Merge branch 'master' of https://github.com/fpvandoorn/Spectral
This commit is contained in:
Robert Rose
commit
85c0ae53a6
9 changed files with 282 additions and 73 deletions
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@ -6,9 +6,9 @@ Authors: Floris van Doorn, Egbert Rijke
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Constructions with groups
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-/
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import .quotient_group .free_commutative_group
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import .quotient_group .free_commutative_group .product_group
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open eq algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
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open eq is_equiv algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
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namespace group
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@ -89,6 +89,35 @@ namespace group
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end
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definition binary_dirsum (G H : AbGroup) : dirsum (bool.rec G H) ≃g G ×ag H :=
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let branch := bool.rec G H in
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let to_hom := (dirsum_elim (bool.rec (product_inl G H) (product_inr G H))
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: dirsum (bool.rec G H) →g G ×ag H) in
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let from_hom := (Group_sum_elim (dirsum (bool.rec G H))
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(dirsum_incl branch bool.ff) (dirsum_incl branch bool.tt)
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: G ×g H →g dirsum branch) in
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begin
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fapply isomorphism.mk,
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{ exact dirsum_elim (bool.rec (product_inl G H) (product_inr G H)) },
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fapply adjointify,
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{ exact from_hom },
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{ intro gh, induction gh with g h,
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exact prod_eq (mul_one (1 * g) ⬝ one_mul g) (ap (λ o, o * h) (mul_one 1) ⬝ one_mul h) },
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{ refine dirsum.rec _ _ _,
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{ intro b x,
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refine ap from_hom (dirsum_elim_compute (bool.rec (product_inl G H) (product_inr G H)) b x) ⬝ _,
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induction b,
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{ exact ap (λ y, dirsum_incl branch bool.ff x * y) (to_respect_one (dirsum_incl branch bool.tt)) ⬝ mul_one _ },
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{ exact ap (λ y, y * dirsum_incl branch bool.tt x) (to_respect_one (dirsum_incl branch bool.ff)) ⬝ one_mul _ }
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},
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{ refine ap from_hom (to_respect_one to_hom) ⬝ to_respect_one from_hom },
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{ intro g h gβ hβ,
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refine ap from_hom (to_respect_mul to_hom _ _) ⬝ to_respect_mul from_hom _ _ ⬝ _,
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exact ap011 mul gβ hβ
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}
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}
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end
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variables {I J : Set} {Y Y' Y'' : I → AbGroup}
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definition dirsum_functor [constructor] (f : Πi, Y i →g Y' i) : dirsum Y →g dirsum Y' :=
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@ -8,7 +8,7 @@ Constructions with groups
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import algebra.group_theory hit.set_quotient types.list types.sum .subgroup .quotient_group
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function
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open eq algebra is_trunc set_quotient relation sigma prod prod.ops sum list trunc function
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equiv
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namespace group
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@ -61,12 +61,19 @@ namespace group
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infix ` ×g `:60 := group.product
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infix ` ×ag `:60 := group.ab_product
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definition product_inl (G H : Group) : G →g G ×g H :=
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definition product_inl [constructor] (G H : Group) : G →g G ×g H :=
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homomorphism.mk (λx, (x, one)) (λx y, prod_eq !refl !one_mul⁻¹)
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definition product_inr (G H : Group) : H →g G ×g H :=
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definition product_inr [constructor] (G H : Group) : H →g G ×g H :=
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homomorphism.mk (λx, (one, x)) (λx y, prod_eq !one_mul⁻¹ !refl)
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definition Group_sum_elim [constructor] {G H : Group} (I : AbGroup) (φ : G →g I) (ψ : H →g I) : G ×g H →g I :=
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homomorphism.mk (λx, φ x.1 * ψ x.2) abstract (λx y, calc
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φ (x.1 * y.1) * ψ (x.2 * y.2) = (φ x.1 * φ y.1) * (ψ x.2 * ψ y.2)
