define parametrized cohomology
This commit is contained in:
Floris van Doorn
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5 changed files with 228 additions and 156 deletions
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@ -1,14 +1,63 @@
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import algebra.group_theory ..pointed eq2
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open pi pointed algebra group eq equiv is_trunc trunc
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/- various groups of maps. Most importantly we define a group structure on trunc 0 (A →* Ω B),
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which is used in the definition of cohomology -/
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--author: Floris van Doorn
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import algebra.group_theory ..pointed ..pointed_pi eq2
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open pi pointed algebra group eq equiv is_trunc trunc susp
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namespace group
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/- We first define the group structure on A →* Ω B (except for truncatedness).
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Instead of Ω B, we could also choose any infinity group. However, we need various 2-coherences,
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so it's easier to just do it for the loop space. -/
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definition pmap_mul [constructor] {A B : Type*} (f g : A →* Ω B) : A →* Ω B :=
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pmap.mk (λa, f a ⬝ g a) (respect_pt f ◾ respect_pt g ⬝ !idp_con)
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definition pmap_inv [constructor] {A B : Type*} (f : A →* Ω B) : A →* Ω B :=
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pmap.mk (λa, (f a)⁻¹ᵖ) (respect_pt f)⁻²
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/- we prove some coherences of the multiplication. We don't need them for the group structure, but they
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are used to show that cohomology satisfies the Eilenberg-Steenrod axioms -/
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definition ap1_pmap_mul {X Y : Type*} (f g : X →* Ω Y) :
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Ω→ (pmap_mul f g) ~* pmap_mul (Ω→ f) (Ω→ g) :=
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begin
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fconstructor,
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{ intro p, esimp,
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refine ap1_gen_con_left (respect_pt f) (respect_pt f)
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(respect_pt g) (respect_pt g) p ⬝ _,
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refine !whisker_right_idp ◾ !whisker_left_idp2, },
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{ refine !con.assoc ⬝ _,
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refine _ ◾ idp ⬝ _, rotate 1,
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rexact ap1_gen_con_left_idp (respect_pt f) (respect_pt g), esimp,
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refine !con.assoc ⬝ _,
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apply whisker_left, apply inv_con_eq_idp,
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refine !con2_con_con2 ⬝ ap011 concat2 _ _:
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refine eq_of_square (!natural_square ⬝hp !ap_id) ⬝ !con_idp }
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end
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definition pmap_mul_pcompose {A B C : Type*} (g h : B →* Ω C) (f : A →* B) :
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pmap_mul g h ∘* f ~* pmap_mul (g ∘* f) (h ∘* f) :=
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begin
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fconstructor,
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{ intro p, reflexivity },
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{ esimp, refine !idp_con ⬝ _, refine !con2_con_con2⁻¹ ⬝ whisker_right _ _,
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refine !ap_eq_ap011⁻¹ }
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end
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definition pcompose_pmap_mul {A B C : Type*} (h : B →* C) (f g : A →* Ω B) :
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Ω→ h ∘* pmap_mul f g ~* pmap_mul (Ω→ h ∘* f) (Ω→ h ∘* g) :=
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begin
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fconstructor,
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{ intro p, exact ap1_con h (f p) (g p) },
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{ refine whisker_left _ !con2_con_con2⁻¹ ⬝ _, refine !con.assoc⁻¹ ⬝ _,
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refine whisker_right _ (eq_of_square !ap1_gen_con_natural) ⬝ _,
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refine !con.assoc ⬝ whisker_left _ _, apply ap1_gen_con_idp }
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end
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definition loop_psusp_intro_pmap_mul {X Y : Type*} (f g : psusp X →* Ω Y) :
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loop_psusp_intro (pmap_mul f g) ~* pmap_mul (loop_psusp_intro f) (loop_psusp_intro g) :=
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pwhisker_right _ !ap1_pmap_mul ⬝* !pmap_mul_pcompose
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definition inf_group_pmap [constructor] [instance] (A B : Type*) : inf_group (A →* Ω B) :=
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begin
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fapply inf_group.mk,
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@ -33,7 +82,7 @@ namespace group
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!trunc_group
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definition Group_trunc_pmap [reducible] [constructor] (A B : Type*) : Group :=
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Group.mk (trunc 0 (A →* Ω (Ω B))) _
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Group.mk (trunc 0 (A →* Ω B)) _
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definition Group_trunc_pmap_homomorphism [constructor] {A A' B : Type*} (f : A' →* A) :
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Group_trunc_pmap A B →g Group_trunc_pmap A' B :=
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@ -73,15 +122,15 @@ namespace group
