minor changes in colimit
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Floris van Doorn
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cca1279f45
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1a35543661
2 changed files with 36 additions and 32 deletions
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@ -37,17 +37,15 @@ begin
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rexact ap (lrep_back f (zero_le k)) (left_inv (@f k) a),
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end
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set_option pp.binder_types true
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section
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variable {f}
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local attribute is_equiv_lrep [instance] --[priority 500]
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definition is_equiv_ι (H : is_equiseq f) : is_equiv (ι' f 0) :=
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definition is_equiv_inclusion0 (H : is_equiseq f) : is_equiv (ι' f 0) :=
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begin
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fapply adjointify,
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{ exact colim_back f},
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{ intro x, induction x with k a k a,
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{ esimp,
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refine (lrep_glue f (zero_le k) (lrep_back f (zero_le k) a))⁻¹ ⬝ _,
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{ refine (lrep_glue f (zero_le k) (lrep_back f (zero_le k) a))⁻¹ ⬝ _,
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exact ap (ι f) (right_inv (lrep f (zero_le k)) a)},
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apply eq_pathover_id_right,
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refine (ap_compose (ι f) (colim_back f) _) ⬝ph _,
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@ -55,19 +53,23 @@ begin
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{ rexact elim_glue f _ _ a},
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refine _ ⬝pv ((natural_square (lrep_glue f (zero_le k))
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(ap (lrep_back f (zero_le k)) (left_inv (@f k) a)))⁻¹ʰ ⬝h _),
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{ exact (glue f (lrep f (zero_le k) (lrep_back f (zero_le (succ k)) (f a))))⁻¹ ⬝
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ap (ι f) (right_inv (lrep f (zero_le (succ k))) (f a))},
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{ rewrite [-con.assoc, -con_inv]},
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{ exact (glue f _)⁻¹ ⬝ ap (ι f) (right_inv (lrep f (zero_le (succ k))) (f a)) },
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{ rewrite [-con.assoc, -con_inv] },
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refine !ap_compose⁻¹ ⬝ ap_compose (ι f) _ _ ⬝ph _,
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refine dconcat (aps (ι' f k) (natural_square (right_inv (lrep f (zero_le k)))
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(left_inv (@f _) a))) _,
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apply move_top_of_left, apply move_left_of_bot,
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refine ap02 _ (whisker_left _ (adj (@f _) a)) ⬝pv _,
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rewrite [-+ap_con, -ap_compose', ap_id],
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apply natural_square_tr},
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apply natural_square_tr },
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intro a,
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reflexivity,
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end
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definition equiv_of_is_equiseq [constructor] (H : is_equiseq f) : seq_colim f ≃ A 0 :=
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(equiv.mk _ (is_equiv_inclusion0 H))⁻¹ᵉ
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variable (f)
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end
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section
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@ -95,7 +97,7 @@ begin
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end
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omit p
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theorem seq_colim_functor_glue {n : ℕ} (a : A n)
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theorem seq_colim_functor_glue {n : ℕ} (a : A n)
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: ap (seq_colim_functor τ p) (glue f a) = ap (ι f') (p a) ⬝ glue f' (τ a) :=
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!elim_glue
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@ -113,8 +115,7 @@ end
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variable (f)
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definition seq_colim_functor_id [constructor] (x : seq_colim f) :
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seq_colim_functor (λn, id) (λn, homotopy.rfl) x =
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x :=
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seq_colim_functor (λn, id) (λn, homotopy.rfl) x = x :=
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begin
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induction x, reflexivity,
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apply eq_pathover, apply hdeg_square,
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@ -125,7 +126,7 @@ variables {f τ τ₂ p p₂}
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definition seq_colim_functor_homotopy [constructor] (q : τ ~2 τ₂)
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(r : Π⦃n⦄ (a : A n), square (q (n+1) (f a)) (ap (@f' n) (q n a)) (p a) (p₂ a))
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(x : seq_colim f) :
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seq_colim_functor τ p x = seq_colim_functor τ₂ p₂ x :=
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seq_colim_functor τ p x = seq_colim_functor τ₂ p₂ x :=
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begin
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induction x,
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exact ap (ι f') (q n a),
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@ -314,9 +315,6 @@ definition kshift_equiv [constructor] (k : ℕ) : seq_colim f ≃ seq_colim (ksh
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-- end
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variable {f}
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definition equiv_of_is_equiseq [constructor] (H : is_equiseq f) : seq_colim f ≃ A 0 :=
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(equiv.mk _ (is_equiv_ι _ H))⁻¹ᵉ
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definition seq_colim_constant_seq [constructor] (X : Type) : seq_colim (constant_seq X) ≃ X :=
