prove that iota is n-truncated/n-connected if the maps in the sequence are
This commit is contained in:
Floris van Doorn
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1a35543661
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899e3cf2e4
4 changed files with 112 additions and 28 deletions
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@ -110,4 +110,4 @@ These projects are done
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- Installation instructions for Lean 2 can be found [here](https://github.com/leanprover/lean2).
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- Some notes on the Emacs mode can be found [here](https://github.com/leanprover/lean2/blob/master/src/emacs/README.md) (for example if some unicode characters don't show up, or increase the spacing between lines by a lot).
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- If you contribute, please use rebase instead of merge (e.g. `git pull -r`).
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- We try to separate the repository into the folders `algebra`, `homotopy`, `homology` and `cohomology`. Homotopy theotic properties of types which do not explicitly mention homotopy, homology or cohomology groups (such as `A ∧ B ≃* B ∧ A`) are part of `homotopy`.
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- We try to separate the repository into the folders `algebra`, `homotopy`, `homology`, `cohomology` and `colimit`. Homotopy theotic properties of types which do not explicitly mention homotopy, homology or cohomology groups (such as `A ∧ B ≃* B ∧ A`) are part of `homotopy`.
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@ -14,7 +14,12 @@ namespace seq_colim
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abbreviation ι [constructor] := @inclusion
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abbreviation ι' [constructor] [parsing_only] {A} (f n) := @inclusion A f n
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universe variable v
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variables {A A' A'' : ℕ → Type} (f : seq_diagram A) (f' : seq_diagram A') (f'' : seq_diagram A'')
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(τ τ₂ : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), τ (f a) = f' (τ a))
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(p₂ : Π⦃n⦄ (a : A n), τ₂ (f a) = f' (τ₂ a))
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(τ' : Π⦃n⦄, A' n → A'' n) (p' : Π⦃n⦄ (a' : A' n), τ' (f' a') = f'' (τ' a'))
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{P : Π⦃n⦄, A n → Type.{v}} (g : seq_diagram_over f P) {n : ℕ} {a : A n}
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definition lrep_glue {n m : ℕ} (H : n ≤ m) (a : A n) : ι f (lrep f H a) = ι f a :=
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begin
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@ -50,7 +55,7 @@ begin
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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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refine ap02 _ _ ⬝ph _, rotate 1,
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{ rexact elim_glue f _ _ a},
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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 _)⁻¹ ⬝ ap (ι f) (right_inv (lrep f (zero_le (succ k))) (f a)) },
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@ -62,8 +67,7 @@ begin
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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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intro a,
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reflexivity,
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{ intro a, 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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@ -73,7 +77,6 @@ variable (f)
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end
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section
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variables {n : ℕ} (a : A n)
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definition rep_glue (k : ℕ) (a : A n) : ι f (rep f k a) = ι f a :=
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begin
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@ -85,9 +88,6 @@ end
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/- functorial action and equivalences -/
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section functor
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variables {f f' f''}
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variables (τ τ₂ : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), τ (f a) = f' (τ a))
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(p₂ : Π⦃n⦄ (a : A n), τ₂ (f a) = f' (τ₂ a))
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(τ' : Π⦃n⦄, A' n → A'' n) (p' : Π⦃n⦄ (a' : A' n), τ' (f' a') = f'' (τ' a'))
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include p
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definition seq_colim_functor [unfold 7] : seq_colim f → seq_colim f' :=
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begin
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@ -334,8 +334,6 @@ kshift_equiv f n ⬝e equiv_of_is_equiseq (λk, H (n+k) !le_add_right)
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section over
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universe variable v
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variables {P : Π⦃n⦄, A n → Type.{v}} (g : seq_diagram_over f P) {n : ℕ} {a : A n}
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variable {f}
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definition rep_f_equiv_natural {k : ℕ} (p : P (rep f k (f a))) :
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@ -809,6 +807,10 @@ equiv.MK (seq_colim_trunc_of_trunc_seq_colim f k) (trunc_seq_colim_of_seq_colim_
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refine !ap_compose'⁻¹ ⬝ !elim_glue }
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end end
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theorem is_conn_seq_colim [instance] (k : ℕ₋₂) [H : Πn, is_conn k (A n)] :
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is_conn k (seq_colim f) :=
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is_trunc_equiv_closed_rev -2 (trunc_seq_colim_equiv f k)
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/- the colimit of a sequence of fibers is the fiber of the functorial action of the colimit -/
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definition domain_seq_colim_functor {A A' : ℕ → Type} {f : seq_diagram A}
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{f' : seq_diagram A'} (τ : Πn, A' n → A n)
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@ -828,7 +830,8 @@ begin
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refine _ ⬝e fiber_pr1 (seq_colim_over (seq_diagram_over_fiber τ p)) (ι f a),
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apply fiber_equiv_of_triangle (domain_seq_colim_functor τ p)⁻¹ᵉ,
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refine _ ⬝hty λx, (colim_sigma_triangle _ _)⁻¹,