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induction f with f, apply ap tr, apply eq_of_phomotopy, apply pcompose_pconst
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end
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definition Group_trunc_pmap_pcompose [constructor] {A A' A'' B : Type*} (f : A' →* A) (f' : A'' →* A')
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(g : Group_trunc_pmap A B) : Group_trunc_pmap_homomorphism (f ∘* f') g =
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definition Group_trunc_pmap_pcompose [constructor] {A A' A'' B : Type*} (f : A' →* A)
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(f' : A'' →* A') (g : Group_trunc_pmap A B) : Group_trunc_pmap_homomorphism (f ∘* f') g =
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Group_trunc_pmap_homomorphism f' (Group_trunc_pmap_homomorphism f g) :=
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begin
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induction g with g, apply ap tr, apply eq_of_phomotopy, exact !passoc⁻¹*
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end
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definition Group_trunc_pmap_phomotopy [constructor] {A A' B : Type*} {f f' : A' →* A} (p : f ~* f') :
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@Group_trunc_pmap_homomorphism _ _ B f ~ Group_trunc_pmap_homomorphism f' :=
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definition Group_trunc_pmap_phomotopy [constructor] {A A' B : Type*} {f f' : A' →* A}
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(p : f ~* f') : @Group_trunc_pmap_homomorphism _ _ B f ~ Group_trunc_pmap_homomorphism f' :=
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begin
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intro g, induction g, exact ap tr (eq_of_phomotopy (pwhisker_left a p))
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end
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@ -95,7 +144,8 @@ namespace group
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apply pwhisker_left_refl
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end
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definition ab_inf_group_pmap [constructor] [instance] (A B : Type*) : ab_inf_group (A →* Ω (Ω B)) :=
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definition ab_inf_group_pmap [constructor] [instance] (A B : Type*) :
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ab_inf_group (A →* Ω (Ω B)) :=
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⦃ab_inf_group, inf_group_pmap A (Ω B), mul_comm :=
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begin
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intro f g, fapply pmap_eq,
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@ -110,8 +160,9 @@ namespace group
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definition AbGroup_trunc_pmap [reducible] [constructor] (A B : Type*) : AbGroup :=
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AbGroup.mk (trunc 0 (A →* Ω (Ω B))) _
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/- Group of functions whose codomain is a group -/
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definition group_pi [instance] [constructor] {A : Type} (P : A → Type) [Πa, group (P a)] : group (Πa, P a) :=
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/- Group of dependent functions whose codomain is a group -/
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definition group_pi [instance] [constructor] {A : Type} (P : A → Type) [Πa, group (P a)] :
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group (Πa, P a) :=
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begin
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fapply group.mk,
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{ apply is_trunc_pi },
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@ -140,62 +191,56 @@ namespace group
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{ intro g h, apply eq_of_homotopy, intro a, exact respect_mul (f a) g h }
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end
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-- definition AbGroup_trunc_pmap_homomorphism [constructor] {A A' B : Type*} (f : A' →* A) :
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-- AbGroup_trunc_pmap A B →g AbGroup_trunc_pmap A' B :=
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-- Group_trunc_pmap_homomorphism f
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/- Group of dependent functions into a loop space -/
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definition ppi_mul [constructor] {A : Type*} {B : A → Type*} (f g : Π*a, Ω (B a)) : Π*a, Ω (B a) :=
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proof ppi.mk (λa, f a ⬝ g a) (ppi_resp_pt f ◾ ppi_resp_pt g ⬝ !idp_con) qed
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definition ppi_inv [constructor] {A : Type*} {B : A → Type*} (f : Π*a, Ω (B a)) : Π*a, Ω (B a) :=
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proof ppi.mk (λa, (f a)⁻¹ᵖ) (ppi_resp_pt f)⁻² qed
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/- Group of functions whose codomain is a group -/
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-- definition group_arrow [instance] (A B : Type) [group B] : group (A → B) :=
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-- begin
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-- fapply group.mk,
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-- { apply is_trunc_arrow },
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-- { intro f g a, exact f a * g a },
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-- { intros, apply eq_of_homotopy, intro a, apply mul.assoc },
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-- { intro a, exact 1 },
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-- { intros, apply eq_of_homotopy, intro a, apply one_mul },
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-- { intros, apply eq_of_homotopy, intro a, apply mul_one },
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-- { intro f a, exact (f a)⁻¹ },
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-- { intros, apply eq_of_homotopy, intro a, apply mul.left_inv }
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-- end
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definition inf_group_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
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inf_group (Π*a, Ω (B a)) :=
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begin
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fapply inf_group.mk,
