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fix after moving stuff to library

Browse source 
also cleanup spectrum.basic a little
This commit is contained in:
Floris van Doorn 2018-09-05 22:56:40 +02:00
parent e1d2392a13
commit fffc3cd03a
12 changed files with 58 additions and 76 deletions
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2
colimit/seq_colim.hlean
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@ -723,7 +723,7 @@ incl_kshift_diag f x
definition seq_colim_eq_equiv0' (a₀ a₁ : A 0) :
ι f a₀ = ι f a₁ ≃ seq_colim (id_seq_diagram f 0 a₀ a₁) :=
begin
refine total_space_method2 (ι f a₀) (seq_colim_over (id0_seq_diagram_over f a₀))
refine total_space_method (ι f a₀) (seq_colim_over (id0_seq_diagram_over f a₀))
_ _ (ι f a₁) ⬝e _,
{ apply @(is_trunc_equiv_closed_rev _ (sigma_seq_colim_over_equiv _ _)),
apply is_contr_seq_colim },

2
colimit/sequence.hlean
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@ -92,7 +92,7 @@ namespace seq_colim
/- lreplace le_of_succ_le with this -/
definition lrep_f {n m : ℕ} (H : succ n ≤ m) (a : A n) :
lrep f H (f a) = lrep f (le_step_left H) a :=
lrep f H (f a) = lrep f (le_of_succ_le H) a :=
begin
induction H with m H p,
{ reflexivity },

2
homology/basic.hlean
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@ -54,7 +54,7 @@ namespace homology
theorem Hh_homotopy (n : ℤ) {A B : Type*} (f g : A →* B) (h : f ~ g) : Hh theory n f ~ Hh theory n g := λ x,
calc Hh theory n f x
= Hh theory n (pmap.mk f (respect_pt f)) x : by exact ap (λ f, Hh theory n f x) (pmap.eta f)⁻¹
= Hh theory n (pmap.mk f (respect_pt f)) x : by exact ap (λ f, Hh theory n f x) (pmap_eta_eq f)⁻¹
... = Hh theory n (pmap.mk f (h pt ⬝ respect_pt g)) x : by exact Hh_homotopy' n f (respect_pt f) (h pt ⬝ respect_pt g) x
... = Hh theory n g x :
begin

4
homotopy/pushout.hlean
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@ -111,7 +111,7 @@ namespace pushout
refine ap (λx, apd10 x _) (ap_compose (λx, x ∘ f) pr1 _ ⬝ ap02 _ !prod_eq_pr1) ⬝ph _
⬝hp ap (λx, apd10 x _) (ap_compose (λx, x ∘ g) pr2 _ ⬝ ap02 _ !prod_eq_pr2)⁻¹,
refine apd10 !apd10_ap_precompose_dependent a ⬝ph _ ⬝hp apd10 !apd10_ap_precompose_dependent⁻¹ a,
refine apd10 !apd10_eq_of_homotopy (f a) ⬝ph _ ⬝hp apd10 !apd10_eq_of_homotopy⁻¹ (g a),
refine !apd10_eq_of_homotopy ⬝ph _ ⬝hp !apd10_eq_of_homotopy⁻¹,
refine ap_compose (pushout.elim h k p) _ _ ⬝pv _,
refine aps (pushout.elim h k p) _ ⬝vp (!elim_glue ⬝ !ap_id⁻¹),
esimp, exact sorry
@ -981,7 +981,7 @@ namespace pushout
assert H : Π w, P w ≃ pushout.elim_type (P ∘ inl) (P ∘ inr) Pglue w,
{ intro w, induction w with x x x,
{ exact erfl }, { exact erfl },
{ apply equiv_pathover, intro pfx pgx q,
{ apply equiv_pathover2, intro pfx pgx q,
apply pathover_of_tr_eq,
apply eq.trans (ap10 (elim_type_glue.{u₁ u₂ u₃ u₄}
(P ∘ inl) (P ∘ inr) Pglue x) pfx), -- why do we need explicit universes here?

