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ext
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[email protected]._hyg.3655
|
∀ (x : R), (iterateFrobenius R p n : R → R) x✝ = (frobenius R p ^ n : R → R) x✝
|
R : Type u_1
inst✝¹ : CommSemiring R
p n : ℕ
inst✝ : ExpChar R p
⊢ iterateFrobenius R p n = frobenius R p ^ n
|
iterateFrobenius_def,
|
[anonymous]
|
x ^ p ^ 1 = x ^ p
|
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
x : R
⊢ (iterateFrobenius R p 1 : R → R) x = x ^ p
|
pow_one
|
[anonymous]
|
x ^ p = x ^ p
|
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
x : R
⊢ x ^ p ^ 1 = x ^ p
|
iterateFrobenius_def,
|
[anonymous]
|
x ^ p ^ 0 = x
|
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
x : R
⊢ (iterateFrobenius R p 0 : R → R) x = x
|
pow_zero,
|
[anonymous]
|
x ^ 1 = x
|
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
x : R
⊢ x ^ p ^ 0 = x
|
pow_one
|
[anonymous]
|
x = x
|
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
x : R
⊢ x ^ 1 = x
|
simp_rw [Algebra.smul_def, map_mul, ← (algebraMap R S).map_frobenius]
|
[anonymous]
|
(algebraMap R S : R → S) ((_root_.frobenius R p : R → R) r) * (_root_.frobenius S p : S → S) s =
(algebraMap R S : R → S) ((_root_.frobenius R p : R → R) r) *
(↑(↑(_root_.frobenius S p) : S →* S) : OneHom S S).toFun s
|
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
f : R →* S
g : R →+* S
p m n : ℕ
inst✝² : ExpChar R p
inst✝¹ : ExpChar S p
x y : R
inst✝ : Algebra R S
r : R
s : S
⊢ (_root_.frobenius S p : S → S) (r • s) =
(_root_.frobenius R p : R → R) r • (↑(↑(_root_.frobenius S p) : S →* S) : OneHom S S).toFun s
|
simp_rw [iterateFrobenius_def, Algebra.smul_def, mul_pow, ← map_pow]
|
[anonymous]
|
(algebraMap R S : R → S) (f ^ p ^ n) * s ^ p ^ n =
(algebraMap R S : R → S) (f ^ p ^ n) * (↑(↑(_root_.iterateFrobenius S p n) : S →* S) : OneHom S S).toFun s
|
R : Type u_1
inst✝⁴ : CommSemiring R
S : Type u_2
inst✝³ : CommSemiring S
f✝ : R →* S
g : R →+* S
p m n : ℕ
inst✝² : ExpChar R p
inst✝¹ : ExpChar S p
x y : R
inst✝ : Algebra R S
f : R
s : S
⊢ (_root_.iterateFrobenius S p n : S → S) (f • s) =
(_root_.iterateFrobenius R p n : R → R) f • (↑(↑(_root_.iterateFrobenius S p n) : S →* S) : OneHom S S).toFun s
|
constructor
|
mp
|
Isδ₀ (SimplexCategory.δ i) → i = 0
|
j : ℕ
i : Fin (j + 2)
⊢ Isδ₀ (SimplexCategory.δ i) ↔ i = 0
|
constructor
|
mpr
|
i = 0 → Isδ₀ (SimplexCategory.δ i)
|
j : ℕ
i : Fin (j + 2)
mp : Isδ₀ (SimplexCategory.δ i) → i = 0
⊢ Isδ₀ (SimplexCategory.δ i) ↔ i = 0
|
rintro ⟨_, h₂⟩
|
mp.intro
|
(Hom.toOrderHom (SimplexCategory.δ i) : Fin (⦋j⦌.len + 1) → Fin (⦋j + 1⦌.len + 1)) 0 ≠ 0 →
⦋j + 1⦌.len = ⦋j⦌.len + 1 → i = 0
|
j : ℕ
i : Fin (j + 2)
⊢ Isδ₀ (SimplexCategory.δ i) → i = 0
|
by_contra h
|
mp.intro
|
¬i = 0 → False
|
j : ℕ
i : Fin (j + 2)
left✝ : ⦋j + 1⦌.len = ⦋j⦌.len + 1
h₂ : (Hom.toOrderHom (SimplexCategory.δ i) : Fin (⦋j⦌.len + 1) → Fin (⦋j + 1⦌.len + 1)) 0 ≠ 0
⊢ i = 0
|
rintro rfl
|
mpr
|
Isδ₀ (SimplexCategory.δ 0)
|
j : ℕ
i : Fin (j + 2)
⊢ i = 0 → Isδ₀ (SimplexCategory.δ i)
|
dsimp
|
[anonymous]
|
¬(Hom.toOrderHom (SimplexCategory.δ 0) : Fin (j + 1) → Fin (j + 1 + 1)) 0 = 0
|
j : ℕ
⊢ (Hom.toOrderHom (SimplexCategory.δ 0) : Fin (⦋j⦌.len + 1) → Fin (⦋j + 1⦌.len + 1)) 0 ≠ 0
|
obtain ⟨j, rfl⟩ := SimplexCategory.eq_δ_of_mono i
|
intro
|
Isδ₀ (SimplexCategory.δ j) →
∀ [Mono (SimplexCategory.δ j)] (j_1 : Fin (n + 2)), SimplexCategory.δ j = SimplexCategory.δ 0
|
n : ℕ
i : ⦋n⦌ ⟶ ⦋n + 1⦌
inst✝ : Mono i
hi : Isδ₀ i
⊢ i = SimplexCategory.δ 0
|
rw [iff] at hi
