW → pi_0_0_S :=
λ g =>
match g with
| GW.zero => Trunc.tm (λ i => Unit.unit)
| GW.form f h => Trunc.tm (λ i => Unit.unit)
| GW.add x y => Trunc.tm (λ i => Unit.unit)
| GW.tensor x y => Trunc.tm (λ i => Unit.unit)
| GW.neg x => Trunc.tm (λ i => Unit.unit)
def iso_pi_0_0_GW : Iso pi_0_0_S GW := {
to := to_GW,
inv := from_GW,
left_inv := λ t =>
match t with
| Trunc.tm γ => Trunc.path (<i> λ j => Unit.unit),
right_inv := λ g =>
match g with
| GW.zero => refl GW.zero
| GW.form f h => <i> GW.form f h -- Simplified, assumes isomorphism classes
| GW.add x y => sorry -- Requires GW group axioms
| GW.tensor x y => sorry
| GW.neg x => sorry
}
def Hpq (X : Type fibrant) (p q n : Nat) : Type fibrant := Trunc 0 (L_A1 (Spq p q) → L_A1 (K n))
def motivic_H (X : Type fibrant) (p q n : Nat) : Hpq (XPlus X) p q n
def H11_P1 : Type fibrant := Trunc 0 (L_A1 S11 → L_A1 (XPlus P1) → L_A1 (K 1))
def to_k_mult : H11_P1 → k_mult :=
λ h =>
let f := h (λ i => S11.base) (λ i => XPlus.inl P1.zero) in
let p := rec_K 1 (0_k) (λ z i => mul_k (match z with
| Z.zero => 0_k
| Z.pos m => iterMul m 1_k
| Z.neg m => iterMul m (inv_k 1_k)))
(λ _ => refl 0_k) f in
⟨p 1, λ h0 => one_neq_zero (mul_inv (p 1) h0)⟩ -- Non-zero via axioms
def from_k_mult : k_mult → H11_P1 :=
λ ⟨g, h⟩ =>
Trunc.tm (λ s => λ x => λ i =>
match s i, x i with
| S11.base, XPlus.inl P1.zero => K.loop_n 1 (Z.pos 1) (λ j => mul_k g (inv_k g))
| _, _ => K.base)
def motivic_H11 : Iso H11_P1 k_mult := {
to := to_k_mult,
inv := from_k_mult,
left_inv := λ h =>
let f := h (λ i => S11.base) (λ i => XPlus.inl P1.zero) in
let p := rec_K 1 (0_k) (λ z i => mul_k (match z with
| Z.zero => 0_k
| Z.pos m => iterMul m 1_k
| Z.neg m => iterMul m (inv_k 1_k)))
(λ _ => refl 0_k) f in
Trunc.path (<i> λ s => λ x => λ j =>
comp (K 1) [k : I] (if s k = S11.base ∧ x k = XPlus.inl P1.zero
then K.loop_n 1 (Z.pos 1) (λ m => mul_k (p m) (inv_k (p m)))
else K.base) (h s x j)),
right_inv := λ ⟨g, h⟩ =>
<j> ⟨g, h⟩ -- mul_k g (inv_k g) = 1_k by mul_inv, h preserved
}
def K1_X : Type fibrant := Trunc 0 (Omega (L_A1 S11) (eta_A1 S11.base) 1)
def K_n_X (X : Type fibrant) (n : Nat) : Type fibrant
:= pi_n_A1 (K n) n (eta_A1 K.base)
def pi1_S11 : Type fibrant := pi_n_A1 S11 1 S11.base
def to_k_mult_pi1 : pi1_S11 → k_mult :=
λ t =>
match t with
| Trunc.tm γ =>
⟨rec_S11 (0_k) (λ a => mul_k (match a with | A1.point x => x))
(λ _ => refl 0_k) (γ 1), λ h0 => one_neq_zero (add_zero (neg_k (γ 1)) h0)⟩
def from_k_mult_pi1 : k_mult → pi1_S11 :=
λ ⟨g, h⟩ => Trunc.tm (λ i => S11.loop (A1.point g))
def k_theory_pi1 : Iso pi1_S11 k_mult := {
to := to_k_mult_pi1,
inv := from_k_mult_pi1,
left_inv := λ t =>
match t with
| Trunc.tm γ =>
let g := rec_S11 (0_k) (λ a => mul_k (match a with | A1.point x => x))
(λ _ => refl 0_k) (γ 1) in
Trunc.path (<i> λ j => S11.loop (a1contr (A1.point g) i)),
right_inv := λ ⟨g, h⟩ =>
<i> ⟨g, h⟩
}
def MGL_2_1 : Type fibrant := Trunc 0 (L_A1 (Spq 2 1) → L_A1 Unit)
def MGL_21_pt : Type fibrant := Trunc 0 (L_A1 (Spq 2 1) → L_A1 MGL)
def to_k_mult_mgl : MGL_21_pt → k_mult :=
λ m =>
let f := m (λ i => S11.base) in
⟨rec_MGL (0_k) (λ s => mul_k (rec_S11 (match s with | S11.base => 0_k | S11.loop a => match a with | A1.point x => x)))
(λ g => refl (mul_k g (inv_k g))) f, λ h0 => one_neq_zero (mul_inv f h0)⟩
def from_k_mult_mgl : k_mult → MGL_21_pt :=
λ ⟨g, h⟩ => Trunc.tm (λ s => MGL.orient g)
def cobordism_MGL : Iso MGL_21_pt k_mult := {
to := to_k_mult_mgl,
inv := from_k_mult_mgl,
left_inv := λ m =>
let f := m (λ i => S11.base) in
let g := rec_MGL (0_k) (λ s => mul_k (rec_S11 (match s with | S11.base => 0_k | S11.loop a => match a with | A1.point x => x)))
(λ g => refl (mul_k g (inv_k g))) f in
Trunc.path (<i> λ s => MGL.orient g),
right_inv := λ ⟨g, h⟩ =>
<i> ⟨g, h⟩
}
def pi_n_A1_BGL (n : Nat) : Type fibrant :=
Trunc 0 (Omega_n (L_A1 BGL) (eta_A1 (sorry : BGL)) n)
Namdak Tonpa