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Merge branch 'master' of github.com:fpvandoorn/Spectral

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Yuri Sulyma 2017-06-08 20:01:47 -06:00
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32
algebra/direct_sum.hlean
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@ -1,7 +1,7 @@
/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Copyright (c) 2015 Floris van Doorn, Egbert Rijke, Favonia. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Egbert Rijke
Authors: Floris van Doorn, Egbert Rijke, Favonia
Constructions with groups
-/
@ -124,7 +124,7 @@ namespace group
dirsum_elim (λi, dirsum_incl Y' i ∘g f i)
theorem dirsum_functor_compose (f' : Πi, Y' i →g Y'' i) (f : Πi, Y i →g Y' i) :
dirsum_functor f' ∘a dirsum_functor f ~ dirsum_functor (λi, f' i ∘a f i) :=
dirsum_functor f' ∘g dirsum_functor f ~ dirsum_functor (λi, f' i ∘g f i) :=
begin
apply dirsum_homotopy,
intro i y, reflexivity,
@ -156,4 +156,30 @@ namespace group
definition dirsum_functor_left [constructor] (f : J → I) : dirsum (Y ∘ f) →g dirsum Y :=
dirsum_elim (λj, dirsum_incl Y (f j))
definition dirsum_isomorphism [constructor] (f : Πi, Y i ≃g Y' i) : dirsum Y ≃g dirsum Y' :=
let to_hom := dirsum_functor (λ i, f i) in
let from_hom := dirsum_functor (λ i, (f i)⁻¹ᵍ) in
begin
fapply isomorphism.mk,
exact dirsum_functor (λ i, f i),
fapply is_equiv.adjointify,
exact dirsum_functor (λ i, (f i)⁻¹ᵍ),
{ intro ds,
refine (homomorphism_comp_compute (dirsum_functor (λ i, f i)) (dirsum_functor (λ i, (f i)⁻¹ᵍ)) _)⁻¹ ⬝ _,
refine dirsum_functor_compose (λ i, f i) (λ i, (f i)⁻¹ᵍ) ds ⬝ _,
refine @dirsum_functor_homotopy I Y' Y' _ (λ i, !gid) (λ i, to_right_inv (equiv_of_isomorphism (f i))) ds ⬝ _,
exact !dirsum_functor_gid
},
{ intro ds,
refine (homomorphism_comp_compute (dirsum_functor (λ i, (f i)⁻¹ᵍ)) (dirsum_functor (λ i, f i)) _)⁻¹ ⬝ _,
refine dirsum_functor_compose (λ i, (f i)⁻¹ᵍ) (λ i, f i) ds ⬝ _,
refine @dirsum_functor_homotopy I Y Y _ (λ i, !gid) (λ i x,
proof
to_left_inv (equiv_of_isomorphism (f i)) x
qed
) ds ⬝ _,
exact !dirsum_functor_gid
}
end
end group