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: by exact ap011 mul (to_respect_mul φ x.1 y.1) (to_respect_mul ψ x.2 y.2)
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... = (φ x.1 * ψ x.2) * (φ y.1 * ψ y.2)
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: by exact interchange I (φ x.1) (φ y.1) (ψ x.2) (ψ y.2)) end
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definition product_functor [constructor] {G G' H H' : Group} (φ : G →g H) (ψ : G' →g H') :
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G ×g G' →g H ×g H' :=
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homomorphism.mk (λx, (φ x.1, ψ x.2)) (λx y, prod_eq !to_respect_mul !to_respect_mul)
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@ -79,4 +86,7 @@ namespace group
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infix ` ×≃g `:60 := group.product_isomorphism
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definition product_group_mul_eq {G H : Group} (g h : G ×g H) : g * h = product_mul g h :=
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idp
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end group
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55
colim.hlean
55
colim.hlean
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@ -285,7 +285,6 @@ namespace seq_colim
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!elim_glue
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omit p
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definition is_equiv_seq_colim_functor [constructor] [H : Πn, is_equiv (@g n)]
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: is_equiv (seq_colim_functor @g p) :=
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adjointify _ (seq_colim_functor (λn, (@g _)⁻¹) (λn a, inv_commute' g f f' p a))
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@ -323,21 +322,30 @@ namespace seq_colim
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: Π(x : seq_colim f), P x :=
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by induction v with Pincl Pglue; exact seq_colim.rec f Pincl Pglue
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definition pseq_colim_pequiv [constructor] {A A' : ℕ → Type*} {f : Π{n}, A n →* A (n+1)}
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{f' : Π{n}, A' n →* A' (n+1)} (g : Π{n}, A n ≃* A' n)
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(p : Π⦃n⦄, g ∘* f ~ f' ∘* g) : pseq_colim @f ≃* pseq_colim @f' :=
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pequiv_of_equiv (seq_colim_equiv @g p) (ap (ι _) (respect_pt g))
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definition pseq_colim_pequiv' [constructor] {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
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{f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
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(p : Π⦃n⦄, g (n+1) ∘* f n ~ f' n ∘* g n) : pseq_colim @f ≃* pseq_colim @f' :=
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pequiv_of_equiv (seq_colim_equiv g p) (ap (ι _) (respect_pt (g _)))
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definition pseq_colim_pequiv [constructor] {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
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{f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
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(p : Πn, g (n+1) ∘* f n ~* f' n ∘* g n) : pseq_colim @f ≃* pseq_colim @f' :=
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pseq_colim_pequiv' g (λn, @p n)
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definition seq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Π⦃n⦄, A n → A (n+1)}
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(p : Π⦃n⦄ (a : A n), f a = f' a) : seq_colim f ≃ seq_colim f' :=
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seq_colim_equiv (λn, erfl) p
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definition pseq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Π{n}, A n →* A (n+1)}
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(p : Π⦃n⦄, f ~ f') : pseq_colim @f ≃* pseq_colim @f' :=
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pseq_colim_pequiv (λn, pequiv.rfl) p
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definition pseq_colim_equiv_constant' [constructor] {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
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(p : Π⦃n⦄, f n ~ f' n) : pseq_colim @f ≃* pseq_colim @f' :=
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pseq_colim_pequiv' (λn, pequiv.rfl) p
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definition pseq_colim_pequiv_pinclusion {A A' : ℕ → Type*} {f : Π(n), A n →* A (n+1)}
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{f' : Π(n), A' n →* A' (n+1)} (g : Π(n), A n ≃* A' n)
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definition pseq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
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(p : Πn, f n ~* f' n) : pseq_colim @f ≃* pseq_colim @f' :=
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pseq_colim_pequiv (λn, pequiv.rfl) (λn, !pid_pcompose ⬝* p n ⬝* !pcompose_pid⁻¹*)
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definition pseq_colim_pequiv_pinclusion {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
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{f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
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(p : Π⦃n⦄, g (n+1) ∘* f n ~* f' n ∘* g n) (n : ℕ) :
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psquare (pinclusion f n) (pinclusion f' n) (g n) (pseq_colim_pequiv g p) :=
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phomotopy.mk homotopy.rfl begin