|
intro
|
j = 0
|
n : ℕ
j : Fin (n + 2)
inst✝ : Mono (SimplexCategory.δ j)
hi : Isδ₀ (SimplexCategory.δ j)
⊢ SimplexCategory.δ j = SimplexCategory.δ 0
|
iff
|
intro
|
j = 0
|
n : ℕ
j : Fin (n + 2)
inst✝ : Mono (SimplexCategory.δ j)
hi : Isδ₀ (SimplexCategory.δ j)
⊢ SimplexCategory.δ j = SimplexCategory.δ 0
|
hi
|
intro
|
SimplexCategory.δ 0 = SimplexCategory.δ 0
|
n : ℕ
j : Fin (n + 2)
inst✝ : Mono (SimplexCategory.δ j)
hi : j = 0
⊢ SimplexCategory.δ j = SimplexCategory.δ 0
|
by_cases Δ = Δ'
|
[email protected]._hyg.484
|
Δ = Δ' → (K.X Δ.len ⟶ K.X Δ'.len)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
K : ChainComplex C ℕ
Δ' Δ : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
⊢ K.X Δ.len ⟶ K.X Δ'.len
|
by_cases Δ = Δ'
|
[email protected]._hyg.484
|
¬Δ = Δ' → (K.X Δ.len ⟶ K.X Δ'.len)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
K : ChainComplex C ℕ
Δ' Δ : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
pos : Δ = Δ' → (K.X Δ.len ⟶ K.X Δ'.len)
⊢ K.X Δ.len ⟶ K.X Δ'.len
|
by_cases Isδ₀ i
|
[email protected]._hyg.529
|
Isδ₀ i → (K.X Δ.len ⟶ K.X Δ'.len)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
K : ChainComplex C ℕ
Δ' Δ : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h✝ : ¬Δ = Δ'
⊢ K.X Δ.len ⟶ K.X Δ'.len
|
by_cases Isδ₀ i
|
[email protected]._hyg.529
|
¬Isδ₀ i → (K.X Δ.len ⟶ K.X Δ'.len)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
K : ChainComplex C ℕ
Δ' Δ : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h✝ : ¬Δ = Δ'
pos : Isδ₀ i → (K.X Δ.len ⟶ K.X Δ'.len)
⊢ K.X Δ.len ⟶ K.X Δ'.len
|
unfold mapMono
|
[anonymous]
|
(if h : Δ = Δ then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ (𝟙 Δ) then K.d Δ.len Δ.len else 0) = 𝟙 (K.X Δ.len)
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
K : ChainComplex C ℕ
Δ : SimplexCategory
⊢ mapMono K (𝟙 Δ) = 𝟙 (K.X Δ.len)
|
unfold mapMono
|
[anonymous]
|
(if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = K.d Δ.len Δ'.len
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
hi : Isδ₀ i
⊢ mapMono K i = K.d Δ.len Δ'.len
|
suffices Δ ≠ Δ' by
simp only [dif_neg this, dif_pos hi]
|
[anonymous]
|
Δ ≠ Δ'
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
hi : Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = K.d Δ.len Δ'.len
|
rintro rfl
|
[anonymous]
|
Isδ₀ i → ∀ [Mono i] (i : Δ ⟶ Δ), False
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
hi : Isδ₀ i
⊢ Δ ≠ Δ'
|
Isδ₀.iff
|
[anonymous]
|
0 = 0
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
K : ChainComplex C ℕ
n : ℕ
⊢ Isδ₀ (SimplexCategory.δ 0)
|
unfold mapMono
|
[anonymous]
|
(if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : Δ ≠ Δ'
h₂ : ¬Isδ₀ i
⊢ mapMono K i = 0
|
rw [Ne] at h₁
|
[anonymous]
|
¬Δ = Δ'
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : Δ ≠ Δ'
h₂ : ¬Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
Ne
|
[anonymous]
|
¬Δ = Δ'
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : Δ ≠ Δ'
h₂ : ¬Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
split_ifs
|
[anonymous]
|
0 = 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
unfold mapMono
|
[anonymous]
|
(if h : Δ' = Δ then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ'.len Δ.len else 0) ≫ f.f Δ.len =
f.f Δ'.len ≫ if h : Δ' = Δ then eqToHom (mapMono._proof_3 K' h) else if h : Isδ₀ i then K'.d Δ'.len Δ.len else 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
⊢ mapMono K i ≫ f.f Δ.len = f.f Δ'.len ≫ mapMono K' i
|
split_ifs with h
|
[email protected]._hyg.1012
|