2
homotopy/realprojective.hlean
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@ -151,7 +151,7 @@ begin
end
definition corollary_II_6 : Π A : BoolType, (pt = A) ≃ A :=
@total_space_method BoolType pt BoolType.carrier theorem_II_2 idp
@total_space_method BoolType pt BoolType.carrier theorem_II_2 pt
definition is_conn_BoolType [instance] : is_conn 0 BoolType :=
begin

6
homotopy/wedge.hlean
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@ -258,9 +258,9 @@ pequiv.MK (nar_of_noo f g h) (noo_of_nar f g h)
-- apply eq_pathover_id_right, refine ap_compose nar_of_noo _ _ ⬝ ap02 _ !elim_glue ⬝ph _
-- apply eq_pathover_id_right, refine ap_compose noo_of_nar _ _ ⬝ ap02 _ !elim_glue ⬝ph _
definition loop_pequiv_of_cross_section {A B : Type*} (f : A →* B) (g : B →* A)
(h : f ∘* g ~* pid B) : Ω A ≃* Ω (pfiber f) ×* Ω B :=
-- definition loop_pequiv_of_cross_section {A B : Type*} (f : A →* B) (g : B →* A)
-- (h : f ∘* g ~* pid B) : Ω A ≃* Ω (pfiber f) ×* Ω B :=
-- sorry

1
move_to_lib.hlean
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@ -23,6 +23,7 @@ is_trunc_loop_of_is_trunc > is_trunc_loopn_of_is_trunc
first two arguments reordered in is_trunc_loopn_nat
reorder pathover arguments of constructions with squareovers (mostly implicit)
-/
namespace eq

7
pointed.hlean
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@ -213,13 +213,6 @@ namespace pointed
/- Homotopy between a function and its eta expansion -/
definition pmap_eta [constructor] {X Y : Type*} (f : X →* Y) : f ~* pmap.mk f (respect_pt f) :=
begin
fapply phomotopy.mk,
reflexivity,
esimp, exact !idp_con
end
definition papply_point [constructor] (A B : Type*) : papply B pt ~* pconst (ppmap A B) B :=
phomotopy.mk (λf, respect_pt f) idp

17
pointed_cubes.hlean
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@ -91,13 +91,22 @@ begin
exact (pconst_pcompose fleft)⁻¹*,
end
definition phsquare_of_phomotopy {A B : Type*} {f g h i : A →* B} {phtpy_top : f ~* g} {phtpy_bot : h ~* i} {phtpy_left : f ~* h} {phtpy_right : g ~* i} (H : phtpy_top ⬝* phtpy_right ~* phtpy_left ⬝* phtpy_bot) : phsquare phtpy_top phtpy_bot phtpy_left phtpy_right :=
eq_of_phomotopy H
definition phsquare_of_phomotopy {A B : Type*} {f g h i : A →* B} {phtpy_top : f ~* g}
{phtpy_bot : h ~* i} {phtpy_left : f ~* h} {phtpy_right : g ~* i}
(H : phtpy_top ⬝* phtpy_right ~* phtpy_left ⬝* phtpy_bot) :
phsquare phtpy_top phtpy_bot phtpy_left phtpy_right :=
eq_of_phomotopy H
definition ptube_v {A B C D : Type*} {ftop ftop' : A →* B} (phtpy_top : ftop ~* ftop') {fbot fbot' : C →* D} (phtpy_bot : fbot ~* fbot') {fleft : A →* C} {fright : B →* D} (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop' fbot' fleft fright) : Type :=
definition ptube_v {A B C D : Type*} {ftop ftop' : A →* B} (phtpy_top : ftop ~* ftop')
{fbot fbot' : C →* D} (phtpy_bot : fbot ~* fbot') {fleft : A →* C} {fright : B →* D}
(psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop' fbot' fleft fright) :
Type :=
phsquare (pwhisker_left fright phtpy_top) (pwhisker_right fleft phtpy_bot) psq_back psq_front
definition ptube_h {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft fleft' : A →* C} (phtpy_left : fleft ~* fleft') {fright fright' : B →* D} (phtpy_right : fright ~* fright') (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop fbot fleft' fright') : Type :=
definition ptube_h {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft fleft' : A →* C}
(phtpy_left : fleft ~* fleft') {fright fright' : B →* D} (phtpy_right : fright ~* fright')
(psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop fbot fleft' fright') :
Type :=
phsquare (pwhisker_right ftop phtpy_right) (pwhisker_left fbot phtpy_left) psq_back psq_front
--print pinv_right_phomotopy_of_phomotopy