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@ -350,25 +358,24 @@ namespace seq_colim
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apply whisker_right, esimp,
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rewrite[(eq_con_inv_of_con_eq (!to_homotopy_pt))],
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rewrite[ap_con], esimp,
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rewrite[-+con.assoc],
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rewrite[ap_con], rewrite[-ap_compose'],
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rewrite[+ap_inv],
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rewrite[-+con.assoc, ap_con, -ap_compose', +ap_inv],
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rewrite[-+con.assoc],
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refine _ ⬝ whisker_right _ (whisker_right _ (whisker_right _ (whisker_right _ !con.left_inv⁻¹))),
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rewrite[idp_con],
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rewrite[+con.assoc], apply whisker_left,
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rewrite[ap_con], rewrite[-ap_compose'],
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rewrite[con_inv],
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rewrite[+con.assoc], apply whisker_left,
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rewrite[idp_con, +con.assoc], apply whisker_left,
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rewrite[ap_con, -ap_compose', con_inv, +con.assoc], apply whisker_left,
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refine eq_inv_con_of_con_eq _,
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symmetry, exact eq_of_square !natural_square
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}
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end
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definition seq_colim_equiv_constant_pinclusion {A : ℕ → Type*} {f f' : Π⦃n⦄, A n →* A (n+1)}
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(p : Π⦃n⦄ (a : A n), f a = f' a) (n : ℕ) :
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definition seq_colim_equiv_constant_pinclusion {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
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(p : Πn, f n ~* f' n) (n : ℕ) :
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pseq_colim_equiv_constant p ∘* pinclusion f n ~* pinclusion f' n :=
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sorry
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begin
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transitivity pinclusion f' n ∘* !pid,
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refine phomotopy_of_psquare !pseq_colim_pequiv_pinclusion,
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exact !pcompose_pid
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end
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definition is_equiv_seq_colim_rec (P : seq_colim f → Type) :
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is_equiv (seq_colim_rec_unc :
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@ -392,7 +399,7 @@ namespace seq_colim
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equiv.mk _ !is_equiv_seq_colim_rec
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end functor
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definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
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definition pseq_colim.elim' [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
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(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~ g n) : pseq_colim @f →* B :=
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begin
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fapply pmap.mk,
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@ -402,6 +409,10 @@ namespace seq_colim
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{ esimp, apply respect_pt }
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end
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definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
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(g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~* g n) : pseq_colim @f →* B :=
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pseq_colim.elim' g p
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definition prep0 [constructor] {A : ℕ → Type*} (f : pseq_diagram A) (k : ℕ) : A 0 →* A k :=
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pmap.mk (rep0 (λn x, f x) k)
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begin induction k with k p, reflexivity, exact ap (@f k) p ⬝ !respect_pt end
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@ -1,3 +1,11 @@
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/-
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Copyright (c) 2017 Yuri Sulyma, Favonia
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Yuri Sulyma, Favonia
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Reduced homology theories
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-/
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import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..choice ..homotopy.pushout ..move_to_lib
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open eq spectrum int pointed group algebra sphere nat equiv susp is_trunc
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@ -10,36 +18,39 @@ namespace homology
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structure homology_theory.{u} : Type.{u+1} :=
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(HH : ℤ → pType.{u} → AbGroup.{u})
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(Hh : Π(n : ℤ) {X Y : Type*} (f : X →* Y), HH n X →g HH n Y)