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{ exact ppi_mul },
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{ intro f g h, fapply ppi_eq,
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{ intro a, exact con.assoc (f a) (g a) (h a) },
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{ rexact eq_of_square (con2_assoc (ppi_resp_pt f) (ppi_resp_pt g) (ppi_resp_pt h)) }},
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{ apply ppi_const },
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{ intros f, fapply ppi_eq,
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{ intro a, exact one_mul (f a) },
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{ esimp, apply eq_of_square, refine _ ⬝vp !ap_id, apply natural_square_tr }},
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{ intros f, fapply ppi_eq,
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{ intro a, exact mul_one (f a) },
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{ reflexivity }},
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{ exact ppi_inv },
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{ intro f, fapply ppi_eq,
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{ intro a, exact con.left_inv (f a) },
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{ exact !con_left_inv_idp⁻¹ }},
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end
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-- definition Group_arrow (A : Type) (G : Group) : Group :=
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-- Group.mk (A → G) _
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definition group_trunc_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
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group (trunc 0 (Π*a, Ω (B a))) :=
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!trunc_group
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-- definition ab_group_arrow [instance] (A B : Type) [ab_group B] : ab_group (A → B) :=
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-- ⦃ab_group, group_arrow A B,
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-- mul_comm := by intros; apply eq_of_homotopy; intro a; apply mul.comm⦄
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definition Group_trunc_ppi [reducible] [constructor] {A : Type*} (B : A → Type*) : Group :=
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Group.mk (trunc 0 (Π*a, Ω (B a))) _
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-- definition AbGroup_arrow (A : Type) (G : AbGroup) : AbGroup :=
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-- AbGroup.mk (A → G) _
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definition ab_inf_group_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
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ab_inf_group (Π*a, Ω (Ω (B a))) :=
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⦃ab_inf_group, inf_group_ppi (λa, Ω (B a)), mul_comm :=
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begin
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intro f g, fapply ppi_eq,
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{ intro a, exact eckmann_hilton (f a) (g a) },
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{ rexact eq_of_square (eckmann_hilton_con2 (ppi_resp_pt f) (ppi_resp_pt g)) }
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end⦄
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-- definition pgroup_ppmap [instance] (A B : Type*) [pgroup B] : pgroup (ppmap A B) :=
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-- begin
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-- fapply pgroup.mk,
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-- { apply is_trunc_pmap },
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-- { intro f g, apply pmap.mk (λa, f a * g a),
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-- exact ap011 mul (respect_pt f) (respect_pt g) ⬝ !one_mul },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul.assoc },
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-- { intro f, apply pmap.mk (λa, (f a)⁻¹), apply inv_eq_one, apply respect_pt },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply one_mul },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul_one },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul.left_inv }
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-- end
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definition ab_group_trunc_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
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ab_group (trunc 0 (Π*a, Ω (Ω (B a)))) :=
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!trunc_ab_group
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-- definition Group_pmap (A : Type*) (G : Group) : Group :=
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-- Group_of_pgroup (ppmap A (pType_of_Group G))
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definition AbGroup_trunc_ppi [reducible] [constructor] {A : Type*} (B : A → Type*) : AbGroup :=
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AbGroup.mk (trunc 0 (Π*a, Ω (Ω (B a)))) _
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-- definition AbGroup_pmap (A : Type*) (G : AbGroup) : AbGroup :=
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-- AbGroup.mk (A →* pType_of_Group G)
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-- ⦃ ab_group, Group.struct (Group_pmap A G),
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-- mul_comm := by intro f g; apply pmap_eq_of_homotopy; intro a; apply mul.comm ⦄
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-- definition Group_pmap_homomorphism [constructor] {A A' : Type*} (f : A' →* A) (G : AbGroup) :
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-- Group_pmap A G →g Group_pmap A' G :=
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-- begin
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-- fapply homomorphism.mk,
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-- { intro g, exact g ∘* f},
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-- { intro g h, apply pmap_eq_of_homotopy, intro a, reflexivity }
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-- end
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end group
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@ -6,87 +6,23 @@ Authors: Floris van Doorn
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Reduced cohomology of spectra and cohomology theories
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-/
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import .spectrum .EM ..algebra.arrow_group .fwedge ..choice .pushout ..move_to_lib ..algebra.product_group