29
algebra/seq_colim.hlean
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@ -6,7 +6,7 @@ namespace group
section
parameters (A : @trunctype.mk 0 ℕ _ → AbGroup) (f : Πi , A i → A (i + 1))
parameters (A : ℕ → AbGroup) (f : Πi , A i → A (i + 1))
variables {A' : AbGroup}
definition seq_colim_carrier : AbGroup := dirsum A
@ -15,17 +15,18 @@ namespace group
definition seq_colim : AbGroup := quotient_ab_group_gen seq_colim_carrier (λa, ∥seq_colim_rel a∥)
parameters {A f}
definition seq_colim_incl [constructor] (i : ℕ) : A i →g seq_colim :=
gqg_map _ _ ∘g dirsum_incl A i
definition seq_colim_quotient (h : Πi, A i →g A') (k : Πi a, h i a = h (succ i) (f i a))
(v : seq_colim_carrier) (r : ∥seq_colim_rel v∥) : dirsum_elim h v = 1 :=
begin
induction r with r, induction r,
induction r with r, induction r,
refine !to_respect_mul ⬝ _,
refine ap (λγ, group_fun (dirsum_elim h) (group_fun (dirsum_incl A i) a) * group_fun (dirsum_elim h) γ)
refine ap (λγ, group_fun (dirsum_elim h) (group_fun (dirsum_incl A i) a) * group_fun (dirsum_elim h) γ)
(!to_respect_inv)⁻¹ ⬝ _,
refine ap (λγ, γ * group_fun (dirsum_elim h) (group_fun (dirsum_incl A (succ i)) (f i a)⁻¹))
refine ap (λγ, γ * group_fun (dirsum_elim h) (group_fun (dirsum_incl A (succ i)) (f i a)⁻¹))
!dirsum_elim_compute ⬝ _,
refine ap (λγ, (h i a) * γ) !dirsum_elim_compute ⬝ _,
refine ap (λγ, γ * group_fun (h (succ i)) (f i a)⁻¹) !k ⬝ _,
@ -38,7 +39,7 @@ namespace group
gqg_elim _ (dirsum_elim h) (seq_colim_quotient h k)
definition seq_colim_compute (h : Πi, A i →g A')
(k : Πi a, h i a = h (succ i) (f i a)) (i : ℕ) (a : A i) :
(k : Πi a, h i a = h (succ i) (f i a)) (i : ℕ) (a : A i) :
(seq_colim_elim h k) (seq_colim_incl i a) = h i a :=
begin
refine gqg_elim_compute (λa, ∥seq_colim_rel a∥) (dirsum_elim h) (seq_colim_quotient h k) (dirsum_incl A i a) ⬝ _,
@ -47,17 +48,17 @@ namespace group
definition seq_colim_glue {i : @trunctype.mk 0 ℕ _} {a : A i} : seq_colim_incl i a = seq_colim_incl (succ i) (f i a) :=
begin
refine !grp_comp_comp ⬝ _,
refine gqg_eq_of_rel _ _ ⬝ (!grp_comp_comp)⁻¹,
exact tr (seq_colim_rel.rmk _ _)
refine !homomorphism_comp_compute ⬝ _,
refine gqg_eq_of_rel _ _ ⬝ (!homomorphism_comp_compute)⁻¹,
exact tr (seq_colim_rel.rmk _ _)
end
section
local abbreviation h (m : seq_colim →g A') : Πi, A i →g A' := λi, m ∘g (seq_colim_incl i)
local abbreviation k (m : seq_colim →g A') : Πi a, h m i a = h m (succ i) (f i a) :=
λ i a, !grp_comp_comp ⬝ ap m (@seq_colim_glue i a) ⬝ !grp_comp_comp⁻¹
local abbreviation k (m : seq_colim →g A') : Πi a, h m i a = h m (succ i) (f i a) :=
λ i a, !homomorphism_comp_compute ⬝ ap m (@seq_colim_glue i a) ⬝ !homomorphism_comp_compute⁻¹
definition seq_colim_unique (m : seq_colim →g A') :
definition seq_colim_unique (m : seq_colim →g A') :
Πv, seq_colim_elim (h m) (k m) v = m v :=
begin
intro v, refine (gqg_elim_unique _ (dirsum_elim (h m)) _ m _ _)⁻¹ ⬝ _,
@ -69,4 +70,10 @@ namespace group
end
definition seq_colim_functor [constructor] {A A' : ℕ → AbGroup}
{f : Πi , A i →g A (i + 1)} {f' : Πi , A' i →g A' (i + 1)}
(h : Πi, A i →g A' i) : seq_colim A f →g seq_colim A' f' :=
sorry --_ ∘g _
end group

36
colim.hlean
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@ -399,8 +399,8 @@ namespace seq_colim
equiv.mk _ !is_equiv_seq_colim_rec
end functor
definition pseq_colim.elim' [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~ g n) : pseq_colim @f →* B :=
definition pseq_colim.elim' [constructor] {A : ℕ → Type*} {B : Type*} {f : Πn, A n →* A (n+1)}
(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f n ~ g n) : pseq_colim f →* B :=
begin
fapply pmap.mk,
{ intro x, induction x with n a n a,
@ -409,10 +409,38 @@ namespace seq_colim
{ esimp, apply respect_pt }
end
definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~* g n) : pseq_colim @f →* B :=
definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Πn, A n →* A (n+1)}
(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f n ~* g n) : pseq_colim @f →* B :=
pseq_colim.elim' g p
definition pseq_colim.elim_pinclusion {A : ℕ → Type*} {B : Type*} {f : Πn, A n →* A (n+1)}
(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f n ~* g n) (n : ℕ) :
pseq_colim.elim g p ∘* pinclusion f n ~* g n :=
begin
refine phomotopy.mk phomotopy.rfl _,
refine !idp_con ⬝ _,
esimp,
induction n with n IH,
{ esimp, esimp[inclusion_pt], exact !idp_con⁻¹ },
{ esimp, esimp[inclusion_pt],
rewrite ap_con, rewrite ap_con,
rewrite elim_glue,
rewrite [-ap_inv],
rewrite [-ap_compose'], esimp,
rewrite [(eq_con_inv_of_con_eq (!to_homotopy_pt))],
rewrite [IH],
rewrite [con_inv],
rewrite [-+con.assoc],
refine _ ⬝ whisker_right _ !con.assoc⁻¹,
rewrite [con.left_inv], esimp,
refine _ ⬝ !con.assoc⁻¹,
rewrite [con.left_inv], esimp,
rewrite [ap_inv],
rewrite [-con.assoc],
refine !idp_con⁻¹ ⬝ whisker_right _ !con.left_inv⁻¹,
}
end
definition prep0 [constructor] {A : ℕ → Type*} (f : pseq_diagram A) (k : ℕ) : A 0 →* A k :=
pmap.mk (rep0 (λn x, f x) k)
begin induction k with k p, reflexivity, exact ap (@f k) p ⬝ !respect_pt end