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(Hid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
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(Hcompose : Π(n : ℤ) {X Y Z : Type*} (f : Y →* Z) (g : X →* Y),
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(Hpid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
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(Hpcompose : Π(n : ℤ) {X Y Z : Type*} (f : Y →* Z) (g : X →* Y),
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Hh n (f ∘* g) ~ Hh n f ∘ Hh n g)
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(Hsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
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(Hsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
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Hsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hsusp n X)
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(Hpsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
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(Hpsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
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Hpsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hpsusp n X)
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(Hexact : Π(n : ℤ) {X Y : Type*} (f : X →* Y), is_exact_g (Hh n f) (Hh n (pcod f)))
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(Hadditive : Π(n : ℤ) {I : Set.{u}} (X : I → Type*), is_equiv
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(dirsum_elim (λi, Hh n (pinl i)) : dirsum (λi, HH n (X i)) → HH n (⋁ X)))
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structure ordinary_homology_theory.{u} extends homology_theory.{u} : Type.{u+1} :=
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(Hdimension : Π(n : ℤ), n ≠ 0 → is_contr (HH n (plift (psphere 0))))
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section
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parameter (theory : homology_theory)
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open homology_theory
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theorem HH_base_indep (n : ℤ) {A : Type} (a b : A)
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: HH theory n (pType.mk A a) ≃g HH theory n (pType.mk A b) :=
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calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hsusp theory n (pType.mk A a)) ⁻¹ᵍ
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... ≃g HH theory n (pType.mk A b) : by exact Hsusp theory n (pType.mk A b)
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calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hpsusp theory n (pType.mk A a)) ⁻¹ᵍ
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... ≃g HH theory n (pType.mk A b) : by exact Hpsusp theory n (pType.mk A b)
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theorem Hh_homotopy' (n : ℤ) {A B : Type*} (f : A → B) (p q : f pt = pt)
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: Hh theory n (pmap.mk f p) ~ Hh theory n (pmap.mk f q) := λ x,
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calc Hh theory n (pmap.mk f p) x
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= Hh theory n (pmap.mk f p) (Hsusp theory n A ((Hsusp theory n A)⁻¹ᵍ x))
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: by exact ap (Hh theory n (pmap.mk f p)) (equiv.to_right_inv (equiv_of_isomorphism (Hsusp theory n A)) x)⁻¹
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... = Hsusp theory n B (Hh theory (succ n) (pmap.mk (susp.functor f) !refl) ((Hsusp theory n A)⁻¹ x))
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: by exact (Hsusp_natural theory n (pmap.mk f p) ((Hsusp theory n A)⁻¹ᵍ x))⁻¹
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... = Hh theory n (pmap.mk f q) (Hsusp theory n A ((Hsusp theory n A)⁻¹ x))
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: by exact Hsusp_natural theory n (pmap.mk f q) ((Hsusp theory n A)⁻¹ x)
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= Hh theory n (pmap.mk f p) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ᵍ x))
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: by exact ap (Hh theory n (pmap.mk f p)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)⁻¹
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... = Hpsusp theory n B (Hh theory (succ n) (pmap.mk (susp.functor f) !refl) ((Hpsusp theory n A)⁻¹ x))
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: by exact (Hpsusp_natural theory n (pmap.mk f p) ((Hpsusp theory n A)⁻¹ᵍ x))⁻¹
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... = Hh theory n (pmap.mk f q) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ x))
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: by exact Hpsusp_natural theory n (pmap.mk f q) ((Hpsusp theory n A)⁻¹ x)
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... = Hh theory n (pmap.mk f q) x
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: by exact ap (Hh theory n (pmap.mk f q)) (equiv.to_right_inv (equiv_of_isomorphism (Hsusp theory n A)) x)
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: by exact ap (Hh theory n (pmap.mk f q)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)
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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,
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calc Hh theory n f x
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@ -56,18 +67,71 @@ namespace homology
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{ exact Hh theory n e⁻¹ᵉ* },
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{ intro x, exact calc
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Hh theory n e (Hh theory n e⁻¹ᵉ* x)
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= Hh theory n (e ∘* e⁻¹ᵉ*) x : by exact (Hcompose theory n e e⁻¹ᵉ* x)⁻¹