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import .spectrum ..algebra.arrow_group .fwedge ..choice .pushout ..algebra.product_group
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open eq spectrum int trunc pointed EM group algebra circle sphere nat EM.ops equiv susp is_trunc
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function fwedge cofiber bool lift sigma is_equiv choice pushout algebra unit pi
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definition is_exact_trunc_functor {A B : Type} {C : Type*} {f : A → B} {g : B → C}
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(H : is_exact_t f g) : @is_exact _ _ (ptrunc 0 C) (trunc_functor 0 f) (trunc_functor 0 g) :=
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begin
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constructor,
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{ intro a, esimp, induction a with a,
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exact ap tr (is_exact_t.im_in_ker H a) },
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{ intro b p, induction b with b, note q := !tr_eq_tr_equiv p, induction q with q,
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induction is_exact_t.ker_in_im H b q with a r,
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exact image.mk (tr a) (ap tr r) }
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end
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definition is_exact_homotopy {A B C : Type*} {f f' : A → B} {g g' : B → C}
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(p : f ~ f') (q : g ~ g') (H : is_exact f g) : is_exact f' g' :=
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begin
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induction p using homotopy.rec_on_idp,
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induction q using homotopy.rec_on_idp,
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assumption
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end
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-- move to arrow group
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definition ap1_pmap_mul {X Y : Type*} (f g : X →* Ω Y) :
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Ω→ (pmap_mul f g) ~* pmap_mul (Ω→ f) (Ω→ g) :=
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begin
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fconstructor,
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{ intro p, esimp,
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refine ap1_gen_con_left (respect_pt f) (respect_pt f)
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(respect_pt g) (respect_pt g) p ⬝ _,
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refine !whisker_right_idp ◾ !whisker_left_idp2, },
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{ refine !con.assoc ⬝ _,
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refine _ ◾ idp ⬝ _, rotate 1,
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rexact ap1_gen_con_left_idp (respect_pt f) (respect_pt g), esimp,
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refine !con.assoc ⬝ _,
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apply whisker_left, apply inv_con_eq_idp,
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refine !con2_con_con2 ⬝ ap011 concat2 _ _:
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refine eq_of_square (!natural_square ⬝hp !ap_id) ⬝ !con_idp }
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end
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definition pmap_mul_pcompose {A B C : Type*} (g h : B →* Ω C) (f : A →* B) :
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pmap_mul g h ∘* f ~* pmap_mul (g ∘* f) (h ∘* f) :=
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begin
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fconstructor,
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{ intro p, reflexivity },
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{ esimp, refine !idp_con ⬝ _, refine !con2_con_con2⁻¹ ⬝ whisker_right _ _,
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refine !ap_eq_ap011⁻¹ }
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end
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definition pcompose_pmap_mul {A B C : Type*} (h : B →* C) (f g : A →* Ω B) :
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Ω→ h ∘* pmap_mul f g ~* pmap_mul (Ω→ h ∘* f) (Ω→ h ∘* g) :=
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begin
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fconstructor,
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{ intro p, exact ap1_con h (f p) (g p) },
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{ refine whisker_left _ !con2_con_con2⁻¹ ⬝ _, refine !con.assoc⁻¹ ⬝ _,
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refine whisker_right _ (eq_of_square !ap1_gen_con_natural) ⬝ _,
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refine !con.assoc ⬝ whisker_left _ _, apply ap1_gen_con_idp }
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end
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definition loop_psusp_intro_pmap_mul {X Y : Type*} (f g : psusp X →* Ω Y) :
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loop_psusp_intro (pmap_mul f g) ~* pmap_mul (loop_psusp_intro f) (loop_psusp_intro g) :=
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pwhisker_right _ !ap1_pmap_mul ⬝* !pmap_mul_pcompose
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definition equiv_glue2 (Y : spectrum) (n : ℤ) : Ω (Ω (Y (n+2))) ≃* Y n :=
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begin
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refine (!equiv_glue ⬝e* loop_pequiv_loop (!equiv_glue ⬝e* loop_pequiv_loop _))⁻¹ᵉ*,
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refine pequiv_of_eq (ap Y _),
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exact add.assoc n 1 1
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end
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namespace cohomology
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definition EM_spectrum /-[constructor]-/ (G : AbGroup) : spectrum :=
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spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)
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/- The cohomology of X with coefficients in Y is
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trunc 0 (A →* Ω[2] (Y (n+2)))
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In the file arrow_group (in algebra) we construct the group structor on this type.