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apply homotopy_inv_of_homotopy_pre (seq_colim_equiv _ _) (seq_colim_functor _ _) (seq_colim_functor _ _),
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apply homotopy_inv_of_homotopy_pre (seq_colim_equiv _ _)
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(seq_colim_functor _ _) (seq_colim_functor _ _),
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refine (λx, !seq_colim_functor_compose⁻¹) ⬝hty _,
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refine seq_colim_functor_homotopy _ _,
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intro n x, exact point_eq x.2,
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@ -840,8 +843,50 @@ definition fiber_seq_colim_functor0 {A A' : ℕ → Type} {f : seq_diagram A}
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{f' : seq_diagram A'} (τ : Πn, A' n → A n)
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(p : Π⦃n⦄, τ (n+1) ∘ @f' n ~ @f n ∘ @τ n) (a : A 0) :
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fiber (seq_colim_functor τ p) (ι f a) ≃ seq_colim (seq_diagram_fiber0 τ p a) :=
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sorry
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fiber_seq_colim_functor τ p a ⬝e
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seq_colim_equiv
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(λn, equiv_apd011 (λx y, fiber (τ x) y) (rep_pathover_rep0 f a))
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(λn x, sorry)
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-- maybe use fn_tro_eq_tro_fn2
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variables {f f'}
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definition fiber_inclusion (x : seq_colim f) :
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fiber (ι' f 0) x ≃ fiber (seq_colim_functor (rep0 f) (λn a, idp)) x :=
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fiber_equiv_of_triangle (seq_colim_constant_seq (A 0))⁻¹ᵉ homotopy.rfl
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theorem is_trunc_fun_seq_colim_functor (k : ℕ₋₂) (H : Πn, is_trunc_fun k (@τ n)) :
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is_trunc_fun k (seq_colim_functor τ p) :=
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begin
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intro x, induction x using seq_colim.rec_prop,
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exact is_trunc_equiv_closed_rev k (fiber_seq_colim_functor τ p a),
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end
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open is_conn
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theorem is_conn_fun_seq_colim_functor (k : ℕ₋₂) (H : Πn, is_conn_fun k (@τ n)) :
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is_conn_fun k (seq_colim_functor τ p) :=
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begin
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intro x, induction x using seq_colim.rec_prop,
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exact is_conn_equiv_closed_rev k (fiber_seq_colim_functor τ p a) _
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end
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variables (f f')
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theorem is_trunc_fun_inclusion (k : ℕ₋₂) (H : Πn, is_trunc_fun k (@f n)) :
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is_trunc_fun k (ι' f 0) :=
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begin
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intro x,
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apply @(is_trunc_equiv_closed_rev k (fiber_inclusion x)),
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apply is_trunc_fun_seq_colim_functor,
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intro n, apply is_trunc_fun_lrep, exact H
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end
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theorem is_conn_fun_inclusion (k : ℕ₋₂) (H : Πn, is_conn_fun k (@f n)) :
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is_conn_fun k (ι' f 0) :=
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begin
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intro x,
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apply is_conn_equiv_closed_rev k (fiber_inclusion x),
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apply is_conn_fun_seq_colim_functor,
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intro n, apply is_conn_fun_lrep, exact H
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end
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/- the sequential colimit of standard finite types is ℕ -/
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open fin
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Egbert Rijke
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import ..move_to_lib types.fin types.trunc
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open nat eq equiv sigma sigma.ops is_equiv is_trunc trunc prod fiber
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open nat eq equiv sigma sigma.ops is_equiv is_trunc trunc prod fiber function
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namespace seq_colim
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@ -28,14 +28,14 @@ namespace seq_colim
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attribute Seq_diagram.carrier [coercion]
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attribute Seq_diagram.struct [coercion]
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variables {A A' : ℕ → Type} (f : seq_diagram A) (f' : seq_diagram A')
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variables {A A' : ℕ → Type} (f : seq_diagram A) (f' : seq_diagram A') {n m k : ℕ}
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include f
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definition lrep {n m : ℕ} (H : n ≤ m) (a : A n) : A m :=
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definition lrep {n m : ℕ} (H : n ≤ m) : A n → A m :=
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begin
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induction H with m H y,
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{ exact a },
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{ exact f y }
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induction H with m H fs,
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{ exact id },
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{ exact @f m ∘ fs }
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end
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definition lrep_irrel_pathover {n m m' : ℕ} (H₁ : n ≤ m) (H₂ : n ≤ m') (p : m = m') (a : A n) :
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@ -85,7 +85,6 @@ namespace seq_colim
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(lrep f H)⁻¹ᶠ
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section generalized_lrep
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variable {n : ℕ}
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-- definition lrep_pathover_lrep0 [reducible] (k : ℕ) (a : A 0) : lrep f k a =[nat.zero_add k] lrep0 f k a :=