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= Hh theory n (e ∘* e⁻¹ᵉ*) x : by exact (Hpcompose theory n e e⁻¹ᵉ* x)⁻¹
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... = Hh theory n !pid x : by exact Hh_homotopy n (e ∘* e⁻¹ᵉ*) !pid (to_right_inv e) x
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... = x : by exact Hid theory n x
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... = x : by exact Hpid theory n x
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},
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{ intro x, exact calc
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Hh theory n e⁻¹ᵉ* (Hh theory n e x)
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= Hh theory n (e⁻¹ᵉ* ∘* e) x : by exact (Hcompose theory n e⁻¹ᵉ* e x)⁻¹
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= Hh theory n (e⁻¹ᵉ* ∘* e) x : by exact (Hpcompose theory n e⁻¹ᵉ* e x)⁻¹
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... = Hh theory n !pid x : by exact Hh_homotopy n (e⁻¹ᵉ* ∘* e) !pid (to_left_inv e) x
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... = x : by exact Hid theory n x
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... = x : by exact Hpid theory n x
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}
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end
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end
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/- homology theory associated to a spectrum -/
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definition homology (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 :=
|
||||
sorry
|
||||
|
||||
definition ordinary_homology [reducible] (X : Type*) (G : AbGroup) (n : ℤ) : AbGroup :=
|
||||
homology X (EM_spectrum G) n
|
||||
|
||||
definition ordinary_homology_Z [reducible] (X : Type*) (n : ℤ) : AbGroup :=
|
||||
ordinary_homology X agℤ n
|
||||
|
||||
notation `H_` n `[`:0 X:0 `, ` E:0 `]`:0 := homology X E n
|
||||
notation `H_` n `[`:0 X:0 `]`:0 := ordinary_homology_Z X n
|
||||
notation `pH_` n `[`:0 binders `, ` r:(scoped E, parametrized_homology E n) `]`:0 := r
|
||||
|
||||
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_theory_spectrum.{u} [constructor] (E : spectrum.{u}) : homology_theory.{u} :=
|
||||
begin
|
||||
refine 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
|
||||
exact sorry,
|
||||
exact sorry,
|
||||
exact sorry,
|
||||
exact sorry,
|
||||
exact sorry
|
||||
-- sorry
|
||||
-- sorry
|
||||
-- sorry
|
||||
-- sorry
|
||||
-- sorry
|
||||
-- sorry
|
||||
-- (λn A, H^n[A, Y])
|
||||
-- (λn A B f, cohomology_isomorphism f Y n)
|
||||
-- (λn A, cohomology_isomorphism_refl A Y n)
|
||||
-- (λn A B f, cohomology_functor f Y n)
|
||||
-- (λn A B f g p, cohomology_functor_phomotopy p Y n)
|
||||
-- (λn A B f x, cohomology_functor_phomotopy_refl f Y n x)
|
||||
-- (λn A x, cohomology_functor_pid A Y n x)
|
||||
-- (λn A B C g f x, cohomology_functor_pcompose g f Y n x)
|
||||
-- (λn A, cohomology_psusp A Y n)
|
||||
-- (λn A B f, cohomology_psusp_natural f Y n)
|
||||
-- (λn A B f, cohomology_exact f Y n)
|
||||
-- (λn I A H, spectrum_additive H A Y n)
|
||||
end
|
||||
end homology
|
||||
|
|
|
|||
|
|
@ -5,7 +5,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Author: Kuen-Bang Hou (Favonia)
|
||||
-/
|
||||
|
||||
import core
|
||||
import .homology
|
||||
|
||||
open eq pointed group algebra circle sphere nat equiv susp
|
||||
|
|
@ -18,7 +17,7 @@ section
|
|||
|
||||
open homology_theory
|
||||
|
||||
theorem Hsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
|
||||
theorem Hpsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
|
||||
begin
|
||||
intros n m, revert n, induction m with m,
|
||||
{ exact λ n, isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_zero n)⁻¹ },
|
||||
|
|
@ -27,7 +26,7 @@ section
|
|||
≃g HH theory n (psusp (plift (psphere m))) : by exact HH_isomorphism theory n (plift_psusp (psphere m))
|
||||
... ≃g HH theory (succ (pred n)) (psusp (plift (psphere m)))
|
||||
: by exact isomorphism_ap (λ n, HH theory n (psusp (plift (psphere m)))) (succ_pred n)⁻¹
|
||||
... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hsusp theory (pred n) (plift (psphere m))
|
||||
... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hpsusp theory (pred n) (plift (psphere m))
|
||||
... ≃g HH theory (pred n - m) (plift (psphere 0)) : by exact v_0 (pred n)
|
||||
... ≃g HH theory (n - succ m) (plift (psphere 0))
|
||||
: by exact isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_sub n 1 m ⬝ ap (λ m, n - m) (add.comm 1 m))
|
||||
|
|
|
|||
|
|
@ -189,7 +189,7 @@ structure cohomology_theory.{u} : Type.{u+1} :=
|
|||
(Hadditive : Π(n : ℤ) {I : Type.{u}} (X : I → Type*), has_choice 0 I →
|
||||
is_equiv (Group_pi_intro (λi, Hh n (pinl i)) : HH n (⋁ X) → Πᵍ i, HH n (X i)))
|
||||
|
||||
structure ordinary_theory.{u} extends cohomology_theory.{u} : Type.{u+1} :=
|
||||
structure ordinary_cohomology_theory.{u} extends cohomology_theory.{u} : Type.{u+1} :=
|
||||
(Hdimension : Π(n : ℤ), n ≠ 0 → is_contr (HH n (plift pbool)))
|
||||
|
||||
attribute cohomology_theory.HH [coercion]
|
||||
|
|
@ -270,7 +270,7 @@ cohomology_theory.mk
|
|||
-- print has_choice_lift
|
||||
-- print equiv_lift
|
||||
-- print has_choice_equiv_closed
|
||||
definition ordinary_theory_EM [constructor] (G : AbGroup) : ordinary_theory :=
|
||||
⦃ordinary_theory, cohomology_theory_spectrum (EM_spectrum G), Hdimension := EM_dimension G ⦄
|
||||
definition ordinary_cohomology_theory_EM [constructor] (G : AbGroup) : ordinary_cohomology_theory :=
|
||||
⦃ordinary_cohomology_theory, cohomology_theory_spectrum (EM_spectrum G), Hdimension := EM_dimension G ⦄
|
||||
|
||||
end cohomology
|
||||
|
|
|
|||
|
|
@ -146,7 +146,7 @@ namespace spectrum
|
|||
-- I guess this manual eta-expansion is necessary because structures lack definitional eta?
