feat(connectedness): is_conn_map -> is_conn_fun, and unbundle the P in elimination principles
This commit is contained in:
Floris van Doorn
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1e10810a1e
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003c11c917
3 changed files with 53 additions and 54 deletions
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@ -21,7 +21,7 @@ namespace homotopy
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assumption
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end
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definition is_conn_map [reducible] (n : ℕ₋₂) {A B : Type} (f : A → B) : Type :=
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definition is_conn_fun [reducible] (n : ℕ₋₂) {A B : Type} (f : A → B) : Type :=
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Πb : B, is_conn n (fiber f b)
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theorem is_conn_of_le (A : Type) {n k : ℕ₋₂} (H : n ≤ k) [is_conn k A] : is_conn n A :=
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@ -30,14 +30,14 @@ namespace homotopy
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apply trunc_trunc_equiv_left _ n k H
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end
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theorem is_conn_map_of_le {A B : Type} (f : A → B) {n k : ℕ₋₂} (H : n ≤ k)
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[is_conn_map k f] : is_conn_map n f :=
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theorem is_conn_fun_of_le {A B : Type} (f : A → B) {n k : ℕ₋₂} (H : n ≤ k)
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[is_conn_fun k f] : is_conn_fun n f :=
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λb, is_conn_of_le _ H
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namespace is_conn_map
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namespace is_conn_fun
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section
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parameters {n : ℕ₋₂} {A B : Type} {h : A → B}
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(H : is_conn_map n h) (P : B → n -Type)
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parameters (n : ℕ₋₂) {A B : Type} {h : A → B}
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(H : is_conn_fun n h) (P : B → Type) [Πb, is_trunc n (P b)]
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private definition rec.helper : (Πa : A, P (h a)) → Πb : B, trunc n (fiber h b) → P b :=
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λt b, trunc.rec (λx, point_eq x ▸ t (point x))
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@ -67,16 +67,16 @@ namespace homotopy
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end
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section
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parameters {n k : ℕ₋₂} {A B : Type} {f : A → B}
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(H : is_conn_map n f) (P : B → (n +2+ k)-Type)
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parameters (n k : ℕ₋₂) {A B : Type} {f : A → B}
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(H : is_conn_fun n f) (P : B → Type) [HP : Πb, is_trunc (n +2+ k) (P b)]
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include H
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include H HP
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-- Lemma 8.6.1
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proposition elim_general : is_trunc_fun k (pi_functor_left f P) :=
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begin
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intro t,
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induction k with k IH,
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{ apply is_contr_fiber_of_is_equiv, apply is_conn_map.rec, exact H },
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revert P HP,
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induction k with k IH: intro P HP t,
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{ apply is_contr_fiber_of_is_equiv, apply is_conn_fun.rec, exact H, exact HP},
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{ apply is_trunc_succ_intro,
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intros x y, cases x with g p, cases y with h q,
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have e : fiber (λr : g ~ h, (λa, r (f a))) (apd10 (p ⬝ q⁻¹))
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@ -104,16 +104,14 @@ namespace homotopy
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apply eq_equiv_eq_symm
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end,
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apply @is_trunc_equiv_closed _ _ k e, clear e,
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apply IH (λb : B, trunctype.mk (g b = h b)
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(@is_trunc_eq (P b) (n +2+ k) (trunctype.struct (P b))
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(g b) (h b))) }
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apply IH (λb : B, (g b = h b)) (λb, @is_trunc_eq (P b) (n +2+ k) (HP b) (g b) (h b))}
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end
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end
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section
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universe variables u v
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parameters {n : ℕ₋₂} {A : Type.{u}} {B : Type.{v}} {h : A → B}
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parameters (n : ℕ₋₂) {A : Type.{u}} {B : Type.{v}} {h : A → B}
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parameter sec : ΠP : B → trunctype.{max u v} n,
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is_retraction (λs : (Πb : B, P b), λ a, s (h a))
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@ -122,7 +120,7 @@ namespace homotopy
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include sec
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-- the other half of Lemma 7.5.7
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definition intro : is_conn_map n h :=
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definition intro : is_conn_fun n h :=
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begin
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intro b,
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apply is_contr.mk (@is_retraction.sect _ _ _ s (λa, tr (fiber.mk a idp)) b),
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@ -134,22 +132,22 @@ namespace homotopy
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exact apd10 (@right_inverse _ _ _ s (λa, tr (fiber.mk a idp))) a
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end
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end
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end is_conn_map
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end is_conn_fun
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-- Connectedness is related to maps to and from the unit type, first to
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section
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parameters (n : ℕ₋₂) (A : Type)
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definition is_conn_of_map_to_unit
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: is_conn_map n (const A unit.star) → is_conn n A :=
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: is_conn_fun n (const A unit.star) → is_conn n A :=
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begin
