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+fabien.groupoid.space
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+# Fabien Morel: 𝔸¹ Type Theory
+
+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) 
+
+## Intro
+
+Targets:
+
+* Motivic Cohomology: `𝐻¹¹(ℙ,ℤ(1))=𝑘ˣ`.
+* Algebraic K-theory: `𝜋₁ᴬ(S¹¹)=𝑘ˣ`.
+* Algebraic Cobordism: `𝑀𝐺𝐿²¹(pt)≅𝑘ˣ`.
+
+Type Formers:
+
+* U,V (Fibrant, Pretypes universes)
+* ΠΣ (Dependent Fibrational Types)
+* Path (Cubical de Morgan Interval)
+* 𝑘 (field)
+* 𝔸¹ (Affine Line)
+* S¹¹ (Motivic Spheres)
+* L_{𝔸¹} (Localization)
+* Suspension
+* Truncⁿ
+* NisCover (Nisnevich Descent)
+* 𝐾(𝑍,𝑛) (Eilenberg-MacLane 𝔸¹ spaces)
+* 𝐵𝐺𝐿
+* 𝑀𝐺𝐿
+* ℕ,ℤ
+
+## Setup
+
+```
+-- 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
+}
+```
+
+### Motivic Cohomology
+
+```
+
+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
+}
+```
+
+### Algebraic K-theory
+
+```
+
+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⟩
+}
+```
+
+### Algebraic Cobordism
+
+```
+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⟩
+}
+```
+
+### Homotopy Groups
+
+```
+def pi_n_A1_BGL (n : Nat) : Type fibrant := 
+  Trunc 0 (Omega_n (L_A1 BGL) (eta_A1 (sorry : BGL)) n)
+```
+
+## Bibliography
+
+* Fabien Morel. <a href="https://web.archive.org/web/20150924032624/http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_49.pdf">𝔸¹-algebraic topology</a>
+* Vladimir Voevodksy. <a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/A1_homotopy_ICM_1998_Berlin_published.pdf">𝔸¹-Homotopy Theory</a>
+
+## Copyright
+
+Namdak Tonpa