|
||||
:= phomotopy.mk (glue_square f n) (to_homotopy_pt (glue_square f n))
|
||||
|
||||
definition sid {N : succ_str} (E : gen_spectrum N) : E →ₛ E :=
|
||||
definition sid {N : succ_str} (E : gen_prespectrum N) : E →ₛ E :=
|
||||
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
|
||||
|
|
@ -373,13 +373,17 @@ namespace spectrum
|
|||
definition spectrify_type_term {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : Type* :=
|
||||
Ω[k] (X (n +' k))
|
||||
|
||||
definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (k : ℕ) (n : N) :
|
||||
definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
|
||||
Ω[k] (X n) →* Ω[k+1] (X (S n)) :=
|
||||
!loopn_succ_in⁻¹ᵉ* ∘* Ω→[k] (glue X n)
|
||||
|
||||
definition spectrify_type_fun {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
|
||||
spectrify_type_term X n k →* spectrify_type_term X n (k+1) :=
|
||||
spectrify_type_fun' X k (n +' k)
|
||||
spectrify_type_fun' X (n +' k) k
|
||||
|
||||
definition spectrify_type_fun_zero {N : succ_str} (X : gen_prespectrum N) (n : N) :
|
||||
spectrify_type_fun X n 0 ~* glue X n :=
|
||||
!pid_pcompose
|
||||
|
||||
definition spectrify_type {N : succ_str} (X : gen_prespectrum N) (n : N) : Type* :=
|
||||
pseq_colim (spectrify_type_fun X n)
|
||||
|
|
@ -393,20 +397,28 @@ namespace spectrum
|
|||
... ≡ Y n
|
||||
-/
|
||||
|
||||
definition spectrify_type_fun'_succ {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
|
||||
spectrify_type_fun' X n (succ k) ~* Ω→ (spectrify_type_fun' X n k) :=
|
||||
begin
|
||||
refine _ ⬝* !ap1_pcompose⁻¹*,
|
||||
apply !pwhisker_right,
|
||||
refine !to_pinv_pequiv_MK2
|
||||
end
|
||||
|
||||
definition spectrify_pequiv {N : succ_str} (X : gen_prespectrum N) (n : N) :
|
||||
spectrify_type X n ≃* Ω (spectrify_type X (S n)) :=
|
||||
begin
|
||||
refine !pshift_equiv ⬝e* _,
|
||||
transitivity pseq_colim (λk, spectrify_type_fun' X (succ k) (S n +' k)),
|
||||
transitivity pseq_colim (λk, spectrify_type_fun' X (S n +' k) (succ k)),
|
||||
fapply pseq_colim_pequiv,
|
||||
{ intro n, apply loopn_pequiv_loopn, apply pequiv_ap X, apply succ_str.add_succ },
|
||||
{ intro k, apply to_homotopy,
|
||||
{ exact abstract begin intro k,
|
||||
refine !passoc⁻¹* ⬝* _, refine pwhisker_right _ (loopn_succ_in_inv_natural (succ k) _) ⬝* _,
|
||||
refine !passoc ⬝* _ ⬝* !passoc⁻¹*, apply pwhisker_left,
|
||||
refine !apn_pcompose⁻¹* ⬝* _ ⬝* !apn_pcompose, apply apn_phomotopy,
|
||||
exact !glue_ptransport⁻¹* },
|
||||
exact !glue_ptransport⁻¹* end end },
|
||||
refine _ ⬝e* !pseq_colim_loop⁻¹ᵉ*,
|
||||
refine pseq_colim_equiv_constant (λn, !ap1_pcompose⁻¹*),
|
||||
exact pseq_colim_equiv_constant (λn, !spectrify_type_fun'_succ),
|
||||
end
|
||||
|
||||
definition spectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_spectrum N :=
|
||||
|
|
@ -419,20 +431,30 @@ namespace spectrum
|
|||
-- note: the forward map is (currently) not definitionally equal to gluen. Is that a problem?