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intro H, unfold is_conn_map at H,
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intro H, unfold is_conn_fun at H,
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rewrite [-(ua (fiber.fiber_star_equiv A))],
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exact (H unit.star)
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end
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-- now maps from unit
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definition is_conn_of_map_from_unit (a₀ : A) (H : is_conn_map n (const unit a₀))
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definition is_conn_of_map_from_unit (a₀ : A) (H : is_conn_fun n (const unit a₀))
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: is_conn n .+1 A :=
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is_contr.mk (tr a₀)
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begin
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@ -158,8 +156,8 @@ namespace homotopy
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(@center _ (H a))
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end
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definition is_conn_map_from_unit (a₀ : A) [H : is_conn n .+1 A]
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: is_conn_map n (const unit a₀) :=
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definition is_conn_fun_from_unit (a₀ : A) [H : is_conn n .+1 A]
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: is_conn_fun n (const unit a₀) :=
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begin
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intro a,
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apply is_conn_equiv_closed n (equiv.symm (fiber_const_equiv A a₀ a)),
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@ -172,15 +170,15 @@ namespace homotopy
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namespace is_conn
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open pointed unit
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section
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parameters {n : ℕ₋₂} {A : Type*}
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[H : is_conn n .+1 A] (P : A → n-Type)
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parameters (n : ℕ₋₂) {A : Type*}
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[H : is_conn n .+1 A] (P : A → Type) [Πa, is_trunc n (P a)]
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include H
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protected definition rec : is_equiv (λs : Πa : A, P a, s (Point A)) :=
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@is_equiv_compose
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(Πa : A, P a) (unit → P (Point A)) (P (Point A))
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(λs x, s (Point A)) (λf, f unit.star)
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(is_conn_map.rec (is_conn_map_from_unit n A (Point A)) P)
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(is_conn_fun.rec n (is_conn_fun_from_unit n A (Point A)) P)
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(to_is_equiv (arrow_unit_left (P (Point A))))
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protected definition elim : P (Point A) → (Πa : A, P a) :=
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@ -191,8 +189,8 @@ namespace homotopy
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end
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section
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parameters {n k : ℕ₋₂} {A : Type*}
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[H : is_conn n .+1 A] (P : A → (n +2+ k)-Type)
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parameters (n k : ℕ₋₂) {A : Type*}
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[H : is_conn n .+1 A] (P : A → Type) [Πa, is_trunc (n +2+ k) (P a)]
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include H
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proposition elim_general (p : P (Point A))
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@ -202,20 +200,20 @@ namespace homotopy
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(fiber (λs, s (Point A)) p)
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k
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(equiv.symm (fiber.equiv_postcompose (to_fun (arrow_unit_left (P (Point A))))))
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(is_conn_map.elim_general (is_conn_map_from_unit n A (Point A)) P (λx, p))
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(is_conn_fun.elim_general n k (is_conn_fun_from_unit n A (Point A)) P (λx, p))
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end
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end is_conn
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-- Lemma 7.5.2
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definition minus_one_conn_of_surjective {A B : Type} (f : A → B)
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: is_surjective f → is_conn_map -1 f :=
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: is_surjective f → is_conn_fun -1 f :=
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begin
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intro H, intro b,
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exact @is_contr_of_inhabited_prop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
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end
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definition is_surjection_of_minus_one_conn {A B : Type} (f : A → B)
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: is_conn_map -1 f → is_surjective f :=
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: is_conn_fun -1 f → is_surjective f :=
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begin
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intro H, intro b,
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exact @center (∥fiber f b∥) (H b),
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@ -234,7 +232,7 @@ namespace homotopy
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-- Lemma 7.5.4
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definition retract_of_conn_is_conn [instance] (r : arrow_hom f g) [H : is_retraction r]
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(n : ℕ₋₂) [K : is_conn_map n f] : is_conn_map n g :=
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(n : ℕ₋₂) [K : is_conn_fun n f] : is_conn_fun n g :=
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begin
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intro b, unfold is_conn,
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apply is_contr_retract (trunc_functor n (retraction_on_fiber r b)),
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@ -245,7 +243,7 @@ namespace homotopy
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-- Corollary 7.5.5
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definition is_conn_homotopy (n : ℕ₋₂) {A B : Type} {f g : A → B}
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(p : f ~ g) (H : is_conn_map n f) : is_conn_map n g :=
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(p : f ~ g) (H : is_conn_fun n f) : is_conn_fun n g :=
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@retract_of_conn_is_conn _ _ (arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H
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-- all types are -2-connected
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@ -274,8 +272,8 @@ namespace homotopy
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{ intros H,