Δ' = Δ → eqToHom (mapMono._proof_3 K h) ≫ f.f Δ.len = f.f Δ'.len ≫ eqToHom (mapMono._proof_3 K' h)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
⊢ (if h : Δ' = Δ then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ'.len Δ.len else 0) ≫ f.f Δ.len =
f.f Δ'.len ≫ if h : Δ' = Δ then eqToHom (mapMono._proof_3 K' h) else if h : Isδ₀ i then K'.d Δ'.len Δ.len else 0
|
split_ifs with h
|
[email protected]._hyg.1039
|
Isδ₀ i → ¬Δ' = Δ → K.d Δ'.len Δ.len ≫ f.f Δ.len = f.f Δ'.len ≫ K'.d Δ'.len Δ.len
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
pos : Δ' = Δ → eqToHom (mapMono._proof_3 K _fvar.15634) ≫ f.f Δ.len = f.f Δ'.len ≫ eqToHom (mapMono._proof_3 K' _fvar.15634)
⊢ (if h : Δ' = Δ then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ'.len Δ.len else 0) ≫ f.f Δ.len =
f.f Δ'.len ≫ if h : Δ' = Δ then eqToHom (mapMono._proof_3 K' h) else if h : Isδ₀ i then K'.d Δ'.len Δ.len else 0
|
split_ifs with h
|
[email protected]._hyg.1039
|
¬Isδ₀ i → ¬Δ' = Δ → 0 ≫ f.f Δ.len = f.f Δ'.len ≫ 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
pos : Δ' = Δ → eqToHom (mapMono._proof_3 K _fvar.15634) ≫ f.f Δ.len = f.f Δ'.len ≫ eqToHom (mapMono._proof_3 K' _fvar.15634)
pos : Isδ₀ i → ¬Δ' = Δ → K.d Δ'.len Δ.len ≫ f.f Δ.len = f.f Δ'.len ≫ K'.d Δ'.len Δ.len
⊢ (if h : Δ' = Δ then eqToHom (mapMono._proof_3 K h) else if h : Isδ₀ i then K.d Δ'.len Δ.len else 0) ≫ f.f Δ.len =
f.f Δ'.len ≫ if h : Δ' = Δ then eqToHom (mapMono._proof_3 K' h) else if h : Isδ₀ i then K'.d Δ'.len Δ.len else 0
|
subst h
|
[email protected]._hyg.1012
|
∀ [Mono i] (i : Δ' ⟶ Δ'),
eqToHom (mapMono._proof_3 K (Eq.refl Δ')) ≫ f.f Δ'.len = f.f Δ'.len ≫ eqToHom (mapMono._proof_3 K' (Eq.refl Δ'))
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
h : Δ' = Δ
⊢ eqToHom (mapMono._proof_3 K h) ≫ f.f Δ.len = f.f Δ'.len ≫ eqToHom (mapMono._proof_3 K' h)
|
HomologicalComplex.Hom.comm
|
[email protected]._hyg.1039
|
K.d Δ'.len Δ.len ≫ f.f Δ.len = K.d Δ'.len Δ.len ≫ f.f Δ.len
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
h : ¬Δ' = Δ
h✝ : Isδ₀ i
⊢ K.d Δ'.len Δ.len ≫ f.f Δ.len = f.f Δ'.len ≫ K'.d Δ'.len Δ.len
|
zero_comp,
|
[email protected]._hyg.1039
|
0 = f.f Δ'.len ≫ 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
h : ¬Δ' = Δ
h✝ : ¬Isδ₀ i
⊢ 0 ≫ f.f Δ.len = f.f Δ'.len ≫ 0
|
comp_zero
|
[email protected]._hyg.1039
|
0 = 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
h : ¬Δ' = Δ
h✝ : ¬Isδ₀ i
⊢ 0 = f.f Δ'.len ≫ 0
|
by_cases h₁ : Δ = Δ'
|
[email protected]._hyg.1219
|
Δ = Δ' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
by_cases h₁ : Δ = Δ'
|
[email protected]._hyg.1219
|
¬Δ = Δ' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
pos : Δ = Δ' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
subst h₁
|
[email protected]._hyg.1219
|
∀ [Mono i] [Mono i'] (i_1 : Δ ⟶ Δ) (i'_1 : Δ'' ⟶ Δ), mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : Δ = Δ'
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
by_cases h₂ : Δ' = Δ''
|
[email protected]._hyg.1256
|
Δ' = Δ'' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
by_cases h₂ : Δ' = Δ''
|
[email protected]._hyg.1256
|
¬Δ' = Δ'' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
pos : Δ' = Δ'' → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
subst h₂
|
[email protected]._hyg.1256
|
∀ [Mono i'] (i'_1 : Δ' ⟶ Δ'), mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : Δ' = Δ''
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_lt (len_lt_of_mono i h₁)
|
[email protected]._hyg.1256
|