|
||||
definition equiv_gluen {N : succ_str} (X : gen_spectrum N) (n : N) (k : ℕ)
|
||||
: X n ≃* Ω[k] (X (n +' k)) :=
|
||||
by induction k with k f; reflexivity; exact f ⬝e* loopn_pequiv_loopn k (equiv_glue X (n +' k))
|
||||
⬝e* !loopn_succ_in⁻¹ᵉ*
|
||||
by induction k with k f; reflexivity; exact f ⬝e* (loopn_pequiv_loopn k (equiv_glue X (n +' k))
|
||||
⬝e* !loopn_succ_in⁻¹ᵉ*)
|
||||
|
||||
definition equiv_gluen_inv_succ {N : succ_str} (X : gen_spectrum N) (n : N) (k : ℕ) :
|
||||
(equiv_gluen X n (k+1))⁻¹ᵉ* ~*
|
||||
(equiv_gluen X n k)⁻¹ᵉ* ∘* Ω→[k] (equiv_glue X (n +' k))⁻¹ᵉ* ∘* !loopn_succ_in :=
|
||||
begin
|
||||
refine !trans_pinv ⬝* pwhisker_left _ _, refine !trans_pinv ⬝* _, refine !to_pinv_pequiv_MK2 ◾* !pinv_pinv
|
||||
end
|
||||
|
||||
definition spectrify_map {N : succ_str} {X : gen_prespectrum N} : X →ₛ spectrify X :=
|
||||
begin
|
||||
fapply smap.mk,
|
||||
{ intro n, exact pinclusion _ 0 },
|
||||
{ intro n, apply phomotopy_of_psquare, refine !pid_pcompose⁻¹* ⬝ph* _,
|
||||
{ intro n, apply phomotopy_of_psquare,
|
||||
refine !pid_pcompose⁻¹* ⬝ph* _,
|
||||
--pshift_equiv_pinclusion (spectrify_type_fun X n) 0
|
||||
refine _ ⬝v* _,
|
||||
rotate 1, exact pshift_equiv_pinclusion (spectrify_type_fun X n) 0,
|
||||
-- refine !passoc⁻¹* ⬝* pwhisker_left _ _ ⬝* _,
|
||||
exact sorry
|
||||
refine !passoc ⬝* pwhisker_left _ (pshift_equiv_pinclusion (spectrify_type_fun X n) 0) ⬝* _,
|
||||
refine !passoc⁻¹* ⬝* _,
|
||||
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 ⬝* _,
|
||||
apply pinv_left_phomotopy_of_phomotopy,
|
||||
exact !pseq_colim_loop_pinclusion⁻¹*
|
||||
}
|
||||
end
|
||||
|
||||
|
|
@ -442,10 +464,25 @@ namespace spectrum
|
|||
fapply smap.mk,
|
||||
{ intro n, fapply pseq_colim.elim,
|
||||
{ intro k, refine !equiv_gluen⁻¹ᵉ* ∘* apn k (f (n +' k)) },
|
||||
{ intro k, apply to_homotopy, exact sorry }},
|
||||
{ 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 sorry }
|
||||
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)
|
||||
|
||||
/- Tensor by spaces -/
|
||||
|
||||
/- Smash product of spectra -/
|
||||
|
|
@ -461,6 +498,28 @@ prespectrum.mk (λ z, X ∧ Y z) begin
|
|||
exact !glue
|
||||
end
|
||||
|
||||
definition smash_prespectrum_fun {X X' : Type*} {Y Y' : prespectrum} (f : X →* X') (g : Y →ₛ Y') : smash_prespectrum X Y →ₛ smash_prespectrum X' Y' :=
|
||||
smap.mk (λn, smash_functor f (g n)) begin
|
||||
intro n,
|
||||
refine susp_to_loop_psquare _ _ _ _ _,
|
||||
refine pvconcat (psquare_transpose (phinverse (smash_psusp_natural f (g n)))) _,
|
||||
refine vconcat_phomotopy _ (smash_functor_split f (g (S n))),
|
||||
refine phomotopy_vconcat (smash_functor_split f (psusp_functor (g n))) _,
|
||||
refine phconcat _ _,
|
||||
let glue_adjoint := psusp_pelim (Y n) (Y (S n)) (glue Y n),
|
||||
exact pid X' ∧→ glue_adjoint,
|
||||
exact smash_functor_psquare (pvrefl f) (phrefl glue_adjoint),
|
||||
refine smash_functor_psquare (phrefl (pid X')) _,
|
||||