16
pointed_pi.hlean
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@ -47,7 +47,7 @@ namespace pointed
apply eq_of_fn_eq_fn eq_equiv_homotopy,
apply eq_of_homotopy, intro b,
refine !apd10_con ⬝ _,
refine whisker_right _ !pi.apd10_eq_of_homotopy ⬝ q b
refine whisker_right _ !apd10_eq_of_homotopy ⬝ q b
end
definition pupi_functor [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
@ -65,9 +65,9 @@ namespace pointed
begin
fapply phomotopy_mk_pupi,
{ intro h a, reflexivity },
{ intro a, refine !idp_con ⬝ _, refine !apd10_con ⬝ _ ⬝ !pi.apd10_eq_of_homotopy⁻¹, esimp,
refine (!apd10_prepostcompose ⬝ ap02 (g' a) !pi.apd10_eq_of_homotopy) ◾
!pi.apd10_eq_of_homotopy }
{ intro a, refine !idp_con ⬝ _, refine !apd10_con ⬝ _ ⬝ !apd10_eq_of_homotopy⁻¹, esimp,
refine (!apd10_prepostcompose ⬝ ap02 (g' a) !apd10_eq_of_homotopy) ◾
!apd10_eq_of_homotopy }
end
definition pupi_functor_pid (A : Type) (B : A → Type*) :
@ -75,7 +75,7 @@ namespace pointed
begin
fapply phomotopy_mk_pupi,
{ intro h a, reflexivity },
{ intro a, refine !idp_con ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
{ intro a, refine !idp_con ⬝ !apd10_eq_of_homotopy⁻¹ }
end
definition pupi_functor_phomotopy {A A' : Type} {B : A → Type*} {B' : A' → Type*}
@ -86,8 +86,8 @@ namespace pointed
fapply phomotopy_mk_pupi,
{ intro h a, exact q a (h (f a)) ⬝ ap (g' a) (apdt h (p a)) },
{ intro a, esimp,
exact whisker_left _ !pi.apd10_eq_of_homotopy ⬝ !con.assoc ⬝
to_homotopy_pt (q a) ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
exact whisker_left _ !apd10_eq_of_homotopy ⬝ !con.assoc ⬝
to_homotopy_pt (q a) ⬝ !apd10_eq_of_homotopy⁻¹ }
end
definition pupi_pequiv [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
@ -634,7 +634,7 @@ namespace pointed
fapply sigma_eq2,
{ refine !sigma_eq_pr1 ⬝ _ ⬝ !ap_sigma_pr1⁻¹,
apply eq_of_fn_eq_fn eq_equiv_homotopy,
refine !apd10_eq_of_homotopy ⬝ _ ⬝ !apd10_to_fun_eq_of_phomotopy⁻¹,
refine !apd10_eq_of_homotopy_fn ⬝ _ ⬝ !apd10_to_fun_eq_of_phomotopy⁻¹,
apply eq_of_homotopy, intro a, reflexivity },
{ exact sorry } }
end