16
homology/homology.hlean
View file

@ -14,7 +14,6 @@ open eq spectrum int pointed group algebra sphere nat equiv susp is_trunc
namespace homology
/- homology theory -/
structure homology_theory.{u} : Type.{u+1} :=
(HH : ℤ → pType.{u} → AbGroup.{u})
(Hh : Π(n : ℤ) {X Y : Type*} (f : X →* Y), HH n X →g HH n Y)
@ -89,8 +88,12 @@ namespace homology
end
/- homology theory associated to a spectrum -/
definition homology (X : Type*) (E : spectrum) (n : ℤ) : AbGroup :=
/- homology theory associated to a prespectrum -/
definition homology (X : Type*) (E : prespectrum) (n : ℤ) : AbGroup :=
pshomotopy_group n (smash_prespectrum X E)
/- an alternative definition, which might be a bit harder to work with -/
definition homology_spectrum (X : Type*) (E : spectrum) (n : ℤ) : AbGroup :=
shomotopy_group n (smash_spectrum X E)
definition parametrized_homology {X : Type*} (E : X → spectrum) (n : ℤ) : AbGroup :=
@ -109,12 +112,13 @@ notation `pH_` n `[`:0 binders `, ` r:(scoped E, parametrized_homology E n) `]`:
definition unpointed_homology (X : Type) (E : spectrum) (n : ℤ) : AbGroup :=
H_ n[X₊, E]
definition homology_functor [constructor] {X Y : Type*} {E F : spectrum} (f : X →* Y) (g : E →ₛ F) (n : ℤ) : homology X E n →g homology Y F n :=
shomotopy_group_fun n (smash_spectrum_fun f g)
definition homology_functor [constructor] {X Y : Type*} {E F : spectrum} (f : X →* Y) (g : E →ₛ F) (n : ℤ)
: homology X E n →g homology Y F n :=
pshomotopy_group_fun n (smash_prespectrum_fun f g)
definition homology_theory_spectrum.{u} [constructor] (E : spectrum.{u}) : homology_theory.{u} :=
begin
refine homology_theory.mk _ _ _ _ _ _ _ _,
fapply homology_theory.mk,
exact (λn X, H_ n[X, E]),
exact (λn X Y f, homology_functor f (sid E) n),
exact sorry, -- Hid is uninteresting but potentially very hard to prove