refine loop_to_susp_square _ _ _ _ _,
|
||||
exact smap.glue_square g n
|
||||
end
|
||||
|
||||
definition smash_spectrum (X : Type*) (Y : spectrum) : spectrum :=
|
||||
spectrify (smash_prespectrum X Y)
|
||||
|
||||
definition smash_spectrum_fun {X X' : Type*} {Y Y' : spectrum} (f : X →* X') (g : Y →ₛ Y') : smash_spectrum X Y →ₛ smash_spectrum X' Y' :=
|
||||
spectrify_fun (smash_prespectrum_fun f g)
|
||||
|
||||
/- Cofibers and stability -/
|
||||
|
||||
/- The Eilenberg-MacLane spectrum -/
|
||||
|
|
|
|||
|
|
@ -1,6 +1,6 @@
|
|||
-- definitions, theorems and attributes which should be moved to files in the HoTT library
|
||||
|
||||
import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc hit.set_quotient eq2
|
||||
import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc hit.set_quotient eq2 types.pointed2 .homotopy.smash_adjoint
|
||||
|
||||
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
|
||||
is_trunc function sphere unit sum prod bool
|
||||
|
|
@ -558,3 +558,32 @@ open trunc fiber image
|
|||
end
|
||||
|
||||
end injective_surjective
|
||||
|
||||
-- Yuri Sulyma's code from HoTT MRC
|
||||
|
||||
notation `⅀→`:(max+5) := psusp_functor
|
||||
|
||||
namespace pointed
|
||||
variables {A₀₀ A₂₀ A₀₂ A₂₂ : Type*}
|
||||
{f₁₀ : A₀₀ →* A₂₀} {f₁₂ : A₀₂ →* A₂₂}
|
||||
{f₀₁ : A₀₀ →* A₀₂} {f₂₁ : A₂₀ →* A₂₂}
|
||||
|
||||
definition psquare_transpose (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare f₀₁ f₂₁ f₁₀ f₁₂ := p⁻¹*
|
||||
|
||||
definition suspend_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare (⅀→ f₁₀) (⅀→ f₁₂) (⅀→ f₀₁) (⅀→ f₂₁) :=
|
||||
sorry
|
||||
|
||||
definition susp_to_loop_psquare (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) (f₀₁ : psusp A₀₀ →* A₀₂) (f₂₁ : psusp A₂₀ →* A₂₂) : (psquare (⅀→ f₁₀) f₁₂ f₀₁ f₂₁) → (psquare f₁₀ (Ω→ f₁₂) ((loop_psusp_pintro A₀₀ A₀₂) f₀₁) ((loop_psusp_pintro A₂₀ A₂₂) f₂₁)) :=
|
||||
begin
|
||||
intro p,
|
||||
refine pvconcat _ (ap1_psquare p),
|
||||
exact loop_psusp_unit_natural f₁₀
|
||||
end
|
||||
|
||||
definition loop_to_susp_square (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂) (f₀₁ : A₀₀ →* Ω A₀₂) (f₂₁ : A₂₀ →* Ω A₂₂) : (psquare f₁₀ (Ω→ f₁₂) f₀₁ f₂₁) → (psquare (⅀→ f₁₀) f₁₂ ((psusp_pelim A₀₀ A₀₂) f₀₁) ((psusp_pelim A₂₀ A₂₂) f₂₁)) :=
|
||||
begin
|
||||
intro p,
|
||||
refine pvconcat (suspend_psquare p) _,
|
||||
exact psquare_transpose (loop_psusp_counit_natural f₁₂)
|
||||
end
|
||||
end pointed
|
||||
|
|
|
|||
|
|
@ -142,5 +142,13 @@ namespace pointed
|
|||
-- definition phomotopy_eq_equiv_phomotopy2 : p = q ≃ p ~*2 q :=
|
||||
-- sorry
|
||||
|
||||
/- Homotopy between a function and its eta expansion -/
|
||||
|
||||
definition pmap_eta {X Y : Type*} (f : X →* Y) : f ~* pmap.mk f (pmap.resp_pt f) :=
|
||||
begin
|
||||
fapply phomotopy.mk,
|
||||
reflexivity,
|
||||
esimp, exact !idp_con
|
||||
end
|
||||
|
||||
end pointed
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Reference in a new issue