73
spectrum/basic.hlean
View file

@ -8,8 +8,8 @@ Authors: Michael Shulman, Floris van Doorn, Egbert Rijke, Stefano Piceghello, Yu
import homotopy.LES_of_homotopy_groups ..algebra.splice ..algebra.seq_colim ..homotopy.EM ..homotopy.fwedge
..pointed_cubes
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
succ_str EM EM.ops function unit lift is_trunc
open eq nat int susp pointed sigma is_equiv equiv fiber algebra trunc trunc_index pi group
succ_str EM EM.ops function unit lift is_trunc sigma.ops
/---------------------
Basic definitions
@ -19,7 +19,7 @@ open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc
structure gen_prespectrum (N : succ_str) :=
(deloop : N → Type*)
(glue : Π(n:N), (deloop n) →* (Ω (deloop (S n))))
(glue : Π(n:N), deloop n →* Ω (deloop (S n)))
attribute gen_prespectrum.deloop [coercion]
@ -182,14 +182,12 @@ namespace spectrum
definition smap_to_sigma [unfold 4] {N : succ_str} {X Y : gen_prespectrum N} (f : X →ₛ Y) : smap_sigma X Y :=
begin
induction f with f fsq,
exact sigma.mk f fsq,
exact sigma.mk (smap.to_fun f) (glue_square f),
end
definition smap_to_struc [unfold 4] {N : succ_str} {X Y : gen_prespectrum N} (f : smap_sigma X Y) : X →ₛ Y :=
begin
induction f with f fsq,
exact smap.mk f fsq,
exact smap.mk f.1 f.2,
end
definition smap_to_sigma_isretr {N : succ_str} {X Y : gen_prespectrum N} (f : smap_sigma X Y) :
@ -220,43 +218,25 @@ namespace spectrum
glue_square f n
definition sid [constructor] [refl] {N : succ_str} (E : gen_prespectrum N) : E →ₛ E :=
smap.mk (λ n, pid (E n)) (λ n, psquare_of_phtpy_bot (ap1_pid) (psquare_of_pid_top_bot (phomotopy.rfl)))
--print sid
-- smap.mk (λn, pid (E n))
-- (λn, calc glue E n ∘* pid (E n) ~* glue E n : pcompose_pid
-- ... ~* pid (Ω(E (S n))) ∘* glue E n : pid_pcompose
-- ... ~* Ω→(pid (E (S n))) ∘* glue E n : pwhisker_right (glue E n) ap1_pid⁻¹*)
begin
apply smap.mk (λ n, pid (E n)),
intro n, exact phrfl ⬝vp* !ap1_pid
end
definition scompose [trans] {N : succ_str} {X Y Z : gen_prespectrum N}
(g : Y →ₛ Z) (f : X →ₛ Y) : X →ₛ Z :=
smap.mk (λn, g n ∘* f n)
(λ n, psquare_of_phtpy_bot
(ap1_pcompose (g (S n)) (f (S n)))
(psquare_hcompose (glue_square f n) (glue_square g n)))
/-
(λn, calc glue Z n ∘* to_fun g n ∘* to_fun f n
~* (glue Z n ∘* to_fun g n) ∘* to_fun f n : passoc
... ~* (Ω→(to_fun g (S n)) ∘* glue Y n) ∘* to_fun f n : pwhisker_right (to_fun f n) (glue_square g n)
... ~* Ω→(to_fun g (S n)) ∘* (glue Y n ∘* to_fun f n) : passoc
... ~* Ω→(to_fun g (S n)) ∘* (Ω→ (f (S n)) ∘* glue X n) : pwhisker_left (Ω→(to_fun g (S n))) (glue_square f n)
... ~* (Ω→(to_fun g (S n)) ∘* Ω→(f (S n))) ∘* glue X n : passoc
... ~* Ω→(to_fun g (S n) ∘* to_fun f (S n)) ∘* glue X n : pwhisker_right (glue X n) (ap1_pcompose _ _))
-/
begin
apply smap.mk (λn, g n ∘* f n),
intro n, exact (glue_square f n ⬝h* glue_square g n) ⬝vp* !ap1_pcompose
end
infixr ` ∘ₛ `:60 := scompose
definition szero [constructor] {N : succ_str} (E F : gen_prespectrum N) : E →ₛ F :=
smap.mk (λn, pconst (E n) (F n))
(λn, psquare_of_phtpy_bot (ap1_pconst (E (S n)) (F (S n)))
(psquare_of_pconst_top_bot (glue E n) (glue F n)))
/-