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equiv_of_is_equiseq (λn, !is_equiv_id)
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@ -325,9 +323,13 @@ definition is_contr_seq_colim {A : ℕ → Type} (f : seq_diagram A)
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[Πk, is_contr (A k)] : is_contr (seq_colim f) :=
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begin
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apply @is_trunc_is_equiv_closed (A 0),
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apply is_equiv_ι, intro n, apply is_equiv_of_is_contr
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apply is_equiv_inclusion0, intro n, apply is_equiv_of_is_contr
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end
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definition seq_colim_equiv_of_is_equiv [constructor] {n : ℕ} (H : Πk, k ≥ n → is_equiv (@f k)) :
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seq_colim f ≃ A n :=
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kshift_equiv f n ⬝e equiv_of_is_equiseq (λk, H (n+k) !le_add_right)
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/- colimits of dependent sequences, sigma's commute with colimits -/
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section over
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@ -697,8 +699,8 @@ equiv.MK (colim_sigma_of_sigma_colim g)
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end over
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definition seq_colim_id_equiv_seq_colim_id0 (x y : A 0) :
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seq_colim (id_seq_diagram f 0 x y) ≃ seq_colim (id0_seq_diagram f x y) :=
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definition seq_colim_id_equiv_seq_colim_id0 (a₀ a₁ : A 0) :
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seq_colim (id_seq_diagram f 0 a₀ a₁) ≃ seq_colim (id0_seq_diagram f a₀ a₁) :=
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seq_colim_equiv
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(λn, !lrep_eq_lrep_irrel (nat.zero_add n))
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(λn p, !lrep_eq_lrep_irrel_natural)
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@ -719,11 +721,13 @@ definition incl_kshift_diag0 {n : ℕ} (x : A n) :
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ι' (kshift_diag f n) 0 x = kshift_equiv f n (ι f x) :=
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incl_kshift_diag f x
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definition seq_colim_eq_equiv0' (a₀ a₁ : A 0) : ι f a₀ = ι f a₁ ≃ seq_colim (id_seq_diagram f 0 a₀ a₁) :=
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definition seq_colim_eq_equiv0' (a₀ a₁ : A 0) :
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ι f a₀ = ι f a₁ ≃ seq_colim (id_seq_diagram f 0 a₀ a₁) :=
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begin
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refine total_space_method2 (ι f a₀) (seq_colim_over (id0_seq_diagram_over f a₀)) _ _ (ι f a₁) ⬝e _,
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{ apply @(is_trunc_equiv_closed_rev _
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(sigma_seq_colim_over_equiv _ _)), apply is_contr_seq_colim },
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refine total_space_method2 (ι f a₀) (seq_colim_over (id0_seq_diagram_over f a₀))
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_ _ (ι f a₁) ⬝e _,
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{ apply @(is_trunc_equiv_closed_rev _ (sigma_seq_colim_over_equiv _ _)),
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apply is_contr_seq_colim },
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{ exact ιo _ idp },
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/-
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In the next equivalence we have to show that
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@ -740,14 +744,14 @@ 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 (x y : A 0) : ι f x = ι f y ≃ seq_colim (id0_seq_diagram f x y) :=
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seq_colim_eq_equiv0' f x y ⬝e seq_colim_id_equiv_seq_colim_id0 f x y
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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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definition seq_colim_eq_equiv {n : ℕ} (x y : A n) :
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ι f x = ι f y ≃ seq_colim (id_seq_diagram f n x y) :=
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eq_equiv_fn_eq_of_equiv (kshift_equiv f n) (ι f x) (ι f y) ⬝e
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eq_equiv_eq_closed (incl_kshift_diag0 f x)⁻¹ (incl_kshift_diag0 f y)⁻¹ ⬝e
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seq_colim_eq_equiv0' (kshift_diag f n) x y ⬝e
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definition seq_colim_eq_equiv {n : ℕ} (a₀ a₁ : A n) :
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ι f a₀ = ι f a₁ ≃ seq_colim (id_seq_diagram f n a₀ a₁) :=
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eq_equiv_fn_eq_of_equiv (kshift_equiv f n) (ι f a₀) (ι f a₁) ⬝e
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eq_equiv_eq_closed (incl_kshift_diag0 f a₀)⁻¹ (incl_kshift_diag0 f a₁)⁻¹ ⬝e
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seq_colim_eq_equiv0' (kshift_diag f n) a₀ a₁ ⬝e
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@seq_colim_equiv _ _ _ (λk, ap (@f _))
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(λm, eq_equiv_eq_closed !lrep_kshift_diag !lrep_kshift_diag)
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(λm p, whisker_right _ (whisker_right _ !ap_inv⁻¹ ⬝ !ap_con⁻¹) ⬝ !ap_con⁻¹) ⬝e
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@ -187,7 +187,7 @@ namespace seq_colim
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lift_succ2
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definition id0_seq [unfold_full] (a₁ a₂ : A 0) : ℕ → Type :=
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λ k, rep0 f k a₁ = lrep f (zero_le k) a₂
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λ k, rep0 f k a₁ = rep0 f k a₂
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definition id0_seq_diagram [unfold_full] (a₁ a₂ : A 0) : seq_diagram (id0_seq f a₁ a₂) :=
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λ (k : ℕ) (p : rep0 f k a₁ = rep0 f k a₂), ap (@f k) p
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@ -215,7 +215,7 @@ namespace seq_colim
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λn a a', f' a'
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definition id0_seq_diagram_over [unfold_full] (a₀ : A 0) :
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seq_diagram_over f (λn a, lrep f (zero_le n) a₀ = a) :=
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seq_diagram_over f (λn a, rep0 f n a₀ = a) :=
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λn a p, ap (@f n) p
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variable (g : seq_diagram_over f P)
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