small changes in colimit
This commit is contained in:
Floris van Doorn
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91c21e9d92
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d4ab6e15ef
4 changed files with 88 additions and 34 deletions
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@ -11,6 +11,9 @@ namespace seq_colim
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definition pseq_colim [constructor] {X : ℕ → Type*} (f : pseq_diagram X) : Type* :=
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pointed.MK (seq_colim f) (@sι _ _ 0 pt)
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variables {A A' : ℕ → Type*} {f : pseq_diagram A} {f' : pseq_diagram A'}
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{τ : Πn, A n →* A' n} {H : Πn, τ (n+1) ∘* f n ~* f' n ∘* τ n}
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definition inclusion_pt {X : ℕ → Type*} (f : pseq_diagram X) (n : ℕ)
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: inclusion f (Point (X n)) = Point (pseq_colim f) :=
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begin
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@ -162,47 +165,45 @@ namespace seq_colim
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theorem prep0_succ_lemma {A : ℕ → Type*} (f : pseq_diagram A) (n : ℕ)
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(p : rep0 (λn x, f n x) n pt = rep0 (λn x, f n x) n pt)
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(q : prep0 f n (Point (A 0)) = Point (A n))
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: loop_equiv_eq_closed (ap (@f n) q ⬝ respect_pt (@f n))
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(ap (@f n) p) = Ω→(@f n) (loop_equiv_eq_closed q p) :=
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by rewrite [▸*, con_inv, ↑ap1_gen, +ap_con, ap_inv, +con.assoc]
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variables {A : ℕ → Type} (f : seq_diagram A)
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definition succ_add_tr_rep {n : ℕ} (k : ℕ) (x : A n)
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: transport A (succ_add n k) (rep f k (f x)) = rep f (succ k) x :=
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begin
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induction k with k p,
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reflexivity,
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exact tr_ap A succ (succ_add n k) _ ⬝ (fn_tr_eq_tr_fn (succ_add n k) f _)⁻¹ ⬝ ap (@f _) p,
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end
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-- definition succ_add_tr_rep {A : ℕ → Type} (f : seq_diagram A) {n : ℕ} (k : ℕ) (x : A n)
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-- : transport A (succ_add n k) (rep f k (f x)) = rep f (succ k) x :=
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-- begin
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-- induction k with k p,
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-- reflexivity,
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-- exact tr_ap A succ (succ_add n k) _ ⬝ (fn_tr_eq_tr_fn (succ_add n k) f _)⁻¹ ⬝ ap (@f _) p,
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-- end
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definition succ_add_tr_rep_succ {n : ℕ} (k : ℕ) (x : A n)
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: succ_add_tr_rep f (succ k) x = tr_ap A succ (succ_add n k) _ ⬝
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(fn_tr_eq_tr_fn (succ_add n k) f _)⁻¹ ⬝ ap (@f _) (succ_add_tr_rep f k x) :=
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by reflexivity
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-- definition succ_add_tr_rep_succ {A : ℕ → Type} (f : seq_diagram A) {n : ℕ} (k : ℕ) (x : A n)
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-- : succ_add_tr_rep f (succ k) x = tr_ap A succ (succ_add n k) _ ⬝
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-- (fn_tr_eq_tr_fn (succ_add n k) f _)⁻¹ ⬝ ap (@f _) (succ_add_tr_rep f k x) :=
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-- by reflexivity
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definition code_glue_equiv [constructor] {n : ℕ} (k : ℕ) (x y : A n)
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: rep f k (f x) = rep f k (f y) ≃ rep f (succ k) x = rep f (succ k) y :=
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begin
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refine eq_equiv_fn_eq_of_equiv (equiv_ap A (succ_add n k)) _ _ ⬝e _,
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apply eq_equiv_eq_closed,
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exact succ_add_tr_rep f k x,
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exact succ_add_tr_rep f k y
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end
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-- definition code_glue_equiv [constructor] {A : ℕ → Type} (f : seq_diagram A) {n : ℕ} (k : ℕ) (x y : A n)
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-- : rep f k (f x) = rep f k (f y) ≃ rep f (succ k) x = rep f (succ k) y :=
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-- begin
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-- refine eq_equiv_fn_eq_of_equiv (equiv_ap A (succ_add n k)) _ _ ⬝e _,
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-- apply eq_equiv_eq_closed,
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-- exact succ_add_tr_rep f k x,