(λn, calc glue F n ∘* pconst (E n) (F n) ~* pconst (E n) (Ω(F (S n))) : pcompose_pconst
... ~* pconst (Ω(E (S n))) (Ω(F (S n))) ∘* glue E n : pconst_pcompose
... ~* Ω→(pconst (E (S n)) (F (S n))) ∘* glue E n : pwhisker_right (glue E n) (ap1_pconst _ _))
-/
begin
apply smap.mk (λn, pconst (E n) (F n)),
intro n, exact !hpconst_square ⬝vp* !ap1_pconst
end
definition stransport [constructor] {N : succ_str} {A : Type} {a a' : A} (p : a = a')
(E : A → gen_prespectrum N) : E a →ₛ E a' :=
@ -270,7 +250,7 @@ namespace spectrum
(to_phomotopy : Πn, f n ~* g n)
(glue_homotopy : Πn, ptube_v
(to_phomotopy n)
(ap1_phomotopy (to_phomotopy (S n)))
(Ω⇒ (to_phomotopy (S n)))
(glue_square f n)
(glue_square g n))
@ -282,6 +262,7 @@ namespace spectrum
-/
infix ` ~ₛ `:50 := shomotopy
attribute [coercion] shomotopy.to_phomotopy
definition shomotopy_compose {N : succ_str} {E F : gen_prespectrum N} {f g h : E →ₛ F} (p : g ~ₛ h) (q : f ~ₛ g) : f ~ₛ h :=
shomotopy.mk
@ -314,10 +295,7 @@ namespace spectrum
(glue_square g n)
definition shomotopy_to_sigma [unfold 6] {N : succ_str} {X Y : gen_prespectrum N} {f g : X →ₛ Y} (H : f ~ₛ g) : shomotopy_sigma f g :=
begin
induction H with H Hsq,
exact sigma.mk H Hsq,
end
⟨H, shomotopy.glue_homotopy H⟩
definition shomotopy_to_struct [unfold 6] {N : succ_str} {X Y : gen_prespectrum N} {f g : X →ₛ Y} (H : shomotopy_sigma f g) : f ~ₛ g :=
begin
@ -423,7 +401,7 @@ namespace spectrum
refine change_path _ _,
-- have p : eq_of_homotopy (λ n, eq_of_phomotopy phomotopy.rfl) = refl f,
reflexivity,
refine (eq_of_homotopy_eta rfl)⁻¹ ⬝ _,
refine (eq_of_homotopy_apd10 rfl)⁻¹ ⬝ _,
fapply ap (eq_of_homotopy), fapply eq_of_homotopy, intro n, refine (eq_of_phomotopy_refl _)⁻¹,
-- fapply eq_of_phomotopy,
fapply pathover_idp_of_eq,
@ -552,6 +530,7 @@ namespace spectrum
smap.mk (λn, ppoint (f n))
begin
intro n,
-- refine _ ⬝vp* !ppoint_loop_pfiber_inv,
refine _ ⬝* !passoc,
refine _ ⬝* pwhisker_right _ !ppoint_loop_pfiber_inv⁻¹*,
rexact (pfiber_pequiv_of_square_ppoint (equiv_glue X n) (equiv_glue Y n) (sglue_square f n))⁻¹*
@ -562,7 +541,7 @@ namespace spectrum
begin
fapply shomotopy.mk,
{ intro n, exact pcompose_ppoint (f n) },
{ intro n, exact sorry }
{ intro n, esimp, exact sorry }
end
/---------------------
@ -750,9 +729,9 @@ namespace spectrum
intro k,
note sq1 := homotopy_group_homomorphism_psquare (k+2) (ptranspose (smap.glue_square f (-n + 2 +[ℤ] k))),
note sq2 := homotopy_group_functor_psquare (k+2) (ap1_psquare (ptransport_natural E F f (add.assoc (-n + 2) k 1))),
note sq3 := (homotopy_group_succ_in_natural (k+2) (f (-n + 2 +[ℤ] (k+1))))⁻¹ʰᵗʸʰ,
note sq4 := hsquare_of_psquare sq2,
note rect := sq1 ⬝htyh sq4 ⬝htyh sq3,
-- note sq3 := (homotopy_group_succ_in_natural (k+2) (f (-n + 2 +[ℤ] (k +[ℕ] 1))))⁻¹ʰᵗʸʰ,
-- note sq4 := hsquare_of_psquare sq2,
-- note rect := sq1 ⬝htyh sq4 ⬝htyh sq3,
exact sorry --sq1 ⬝htyh sq4 ⬝htyh sq3,
end
qed

2
spectrum/trunc.hlean
View file

@ -7,7 +7,7 @@ Truncatedness and truncation of spectra
-/
import .basic
open int trunc eq is_trunc lift unit pointed equiv is_equiv algebra EM
open int trunc eq is_trunc lift unit pointed equiv is_equiv algebra EM trunc_index
namespace spectrum