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-/
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definition cohomology (X : Type*) (Y : spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_pmap X (Y (n+2))
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definition parametrized_cohomology {X : Type*} (Y : X → spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_ppi (λx, Y x (n+2))
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definition ordinary_cohomology [reducible] (X : Type*) (G : AbGroup) (n : ℤ) : AbGroup :=
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cohomology X (EM_spectrum G) n
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@ -95,9 +31,17 @@ ordinary_cohomology X agℤ n
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notation `H^` n `[`:0 X:0 `, ` Y:0 `]`:0 := cohomology X Y n
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notation `H^` n `[`:0 X:0 `]`:0 := ordinary_cohomology_Z X n
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notation `pH^` n `[`:0 binders `, ` r:(scoped Y, parametrized_cohomology Y n) `]`:0 := r
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-- check H^3[S¹*,EM_spectrum agℤ]
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-- check H^3[S¹*]
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-- check pH^3[(x : S¹*), EM_spectrum agℤ]
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/- an alternate definition of cohomology -/
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definition cohomology_equiv_shomotopy_group_cotensor (X : Type*) (Y : spectrum) (n : ℤ) :
|
||||
H^n[X, Y] ≃ πₛ[-n] (sp_cotensor X Y) :=
|
||||
trunc_equiv_trunc 0 (!pfunext ⬝e loop_pequiv_loop !pfunext ⬝e loopn_pequiv_loopn 2
|
||||
(pequiv_of_eq (ap (λn, ppmap X (Y n)) (add.comm n 2 ⬝ ap (add 2) !neg_neg⁻¹))))
|
||||
|
||||
definition unpointed_cohomology (X : Type) (Y : spectrum) (n : ℤ) : AbGroup :=
|
||||
cohomology X₊ Y n
|
||||
|
|
|
|||
|
|
@ -5,9 +5,9 @@ Authors: Michael Shulman, Floris van Doorn
|
|||
|
||||
-/
|
||||
|
||||
import homotopy.LES_of_homotopy_groups .splice homotopy.susp ..colim ..pointed
|
||||
import homotopy.LES_of_homotopy_groups .splice homotopy.susp ..colim ..pointed .EM ..pointed_pi
|
||||
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
|
||||
seq_colim succ_str
|
||||
seq_colim succ_str EM EM.ops
|
||||
|
||||
/---------------------
|
||||
Basic definitions
|
||||
|
|
@ -48,6 +48,13 @@ namespace spectrum
|
|||
definition equiv_glue {N : succ_str} (E : gen_prespectrum N) [H : is_spectrum E] (n:N) : (E n) ≃* (Ω (E (S n))) :=
|
||||
pequiv_of_pmap (glue E n) (is_spectrum.is_equiv_glue E n)
|
||||
|
||||
definition equiv_glue2 (Y : spectrum) (n : ℤ) : Ω (Ω (Y (n+2))) ≃* Y n :=
|
||||
begin
|
||||
refine (!equiv_glue ⬝e* loop_pequiv_loop (!equiv_glue ⬝e* loop_pequiv_loop _))⁻¹ᵉ*,
|
||||
refine pequiv_of_eq (ap Y _),
|
||||
exact add.assoc n 1 1
|
||||
end
|
||||
|
||||
-- a square when we compose glue with transporting over a path in N
|
||||
definition glue_ptransport {N : succ_str} (X : gen_prespectrum N) {n n' : N} (p : n = n') :
|
||||
glue X n' ∘* ptransport X p ~* Ω→ (ptransport X (ap S p)) ∘* glue X n :=
|
||||
|
|
@ -224,7 +231,8 @@ namespace spectrum
|
|||
|
||||
-- Makes sense for any indexing succ_str. Could be done for
|
||||
-- prespectra too, but as with truncation, why bother?