Δ.len = Δ'.len + k + 1 → ∀ (k : ℕ), mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
obtain ⟨k', hk'⟩ := Nat.exists_eq_add_of_lt (len_lt_of_mono i' h₂)
|
neg.intro.intro._@.Mathlib.AlgebraicTopology.DoldKan.FunctorGamma._hyg.1256
|
Δ'.len = Δ''.len + k' + 1
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
have eq : Δ.len = Δ''.len + (k + k' + 2) := by omega
|
neg.intro.intro._@.Mathlib.AlgebraicTopology.DoldKan.FunctorGamma._hyg.1256
|
Δ.len = Δ''.len + (k + k' + 2) → mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
rw [mapMono_eq_zero K (i' ≫ i) _ _]
|
neg.intro.intro._@.Mathlib.AlgebraicTopology.DoldKan.FunctorGamma._hyg.1256
|
mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
rw [mapMono_eq_zero K (i' ≫ i) _ _]
|
[anonymous]
|
Δ ≠ Δ''
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
neg.intro.intro : mapMono K i ≫ mapMono K i' = 0
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
rw [mapMono_eq_zero K (i' ≫ i) _ _]
|
[anonymous]
|
¬Isδ₀ (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
neg.intro.intro : mapMono K i ≫ mapMono K i' = 0
[anonymous] : Δ ≠ Δ''
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
mapMono_eq_zero K (i' ≫ i) _ _
|
neg.intro.intro._@.Mathlib.AlgebraicTopology.DoldKan.FunctorGamma._hyg.1256
|
mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
mapMono_eq_zero K (i' ≫ i) _ _
|
[anonymous]
|
Δ ≠ Δ''
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
neg.intro.intro : mapMono K i ≫ mapMono K i' = 0
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
mapMono_eq_zero K (i' ≫ i) _ _
|
[anonymous]
|
¬Isδ₀ (i' ≫ i)
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
neg.intro.intro : mapMono K i ≫ mapMono K i' = 0
[anonymous] : Δ ≠ Δ''
⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)
|
by_contra h
|
[anonymous]
|
Δ = Δ'' → False
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ Δ ≠ Δ''
|
by_contra h
|
[anonymous]
|
Isδ₀ (i' ≫ i) → False
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ ¬Isδ₀ (i' ≫ i)
|
simp only [h.1, add_right_inj] at eq
|
[anonymous]
|
1 = k + k' + 2
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h : Isδ₀ (i' ≫ i)
⊢ False
|
by_cases h₃ : Isδ₀ i
|
[email protected]._hyg.1409
|
Isδ₀ i → mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
⊢ mapMono K i ≫ mapMono K i' = 0
|
by_cases h₃ : Isδ₀ i
|
[email protected]._hyg.1409
|
¬Isδ₀ i → mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
pos : Isδ₀ i → mapMono K i ≫ mapMono K i' = 0
⊢ mapMono K i ≫ mapMono K i' = 0
|
by_cases h₄ : Isδ₀ i'
|
[email protected]._hyg.1437
|
Isδ₀ i' → mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h₃ : Isδ₀ i
⊢ mapMono K i ≫ mapMono K i' = 0
|
by_cases h₄ : Isδ₀ i'
|
[email protected]._hyg.1437
|
¬Isδ₀ i' → mapMono K i ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h₃ : Isδ₀ i
pos : Isδ₀ i' → mapMono K i ≫ mapMono K i' = 0
⊢ mapMono K i ≫ mapMono K i' = 0
|
mapMono_δ₀' K i h₃,
|
[email protected]._hyg.1437
|
K.d Δ.len Δ'.len ≫ mapMono K i' = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h₃ : Isδ₀ i
h₄ : Isδ₀ i'
⊢ mapMono K i ≫ mapMono K i' = 0
|
mapMono_δ₀' K i' h₄,
|
[email protected]._hyg.1437
|
K.d Δ.len Δ'.len ≫ K.d Δ'.len Δ''.len = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h₃ : Isδ₀ i
h₄ : Isδ₀ i'
⊢ K.d Δ.len Δ'.len ≫ mapMono K i' = 0
|
HomologicalComplex.d_comp_d
|
[email protected]._hyg.1437
|
0 = 0
|
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
K : ChainComplex C ℕ
Δ Δ' Δ'' : SimplexCategory
i' : Δ'' ⟶ Δ'
i : Δ' ⟶ Δ