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-- exact succ_add_tr_rep f k y
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-- end
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theorem code_glue_equiv_ap {n : ℕ} {k : ℕ} {x y : A n} (p : rep f k (f x) = rep f k (f y))
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: code_glue_equiv f (succ k) x y (ap (@f _) p) = ap (@f _) (code_glue_equiv f k x y p) :=
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begin
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rewrite [▸*, +ap_con, ap_inv, +succ_add_tr_rep_succ, con_inv, inv_con_inv_right, +con.assoc],
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apply whisker_left,
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rewrite [- +con.assoc], apply whisker_right, rewrite [- +ap_compose'],
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note s := (eq_top_of_square (natural_square_tr
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(λx, fn_tr_eq_tr_fn (succ_add n k) f x ⬝ (tr_ap A succ (succ_add n k) (f x))⁻¹) p))⁻¹ᵖ,
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rewrite [inv_con_inv_right at s, -con.assoc at s], exact s
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end
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-- theorem code_glue_equiv_ap {n : ℕ} {k : ℕ} {x y : A n} (p : rep f k (f x) = rep f k (f y))
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-- : code_glue_equiv f (succ k) x y (ap (@f _) p) = ap (@f _) (code_glue_equiv f k x y p) :=
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-- begin
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-- rewrite [▸*, +ap_con, ap_inv, +succ_add_tr_rep_succ, con_inv, inv_con_inv_right, +con.assoc],
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-- apply whisker_left,
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-- rewrite [- +con.assoc], apply whisker_right, rewrite [- +ap_compose'],
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-- note s := (eq_top_of_square (natural_square_tr
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-- (λx, fn_tr_eq_tr_fn (succ_add n k) f x ⬝ (tr_ap A succ (succ_add n k) (f x))⁻¹) p))⁻¹ᵖ,
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-- rewrite [inv_con_inv_right at s, -con.assoc at s], exact s
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-- end
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definition pseq_colim_loop {X : ℕ → Type*} (f : Πn, X n →* X (n+1)) :
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Ω (pseq_colim f) ≃* pseq_colim (λn, Ω→(f n)) :=
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Ω (pseq_colim f) ≃* pseq_colim (λn, Ω→ (f n)) :=
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begin
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fapply pequiv_of_equiv,
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{ refine !seq_colim_eq_equiv0 ⬝e _,
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@ -216,6 +217,10 @@ namespace seq_colim
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pseq_colim_loop f ∘* Ω→ (pinclusion f n) ~* pinclusion (λn, Ω→(f n)) n :=
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sorry
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definition pseq_colim_loop_natural (n : ℕ) : psquare (pseq_colim_loop f) (pseq_colim_loop f')
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(Ω→ (pseq_colim_functor τ H)) (pseq_colim_functor (λn, Ω→ (τ n)) (λn, ap1_psquare (H n))) :=
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sorry
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definition pseq_diagram_pfiber {A A' : ℕ → Type*} {f : pseq_diagram A} {f' : pseq_diagram A'}
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(g : Πn, A n →* A' n) (p : Πn, g (succ n) ∘* f n ~* f' n ∘* g n) :
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pseq_diagram (λk, pfiber (g k)) :=
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@ -602,6 +602,7 @@ begin
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exact (g_star_path_right_step g e w k x @(g_star_path_right g e w k)).2 }}
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end
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/- We now define the map back, and show using this induction principle that the composites are the identity -/
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variable {P}
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@ -742,6 +743,13 @@ begin
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intro n p, refine whisker_right _ (!lrep_irrel2⁻² ⬝ !ap_inv⁻¹) ⬝ !ap_con⁻¹ }
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end
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-- definition seq_colim_eq_equiv0'_natural {a₀ a₁ : A 0} {a₀' a₁' : A' 0} (p₀ : τ a₀ = a₀')
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-- (p₁ : τ a₁ = a₁') :
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-- hsquare (seq_colim_eq_equiv0' f a₀ a₁) (seq_colim_eq_equiv0' f' a₀' a₁')
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-- (pointed.ap1_gen (seq_colim_functor τ p) (ap (ι' f' 0) p₀) (ap (ι' f' 0) p₁))
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-- (seq_colim_functor (λn, pointed.ap1_gen (@τ _)) _) :=
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-- _
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definition seq_colim_eq_equiv0 (a₀ a₁ : A 0) : ι f a₀ = ι f a₁ ≃ seq_colim (id0_seq_diagram f a₀ a₁) :=