|
||||
definition sp_cotensor {N : succ_str} (A : Type*) (B : gen_spectrum N) : gen_spectrum N :=
|
||||
|
||||
definition sp_cotensor [constructor] {N : succ_str} (A : Type*) (B : gen_spectrum N) : gen_spectrum N :=
|
||||
spectrum.MK (λn, ppmap A (B n))
|
||||
(λn, (loop_pmap_commute A (B (S n)))⁻¹ᵉ* ∘*ᵉ (pequiv_ppcompose_left (equiv_glue B n)))
|
||||
|
||||
|
|
@ -232,10 +240,9 @@ namespace spectrum
|
|||
-- Sections of parametrized spectra
|
||||
----------------------------------------
|
||||
|
||||
-- this definition must be changed to use dependent maps respecting the basepoint, presumably
|
||||
-- definition spi {N : succ_str} (A : Type) (E : A -> gen_spectrum N) : gen_spectrum N :=
|
||||
-- spectrum.MK (λn, ppi (λa, E a n))
|
||||
-- (λn, (loop_ppi_commute (λa, E a (S n)))⁻¹ᵉ* ∘*ᵉ equiv_ppi_right (λa, equiv_glue (E a) n))
|
||||
definition spi [constructor] {N : succ_str} (A : Type*) (E : A -> gen_spectrum N) : gen_spectrum N :=
|
||||
spectrum.MK (λn, Π*a, E a n)
|
||||
(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
|
||||
|
||||
/-----------------------------------------
|
||||
Fibers and long exact sequences
|
||||
|
|
@ -355,17 +362,8 @@ namespace spectrum
|
|||
end
|
||||
|
||||
/- Mapping spectra -/
|
||||
-- note: see also cotensor above
|
||||
|
||||
definition mapping_prespectrum [constructor] {N : succ_str} (X : Type*) (Y : gen_prespectrum N) :
|
||||
gen_prespectrum N :=
|
||||
gen_prespectrum.mk (λn, ppmap X (Y n)) (λn, pfunext X (Y (S n)) ∘* ppcompose_left (glue Y n))
|
||||
|
||||
definition mapping_spectrum [constructor] {N : succ_str} (X : Type*) (Y : gen_spectrum N) :
|
||||
gen_spectrum N :=
|
||||
gen_spectrum.mk
|
||||
(mapping_prespectrum X Y)
|
||||
(is_spectrum.mk (λn, to_is_equiv (pequiv_ppcompose_left (equiv_glue Y n) ⬝e
|
||||
pfunext X (Y (S n)))))
|
||||
|
||||
/- Spectrification -/
|
||||
|
||||
|
|
@ -446,4 +444,10 @@ namespace spectrum
|
|||
|
||||
/- Cofibers and stability -/
|
||||
|
||||
/- The Eilenberg-MacLane spectrum -/
|
||||
|
||||
definition EM_spectrum /-[constructor]-/ (G : AbGroup) : spectrum :=
|
||||
spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)
|
||||
|
||||
|
||||
end spectrum
|
||||
|
|
|
|||
|
|
@ -44,6 +44,17 @@ begin
|
|||
exact H
|
||||
end
|
||||
|
||||
definition is_exact_trunc_functor {A B : Type} {C : Type*} {f : A → B} {g : B → C}
|
||||
(H : is_exact_t f g) : @is_exact _ _ (ptrunc 0 C) (trunc_functor 0 f) (trunc_functor 0 g) :=
|
||||
begin
|
||||
constructor,
|
||||
{ intro a, esimp, induction a with a,
|
||||
exact ap tr (is_exact_t.im_in_ker H a) },
|
||||
{ intro b p, induction b with b, note q := !tr_eq_tr_equiv p, induction q with q,
|
||||
induction is_exact_t.ker_in_im H b q with a r,
|
||||
exact image.mk (tr a) (ap tr r) }
|
||||
end
|
||||
|
||||
definition is_contr_middle_of_is_exact {A B : Type} {C : Type*} {f : A → B} {g : B → C} (H : is_exact f g)
|
||||
[is_contr A] [is_set B] [is_contr C] : is_contr B :=
|
||||
begin
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ Authors: Ulrik Buchholtz
|
|||
|
||||
import .move_to_lib
|
||||
|
||||
open eq pointed equiv sigma
|
||||
open eq pointed equiv sigma is_equiv
|
||||
|
||||
/-
|
||||
The goal of this file is to give the truncation level
|
||||
|
|
@ -18,17 +18,18 @@ open eq pointed equiv sigma
|
|||
as a tool, because these appear in the more general
|
||||
statement is_trunc_ppi (the generality needed for induction).