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-- begin induction k with k p, constructor, exact pathover_ap A succ (apo f p) end
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@ -128,6 +127,9 @@ namespace seq_colim
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definition rep0 (m : ℕ) (a : A 0) : A m :=
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lrep f (zero_le m) a
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definition rep_pathover_rep0 {n : ℕ} (a : A 0) : rep f n a =[nat.zero_add n] rep0 f n a :=
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!lrep_irrel_pathover
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-- definition old_rep (m : ℕ) (a : A n) : A (n + m) :=
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-- by induction m with m y; exact a; exact f y
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@ -150,6 +152,17 @@ 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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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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definition is_conn_fun_lrep (k : ℕ₋₂) (H : n ≤ m) (H2 : Πn, is_conn_fun k (@f n)) :
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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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end generalized_lrep
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section shift
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@ -228,7 +241,7 @@ namespace seq_colim
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definition seq_diagram_sigma [unfold 6] : seq_diagram (λn, Σ(x : A n), P x) :=
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λn v, ⟨f v.1, g v.2⟩
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variables {n : ℕ} (f P)
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variables (f P)
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theorem rep_f_equiv [constructor] (a : A n) (k : ℕ) :
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P (lrep f (le_add_right (succ n) k) (f a)) ≃ P (lrep f (le_add_right n (succ k)) a) :=
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@ -247,17 +260,17 @@ namespace seq_colim
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λn f x, g (f x)
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variables {f f'}
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definition seq_diagram_over_fiber (g : Π⦃n⦄, A n → A' n)
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(p : Π⦃n⦄ (a : A n), g (f a) = f' (g a)) : seq_diagram_over f' (λn, fiber (@g n)) :=
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λk a, fiber_functor (@f k) (@f' k) (@p k) idp
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definition seq_diagram_over_fiber (g : Π⦃n⦄, A' n → A n)
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(p : Π⦃n⦄ (a : A' n), g (f' a) = f (g a)) : seq_diagram_over f (λn, fiber (@g n)) :=
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λk a, fiber_functor (@f' k) (@f k) (@p k) idp
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definition seq_diagram_fiber (g : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), g (f a) = f' (g a))
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{n : ℕ} (a : A' n) : seq_diagram (λk, fiber (@g (n + k)) (rep f' k a)) :=
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definition seq_diagram_fiber (g : Π⦃n⦄, A' n → A n) (p : Π⦃n⦄ (a : A' n), g (f' a) = f (g a))
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{n : ℕ} (a : A n) : seq_diagram (λk, fiber (@g (n + k)) (rep f k a)) :=
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seq_diagram_of_over (seq_diagram_over_fiber g p) a
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definition seq_diagram_fiber0 (g : Π⦃n⦄, A n → A' n) (p : Π⦃n⦄ (a : A n), g (f a) = f' (g a))
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(a : A' 0) : seq_diagram (λk, fiber (@g k) (rep0 f' k a)) :=
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λk, fiber_functor (@f k) (@f' k) (@p k) idp
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definition seq_diagram_fiber0 (g : Π⦃n⦄, A' n → A n) (p : Π⦃n⦄ (a : A' n), g (f' a) = f (g a))
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(a : A 0) : seq_diagram (λk, fiber (@g k) (rep0 f k a)) :=
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λk, fiber_functor (@f' k) (@f k) (@p k) idp
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end seq_colim
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@ -826,6 +826,32 @@ definition fiber_equiv_of_triangle {A B C : Type} {b : B} {f : A → B} {g : C
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(s : f ~ g ∘ h) : fiber f b ≃ fiber g b :=
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fiber_equiv_of_square h erfl s idp
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definition is_trunc_fun_id (k : ℕ₋₂) (A : Type) : is_trunc_fun k (@id A) :=
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λa, is_trunc_of_is_contr _ _
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definition is_conn_fun_id (k : ℕ₋₂) (A : Type) : is_conn_fun k (@id A) :=
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λa, _
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open sigma.ops is_conn
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definition fiber_compose {A B C : Type} (g : B → C) (f : A → B) (c : C) :
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fiber (g ∘ f) c ≃ Σ(x : fiber g c), fiber f (point x) :=
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begin
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fapply equiv.MK,
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{ intro x, exact ⟨fiber.mk (f (point x)) (point_eq x), fiber.mk (point x) idp⟩ },
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{ intro x, exact fiber.mk (point x.2) (ap g (point_eq x.2) ⬝ point_eq x.1) },
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{ intro x, induction x with x₁ x₂, induction x₁ with b p, induction x₂ with a q,
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induction p, esimp at q, induction q, reflexivity },
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{ intro x, induction x with a p, induction p, reflexivity }
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end
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definition is_trunc_fun_compose (k : ℕ₋₂) {A B C : Type} {g : B → C} {f : A → B}
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(Hg : is_trunc_fun k g) (Hf : is_trunc_fun k f) : is_trunc_fun k (g ∘ f) :=
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λc, is_trunc_equiv_closed_rev k (fiber_compose g f c)
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definition is_conn_fun_compose (k : ℕ₋₂) {A B C : Type} {g : B → C} {f : A → B}
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(Hg : is_conn_fun k g) (Hf : is_conn_fun k f) : is_conn_fun k (g ∘ f) :=
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λc, is_conn_equiv_closed_rev k (fiber_compose g f c) _
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end fiber
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namespace fin
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