inst✝¹ : Mono i'
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Δ' = Δ''
k : ℕ
hk : Δ.len = Δ'.len + k + 1
k' : ℕ
hk' : Δ'.len = Δ''.len + k' + 1
eq : Δ.len = Δ''.len + (k + k' + 2)
h₃ : Isδ₀ i
h₄ : Isδ₀ i'
⊢ K.d Δ.len Δ'.len ≫ K.d Δ'.len Δ''.len = 0
|
simp only [map, colimit.ι_desc, Cofan.mk_ι_app]
|
[anonymous]
|
Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Sigma.ι (summand K Δ) A ≫ map K θ = Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
have h := SimplexCategory.image_eq fac
|
[anonymous]
|
image (θ.unop ≫ A.e) = Δ'' →
Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
subst h
|
[anonymous]
|
e ≫ i = θ.unop ≫ A.e →
∀ [Mono i] [Epi e] {i_1 : image (θ.unop ≫ A.e) ⟶ unop A.fst} {e_1 : unop Δ' ⟶ image (θ.unop ≫ A.e)},
Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
h : image (θ.unop ≫ A.e) = Δ''
⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
congr
|
[email protected]._hyg.25
|
image.ι (θ.unop ≫ A.e) = i
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
e : unop Δ' ⟶ image (θ.unop ≫ A.e)
i : image (θ.unop ≫ A.e) ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
congr
|
[email protected]._hyg.33
|
A.pull θ = Splitting.IndexSet.mk e
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
e : unop Δ' ⟶ image (θ.unop ≫ A.e)
i : image (θ.unop ≫ A.e) ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
e_a.e_i : image.ι (θ.unop ≫ A.e) = i
⊢ Termwise.mapMono K (image.ι (θ.unop ≫ A.e)) ≫ Sigma.ι (summand K Δ') (A.pull θ) =
Termwise.mapMono K i ≫ Sigma.ι (summand K Δ') (Splitting.IndexSet.mk e)
|
dsimp only [SimplicialObject.Splitting.IndexSet.pull]
|
[email protected]._hyg.33
|
Splitting.IndexSet.mk (factorThruImage (θ.unop ≫ A.e)) = Splitting.IndexSet.mk e
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
e : unop Δ' ⟶ image (θ.unop ≫ A.e)
i : image (θ.unop ≫ A.e) ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ A.pull θ = Splitting.IndexSet.mk e
|
congr
|
[email protected]._hyg.33
|
factorThruImage (θ.unop ≫ A.e) = e
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
e : unop Δ' ⟶ image (θ.unop ≫ A.e)
i : image (θ.unop ≫ A.e) ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ Splitting.IndexSet.mk (factorThruImage (θ.unop ≫ A.e)) = Splitting.IndexSet.mk e
|
have fac : A.e ≫ 𝟙 A.1.unop = (𝟙 Δ).unop ≫ A.e := by rw [unop_id, comp_id, id_comp]
|
[anonymous]
|
A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e →
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ Obj.map K (𝟙 Δ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ Obj.map K (𝟙 Δ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
unop_id,
|
[anonymous]
|
A.e ≫ 𝟙 (unop A.fst) = 𝟙 (unop Δ) ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
⊢ A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
|
comp_id,
|
[anonymous]
|
A.e = 𝟙 (unop Δ) ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
⊢ A.e ≫ 𝟙 (unop A.fst) = 𝟙 (unop Δ) ≫ A.e
|
id_comp
|
[anonymous]
|
A.e = A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
⊢ A.e = 𝟙 (unop Δ) ≫ A.e
|
rw [Obj.map_on_summand₀ K A fac, Obj.Termwise.mapMono_id, id_comp]
|
[anonymous]
|
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ Obj.map K (𝟙 Δ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
Obj.map_on_summand₀ K A fac,
|
[anonymous]
|
Obj.Termwise.mapMono K (𝟙 (unop A.fst)) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ Obj.map K (𝟙 Δ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
Obj.Termwise.mapMono_id,
|
[anonymous]
|
𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ Obj.Termwise.mapMono K (𝟙 (unop A.fst)) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
id_comp
|
[anonymous]
|
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ 𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
dsimp only [Obj.obj₂]
|
[anonymous]
|
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (∐ fun A ↦ Obj.summand K Δ A)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (Obj.obj₂ K Δ)
|
rw [comp_id]
|
[anonymous]
|
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A }
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (∐ fun A ↦ Obj.summand K Δ A)
|
comp_id
|
[anonymous]
|
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A }
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
x✝ : Discrete (Splitting.IndexSet Δ)
A : Splitting.IndexSet Δ
fac : A.e ≫ 𝟙 (unop A.fst) = (𝟙 Δ).unop ≫ A.e
⊢ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ A) { as := A } ≫ 𝟙 (∐ fun A ↦ Obj.summand K Δ A)
|
have fac : θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e := by rw [unop_comp, assoc]
|
[anonymous]
|
θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e →
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
unop_comp,
|
[anonymous]
|
θ.unop ≫ θ'.unop ≫ A.e = (θ.unop ≫ θ'.unop) ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
⊢ θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e
|
assoc
|
[anonymous]
|
θ.unop ≫ θ'.unop ≫ A.e = θ.unop ≫ θ'.unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
⊢ θ.unop ≫ θ'.unop ≫ A.e = (θ.unop ≫ θ'.unop) ≫ A.e
|
rw [← image.fac (θ'.unop ≫ A.e), ← assoc, ←
image.fac (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)), assoc] at fac
|
[anonymous]
|
factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫
image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e) =
(θ' ≫ θ).unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac : θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
← image.fac (θ'.unop ≫ A.e),
|
[anonymous]
|
θ.unop ≫ factorThruImage (θ'.unop ≫ A.e) ≫ image.ι (θ'.unop ≫ A.e) = (θ' ≫ θ).unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac : θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
← assoc,
|
[anonymous]
|
(θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e) = (θ' ≫ θ).unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac : θ.unop ≫ factorThruImage (θ'.unop ≫ A.e) ≫ image.ι (θ'.unop ≫ A.e) = (θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
←
image.fac (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)),
|
[anonymous]
|
(factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e))) ≫
image.ι (θ'.unop ≫ A.e) =
(θ' ≫ θ).unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac : (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e) = (θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
assoc
|
[anonymous]
|
factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫
image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e) =
(θ' ≫ θ).unop ≫ A.e
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac :
(factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e))) ≫
image.ι (θ'.unop ≫ A.e) =
(θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
simp only [Obj.map_on_summand₀'_assoc K A θ', Obj.map_on_summand₀' K _ θ,
Obj.Termwise.mapMono_comp_assoc, Obj.map_on_summand₀ K A fac]
|
[anonymous]
|
Obj.Termwise.mapMono K (image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e)) ≫
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk (factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)))) =