|
||||
|
||||
Unfortunately, some of these names (ppi) and notations (Π*)
|
||||
clash with those in types.pi in the pi namespace.
|
||||
-/
|
||||
|
||||
namespace pointed
|
||||
|
||||
abbreviation ppi_resp_pt [unfold 3] := @ppi.resp_pt
|
||||
|
||||
definition ppi_const [constructor] {A : Type*} (P : A → Type*) : Π*(a : A), P a :=
|
||||
ppi.mk (λa, pt) idp
|
||||
|
||||
definition pointed_ppi [instance] [constructor] {A : Type*}
|
||||
(P : A → Type*) : pointed (ppi A P) :=
|
||||
pointed.mk (ppi.mk (λ a, pt) idp)
|
||||
pointed.mk (ppi_const P)
|
||||
|
||||
definition pppi [constructor] {A : Type*} (P : A → Type*) : Type* :=
|
||||
pointed.mk' (ppi A P)
|
||||
|
|
@ -39,13 +40,29 @@ namespace pointed
|
|||
(homotopy : f ~ g)
|
||||
(homotopy_pt : homotopy pt ⬝ ppi_resp_pt g = ppi_resp_pt f)
|
||||
|
||||
variables {A : Type*} {P : A → Type*} {f g : Π*(a : A), P a}
|
||||
variables {A : Type*} {P Q R : A → Type*} {f g h : Π*(a : A), P a}
|
||||
|
||||
infix ` ~~* `:50 := ppi_homotopy
|
||||
abbreviation ppi_to_homotopy_pt [unfold 5] := @ppi_homotopy.homotopy_pt
|
||||
abbreviation ppi_to_homotopy [coercion] [unfold 5] (p : f ~~* g) : Πa, f a = g a :=
|
||||
ppi_homotopy.homotopy p
|
||||
|
||||
variable (f)
|
||||
protected definition ppi_homotopy.refl : f ~~* f :=
|
||||
sorry
|
||||
variable {f}
|
||||
protected definition ppi_homotopy.rfl [refl] : f ~~* f :=
|
||||
ppi_homotopy.refl f
|
||||
|
||||
protected definition ppi_homotopy.symm [symm] (p : f ~~* g) : g ~~* f :=
|
||||
sorry
|
||||
|
||||
protected definition ppi_homotopy.trans [trans] (p : f ~~* g) (q : g ~~* h) : f ~~* h :=
|
||||
sorry
|
||||
|
||||
infix ` ⬝*' `:75 := ppi_homotopy.trans
|
||||
postfix `⁻¹*'`:(max+1) := ppi_homotopy.symm
|
||||
|
||||
definition ppi_equiv_pmap (A B : Type*) : (Π*(a : A), B) ≃ (A →* B) :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
|
|
@ -92,14 +109,14 @@ namespace pointed
|
|||
definition ppi_eq_equiv : (f = g) ≃ (f ~~* g) :=
|
||||
calc (f = g) ≃ ppi.sigma_char P f = ppi.sigma_char P g
|
||||
: eq_equiv_fn_eq (ppi.sigma_char P) f g
|
||||
... ≃ Σ(p : ppi.to_fun f = ppi.to_fun g),
|
||||
... ≃ Σ(p : f = g),
|
||||
pathover (λh, h pt = pt) (ppi_resp_pt f) p (ppi_resp_pt g)
|
||||
: sigma_eq_equiv _ _
|
||||
... ≃ Σ(p : ppi.to_fun f = ppi.to_fun g),
|
||||
... ≃ Σ(p : f = g),
|
||||
ppi_resp_pt f = ap (λh, h pt) p ⬝ ppi_resp_pt g
|
||||
: sigma_equiv_sigma_right
|
||||
(λp, eq_pathover_equiv_Fl p (ppi_resp_pt f) (ppi_resp_pt g))
|
||||
... ≃ Σ(p : ppi.to_fun f = ppi.to_fun g),
|
||||
... ≃ Σ(p : f = g),
|
||||
ppi_resp_pt f = apd10 p pt ⬝ ppi_resp_pt g
|
||||
: sigma_equiv_sigma_right
|
||||