70
homotopy/spectrum.hlean
View file

@ -5,9 +5,9 @@ Authors: Michael Shulman, Floris van Doorn
-/
import homotopy.LES_of_homotopy_groups .splice ..colim types.pointed2 .EM ..pointed_pi .smash_adjoint
import homotopy.LES_of_homotopy_groups .splice ..colim types.pointed2 .EM ..pointed_pi .smash_adjoint ..algebra.seq_colim
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
seq_colim succ_str EM EM.ops
seq_colim succ_str EM EM.ops function
/---------------------
Basic definitions
@ -235,6 +235,25 @@ namespace spectrum
notation `πₛ→[`:95 n:0 `]`:0 := shomotopy_group_fun n
/- homotopy group of a prespectrum -/
definition pshomotopy_group (n : ℤ) (E : prespectrum) : AbGroup :=
group.seq_colim (λ(k : ℕ), πag[k+2] (E (-n - 2 + k)))
begin
intro k,
refine _ ∘ π→g[k+2] (glue E _),
refine (homotopy_group_succ_in _ (k+2))⁻¹ᵉ* ∘ _,
refine homotopy_group_pequiv (k+2) (loop_pequiv_loop (pequiv_of_eq (ap E !add.assoc)))
end
notation `πₚₛ[`:95 n:0 `]`:0 := pshomotopy_group n
definition pshomotopy_group_fun (n : ℤ) {E F : prespectrum} (f : E →ₛ F) :
πₚₛ[n] E →g πₚₛ[n] F :=
sorry --group.seq_colim_functor _ _
notation `πₚₛ→[`:95 n:0 `]`:0 := pshomotopy_group_fun n
/-------------------------------
Cotensor of spectra by types
-------------------------------/
@ -409,7 +428,7 @@ namespace spectrum
spectrify_type_fun' X n (succ k) ~* Ω→ (spectrify_type_fun' X n k) :=
begin
refine _ ⬝* !ap1_pcompose⁻¹*,
apply !pwhisker_right,
apply !pwhisker_right,
refine !to_pinv_pequiv_MK2
end
@ -460,34 +479,47 @@ namespace spectrum
refine _ ◾* (spectrify_type_fun_zero X n ⬝* !pid_pcompose⁻¹*),
refine !passoc ⬝* pwhisker_left _ !pseq_colim_pequiv_pinclusion ⬝* _,
refine pwhisker_left _ (pwhisker_left _ (ap1_pid) ⬝* !pcompose_pid) ⬝* _,
refine !passoc ⬝* pwhisker_left _ !seq_colim_equiv_constant_pinclusion ⬝* _,
refine !passoc ⬝* pwhisker_left _ !seq_colim_equiv_constant_pinclusion ⬝* _,
apply pinv_left_phomotopy_of_phomotopy,
exact !pseq_colim_loop_pinclusion⁻¹*
}
end
definition spectrify.elim_n {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
(f : X →ₛ Y) (n : N) : (spectrify X) n →* Y n :=
begin
fapply pseq_colim.elim,
{ intro k, refine !equiv_gluen⁻¹ᵉ* ∘* apn k (f (n +' k)) },
{ intro k, refine !passoc ⬝* pwhisker_right _ !equiv_gluen_inv_succ ⬝* _,
refine !passoc ⬝* _, apply pwhisker_left,
refine !passoc ⬝* _,
refine pwhisker_left _ ((passoc _ _ (_ ∘* _))⁻¹*) ⬝* _,
refine pwhisker_left _ !passoc⁻¹* ⬝* _,
refine pwhisker_left _ (pwhisker_right _ (phomotopy_pinv_right_of_phomotopy (!loopn_succ_in_natural)⁻¹*)⁻¹*) ⬝* _,
refine pwhisker_right _ !apn_pinv ⬝* _,
refine (phomotopy_pinv_left_of_phomotopy _)⁻¹*,
refine pwhisker_right _ !pmap_eta⁻¹* ⬝* _,
refine apn_psquare k _,
refine pwhisker_right _ _ ⬝* psquare_of_phomotopy !smap.glue_square,
exact !pmap_eta⁻¹*
}
end
definition spectrify.elim {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
(f : X →ₛ Y) : spectrify X →ₛ Y :=
begin
fapply smap.mk,
{ intro n, fapply pseq_colim.elim,
{ intro k, refine !equiv_gluen⁻¹ᵉ* ∘* apn k (f (n +' k)) },
{ intro k, refine !passoc ⬝* pwhisker_right _ !equiv_gluen_inv_succ ⬝* _,
refine !passoc ⬝* _, apply pwhisker_left,
refine !passoc ⬝* _,
refine pwhisker_left _ ((passoc _ _ (_ ∘* _))⁻¹*) ⬝* _,
refine pwhisker_left _ !passoc⁻¹* ⬝* _,
refine pwhisker_left _ (pwhisker_right _ (phomotopy_pinv_right_of_phomotopy (!loopn_succ_in_natural)⁻¹*)⁻¹*) ⬝* _,
refine pwhisker_right _ !apn_pinv ⬝* _,
refine (phomotopy_pinv_left_of_phomotopy _)⁻¹*,
refine pwhisker_right _ !pmap_eta⁻¹* ⬝* _,
refine apn_psquare k _,
refine pwhisker_right _ _ ⬝* psquare_of_phomotopy !smap.glue_square,
exact !pmap_eta⁻¹*
}},
{ intro n, exact spectrify.elim_n f n },
{ intro n, exact sorry }
end
definition phomotopy_spectrify.elim {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
(f : X →ₛ Y) (n : N) : spectrify.elim_n f n ∘* spectrify_map n ~* f n :=
begin
refine pseq_colim.elim_pinclusion _ _ 0 ⬝* _,
exact !pid_pcompose
end
definition spectrify_fun {N : succ_str} {X Y : gen_prespectrum N} (f : X →ₛ Y) : spectrify X →ₛ spectrify Y :=
spectrify.elim ((@spectrify_map _ Y) ∘ₛ f)