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change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
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generalize a',
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apply is_conn_map.elim
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(is_conn_map_from_unit n A a)
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apply is_conn_fun.elim n
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(is_conn_fun_from_unit n A a)
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(λx : A, trunctype.mk' n (ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid x))),
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intros,
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change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a),
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@ -285,8 +283,8 @@ namespace homotopy
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end
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-- Lemma 7.5.14
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theorem is_equiv_trunc_functor_of_is_conn_map {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_map n f] : is_equiv (trunc_functor n f) :=
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theorem is_equiv_trunc_functor_of_is_conn_fun {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_fun n f] : is_equiv (trunc_functor n f) :=
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begin
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fapply adjointify,
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{ intro b, induction b with b, exact trunc_functor n point (center (trunc n (fiber f b)))},
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@ -295,9 +293,9 @@ namespace homotopy
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{ intro a, induction a with a, esimp, rewrite [center_eq (tr (fiber.mk a idp))]}
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end
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theorem trunc_equiv_trunc_of_is_conn_map {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_map n f] : trunc n A ≃ trunc n B :=
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equiv.mk (trunc_functor n f) (is_equiv_trunc_functor_of_is_conn_map n f)
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theorem trunc_equiv_trunc_of_is_conn_fun {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_fun n f] : trunc n A ≃ trunc n B :=
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equiv.mk (trunc_functor n f) (is_equiv_trunc_functor_of_is_conn_fun n f)
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open trunc_index pointed sphere.ops
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-- Corollary 8.2.2
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@ -39,13 +39,12 @@ namespace is_trunc
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cases n with n,
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{ exfalso, apply not_lt_zero, exact H},
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{ have H2 : k ≤ n, from le_of_lt_succ H,
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apply @(trivial_homotopy_group_of_is_conn _ H2),
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rewrite [-trunc_index.of_sphere_index_of_nat, -trunc_index.succ_sub_one], apply is_conn_sphere}
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apply @(trivial_homotopy_group_of_is_conn _ H2)}
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end
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end
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theorem is_contr_HG_fiber_of_is_connected {A B : Type*} (k n : ℕ) (f : A →* B)
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[H : is_conn_map n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
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[H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
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@(trivial_homotopy_group_of_is_conn (pfiber f) H2) (H pt)
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@ -36,22 +36,24 @@ section
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-- The wedge connectivity lemma (Lemma 8.6.2)
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parameters {A B : Type*} (n m : ℕ)
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[cA : is_conn n A] [cB : is_conn m B]
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(P : A → B → (m + n)-Type)
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(P : A → B → Type) [HP : Πa b, is_trunc (m + n) (P a b)]
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(f : Πa : A, P a pt)
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(g : Πb : B, P pt b)
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(p : f pt = g pt)
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include cA cB
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private definition Q (a : A) : (n.-1)-Type :=
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trunctype.mk
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(fiber (λs : (Πb : B, P a b), s (Point B)) (f a))
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abstract begin
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refine @is_conn.elim_general (m.-1) _ _ _ (λb, trunctype.mk (P a b) _) (f a),
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rewrite [-succ_add_succ, of_nat_add_of_nat], intro b, apply trunctype.struct
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end end
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include cA cB HP
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private definition Q (a : A) : Type :=
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fiber (λs : (Πb : B, P a b), s (Point B)) (f a)
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private definition is_trunc_Q (a : A) : is_trunc (n.-1) (Q a) :=
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begin
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refine @is_conn.elim_general (m.-1) _ _ _ (P a) _ (f a),
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rewrite [-succ_add_succ, of_nat_add_of_nat], intro b, apply HP
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end
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local attribute is_trunc_Q [instance]
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private definition Q_sec : Πa : A, Q a :=
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is_conn.elim Q (fiber.mk g p⁻¹)
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is_conn.elim (n.-1) Q (fiber.mk g p⁻¹)
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protected definition ext : Π(a : A)(b : B), P a b :=
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λa, fiber.point (Q_sec a)
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@ -62,7 +64,7 @@ section
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private definition coh_aux : Σq : ext (Point A) = g,
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β_left (Point A) = ap (λs : (Πb : B, P (Point A) b), s (Point B)) q ⬝ p⁻¹ :=
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equiv.to_fun (fiber.fiber_eq_equiv (Q_sec (Point A)) (fiber.mk g p⁻¹))
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(is_conn.elim_β Q (fiber.mk g p⁻¹))
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(is_conn.elim_β (n.-1) Q (fiber.mk g p⁻¹))
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protected definition β_right (b : B) : ext (Point A) b = g b :=
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apd10 (sigma.pr1 coh_aux) b
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