(λp, equiv_eq_closed_right _ (whisker_right _ (ap_eq_apd10 p _)))
|
||||
|
|
@ -130,6 +147,57 @@ namespace pointed
|
|||
... ≃ Π*(a : A), Ω (pType.mk (P a) (f a))
|
||||
: erfl
|
||||
|
||||
variables {f g}
|
||||
definition eq_of_ppi_homotopy (h : f ~~* g) : f = g :=
|
||||
(ppi_eq_equiv f g)⁻¹ᵉ h
|
||||
|
||||
definition ppi_loop_pequiv : Ω (Π*(a : A), P a) ≃* Π*(a : A), Ω (P a) :=
|
||||
pequiv_of_equiv (ppi_loop_equiv pt) idp
|
||||
|
||||
definition pmap_compose_ppi [constructor] (g : Π(a : A), ppmap (P a) (Q a))
|
||||
(f : Π*(a : A), P a) : Π*(a : A), Q a :=
|
||||
proof ppi.mk (λa, g a (f a)) (ap (g pt) (ppi.resp_pt f) ⬝ respect_pt (g pt)) qed
|
||||
|
||||
definition pmap_compose_ppi_const_right (g : Π(a : A), ppmap (P a) (Q a)) :
|
||||
pmap_compose_ppi g (ppi_const P) ~~* ppi_const Q :=
|
||||
proof ppi_homotopy.mk (λa, respect_pt (g a)) !idp_con⁻¹ qed
|
||||
|
||||
definition pmap_compose_ppi_const_left (f : Π*(a : A), P a) :
|
||||
pmap_compose_ppi (λa, pconst (P a) (Q a)) f ~~* ppi_const Q :=
|
||||
sorry
|
||||
|
||||
definition ppi_compose_left [constructor] (g : Π(a : A), ppmap (P a) (Q a)) :
|
||||
(Π*(a : A), P a) →* Π*(a : A), Q a :=
|
||||
pmap.mk (pmap_compose_ppi g) (eq_of_ppi_homotopy (pmap_compose_ppi_const_right g))
|
||||
|
||||
definition pmap_compose_ppi_phomotopy_left [constructor] {g g' : Π(a : A), ppmap (P a) (Q a)}
|
||||
(f : Π*(a : A), P a) (p : Πa, g a ~* g' a) : pmap_compose_ppi g f ~~* pmap_compose_ppi g' f :=
|
||||
sorry
|
||||
|
||||
definition pmap_compose_ppi_pid_left [constructor]
|
||||
(f : Π*(a : A), P a) : pmap_compose_ppi (λa, pid (P a)) f ~~* f :=
|
||||
sorry
|
||||
|
||||
definition pmap_compose_pmap_compose_ppi [constructor] (h : Π(a : A), ppmap (Q a) (R a))
|
||||
(g : Π(a : A), ppmap (P a) (Q a)) :
|
||||
pmap_compose_ppi h (pmap_compose_ppi g f) ~~* pmap_compose_ppi (λa, h a ∘* g a) f :=
|
||||
sorry
|
||||
|
||||
definition ppi_pequiv_right [constructor] (g : Π(a : A), P a ≃* Q a) :
|
||||
(Π*(a : A), P a) ≃* Π*(a : A), Q a :=
|
||||
begin
|
||||
apply pequiv_of_pmap (ppi_compose_left g),
|
||||
apply adjointify _ (ppi_compose_left (λa, (g a)⁻¹ᵉ*)),
|
||||
{ intro f, apply eq_of_ppi_homotopy,
|
||||
refine !pmap_compose_pmap_compose_ppi ⬝*' _,
|
||||
refine pmap_compose_ppi_phomotopy_left _ (λa, !pright_inv) ⬝*' _,
|
||||
apply pmap_compose_ppi_pid_left },
|
||||
{ intro f, apply eq_of_ppi_homotopy,
|
||||
refine !pmap_compose_pmap_compose_ppi ⬝*' _,
|
||||
refine pmap_compose_ppi_phomotopy_left _ (λa, !pleft_inv) ⬝*' _,
|
||||
apply pmap_compose_ppi_pid_left }
|
||||
end
|
||||
|
||||
end pointed open pointed
|
||||
|
||||
open is_trunc is_conn
|
||||
|
|
|
|||
Loading…
Reference in a new issue