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seq_colim_eq_equiv0' f a₀ a₁ ⬝e seq_colim_id_equiv_seq_colim_id0 f a₀ a₁
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@ -152,7 +152,7 @@ namespace seq_colim
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{ apply pathover_ap, exact apo f IH}
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end
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variable {f}
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variables {f f'}
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definition is_trunc_fun_lrep (k : ℕ₋₂) (H : n ≤ m) (H2 : Πn, is_trunc_fun k (@f n)) :
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is_trunc_fun k (lrep f H) :=
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begin induction H with m H IH, apply is_trunc_fun_id, exact is_trunc_fun_compose k (H2 m) IH end
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@ -161,7 +161,21 @@ namespace seq_colim
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is_conn_fun k (lrep f H) :=
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begin induction H with m H IH, apply is_conn_fun_id, exact is_conn_fun_compose k (H2 m) IH end
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variable (f)
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definition lrep_natural (τ : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), τ (f a) = f' (τ a))
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{n m : ℕ} (H : n ≤ m) (a : A n) : τ (lrep f H a) = lrep f' H (τ a) :=
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begin
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induction H with m H IH, reflexivity, exact p (lrep f H a) ⬝ ap (@f' m) IH
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end
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definition rep_natural (τ : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), τ (f a) = f' (τ a))
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{n : ℕ} (k : ℕ) (a : A n) : τ (rep f k a) = rep f' k (τ a) :=
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lrep_natural τ p _ a
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definition rep0_natural (τ : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), τ (f a) = f' (τ a))
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(k : ℕ) (a : A 0) : τ (rep0 f k a) = rep0 f' k (τ a) :=
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lrep_natural τ p _ a
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variables (f f')
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end generalized_lrep
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@ -732,6 +732,33 @@ by induction q'; reflexivity
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-- end
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--rexact @(ap pathover_pr1) _ idpo _,
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definition sigma_functor_compose {A A' A'' : Type} {B : A → Type} {B' : A' → Type}
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{B'' : A'' → Type} {f' : A' → A''} {f : A → A'} (g' : Πa, B' a → B'' (f' a))
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(g : Πa, B a → B' (f a)) (x : Σa, B a) :
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sigma_functor f' g' (sigma_functor f g x) = sigma_functor (f' ∘ f) (λa, g' (f a) ∘ g a) x :=
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begin
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reflexivity
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end
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definition sigma_functor_homotopy {A A' : Type} {B : A → Type} {B' : A' → Type}
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{f f' : A → A'} {g : Πa, B a → B' (f a)} {g' : Πa, B a → B' (f' a)} (h : f ~ f')
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(k : Πa b, g a b =[h a] g' a b) (x : Σa, B a) : sigma_functor f g x = sigma_functor f' g' x :=
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sigma_eq (h x.1) (k x.1 x.2)
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variables {A₀₀ A₂₀ A₀₂ A₂₂ : Type}
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{B₀₀ : A₀₀ → Type} {B₂₀ : A₂₀ → Type} {B₀₂ : A₀₂ → Type} {B₂₂ : A₂₂ → Type}
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{f₁₀ : A₀₀ → A₂₀} {f₁₂ : A₀₂ → A₂₂} {f₀₁ : A₀₀ → A₀₂} {f₂₁ : A₂₀ → A₂₂}
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{g₁₀ : Πa, B₀₀ a → B₂₀ (f₁₀ a)} {g₁₂ : Πa, B₀₂ a → B₂₂ (f₁₂ a)}
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{g₀₁ : Πa, B₀₀ a → B₀₂ (f₀₁ a)} {g₂₁ : Πa, B₂₀ a → B₂₂ (f₂₁ a)}
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definition sigma_functor_hsquare (h : hsquare f₁₀ f₁₂ f₀₁ f₂₁)
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(k : Πa (b : B₀₀ a), g₂₁ _ (g₁₀ _ b) =[h a] g₁₂ _ (g₀₁ _ b)) :
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hsquare (sigma_functor f₁₀ g₁₀) (sigma_functor f₁₂ g₁₂)
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(sigma_functor f₀₁ g₀₁) (sigma_functor f₂₁ g₂₁) :=
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λx, sigma_functor_compose g₂₁ g₁₀ x ⬝
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sigma_functor_homotopy h k x ⬝
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(sigma_functor_compose g₁₂ g₀₁ x)⁻¹
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end sigma open sigma
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namespace group
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