Obj.Termwise.mapMono K (image.ι (θ.unop ≫ (A.pull θ').e) ≫ image.ι (θ'.unop ≫ A.e)) ≫
Sigma.ι (Obj.summand K Δ) ((A.pull θ').pull θ)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
fac :
factorThruImage (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫
image.ι (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)) ≫ image.ι (θ'.unop ≫ A.e) =
(θ' ≫ θ).unop ≫ A.e
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ
|
intro A
|
[anonymous]
|
∀ (A_1 : Splitting.IndexSet Δ),
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A ≫
(Iso.refl (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))).pt).hom =
(Splitting.cofan' (fun n ↦ K.X n) (obj K)
(fun n ↦ Sigma.ι (Obj.summand K (op ⦋n⦌)) (Splitting.IndexSet.id (op ⦋n⦌))) Δ).inj
A
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
⊢ ∀ (b : Splitting.IndexSet Δ),
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) b ≫
(Iso.refl (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))).pt).hom =
(Splitting.cofan' (fun n ↦ K.X n) (obj K)
(fun n ↦ Sigma.ι (Obj.summand K (op ⦋n⦌)) (Splitting.IndexSet.id (op ⦋n⦌))) Δ).inj
b
|
dsimp [Splitting.cofan']
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A ≫
𝟙 (colimit (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) =
Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K A.e.op
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A ≫
(Iso.refl (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))).pt).hom =
(Splitting.cofan' (fun n ↦ K.X n) (obj K)
(fun n ↦ Sigma.ι (Obj.summand K (op ⦋n⦌)) (Splitting.IndexSet.id (op ⦋n⦌))) Δ).inj
A
|
rw [comp_id, Γ₀.Obj.map_on_summand₀ K (SimplicialObject.Splitting.IndexSet.id A.1)
(show A.e ≫ 𝟙 _ = A.e.op.unop ≫ 𝟙 _ by rfl), Γ₀.Obj.Termwise.mapMono_id]
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A ≫
𝟙 (colimit (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) =
Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K A.e.op
|
comp_id,
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K A.e.op
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A ≫
𝟙 (colimit (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) =
Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K A.e.op
|
Γ₀.Obj.map_on_summand₀ K (SimplicialObject.Splitting.IndexSet.id A.1)
(show A.e ≫ 𝟙 _ = A.e.op.unop ≫ 𝟙 _ by rfl),
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Obj.Termwise.mapMono K (𝟙 (unop A.fst)) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Sigma.ι (Obj.summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ Obj.map K A.e.op
|
Γ₀.Obj.Termwise.mapMono_id
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Obj.Termwise.mapMono K (𝟙 (unop A.fst)) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
rw [id_comp]
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
id_comp
|
[anonymous]
|
Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ' Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summand (fun n ↦ K.X n) Δ))) A =
𝟙 (K.X (unop A.fst).len) ≫ Sigma.ι (Obj.summand K Δ) (Splitting.IndexSet.mk A.e)
|
dsimp [Splitting.cofan]
|
[anonymous]
|
((splitting K).ι (unop A.fst).len ≫ map K A.e.op) ≫ map K θ =
Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K : ChainComplex C ℕ
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ ((splitting K).cofan Δ).inj A ≫ (obj K).map θ =
Termwise.mapMono K i ≫ ((splitting K).cofan Δ').inj (Splitting.IndexSet.mk e)
|
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