work of fiber of maps between EM-spaces
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Floris van Doorn
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2 changed files with 163 additions and 30 deletions
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@ -5,8 +5,8 @@ Authors: Jeremy Avigad
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Short exact sequences
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-/
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import homotopy.chain_complex eq2
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open pointed is_trunc equiv is_equiv eq algebra group trunc function
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import homotopy.chain_complex eq2 .quotient_group
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open pointed is_trunc equiv is_equiv eq algebra group trunc function fiber sigma
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structure is_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
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( im_in_ker : Π(a:A), g (f a) = pt)
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@ -34,7 +34,6 @@ begin
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induction h with a p, exact ap g p⁻¹ ⬝ is_exact.im_in_ker H a
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end
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-- TO DO: give less univalency proof
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definition is_exact_homotopy {A B : Type} {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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@ -54,8 +53,8 @@ begin
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exact image.mk (tr a) (ap tr r) }
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end
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definition is_contr_middle_of_is_exact {A B : Type} {C : Type*} {f : A → B} {g : B → C} (H : is_exact f g)
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[is_contr A] [is_set B] [is_contr C] : is_contr B :=
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definition is_contr_middle_of_is_exact {A B : Type} {C : Type*} {f : A → B} {g : B → C}
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(H : is_exact f g) [is_contr A] [is_set B] [is_contr C] : is_contr B :=
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begin
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apply is_contr.mk (f pt),
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intro b,
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@ -67,6 +66,99 @@ definition is_surjective_of_is_exact_of_is_contr {A B : Type} {C : Type*} {f : A
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(H : is_exact f g) [is_contr C] : is_surjective f :=
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λb, is_exact.ker_in_im H b !is_prop.elim
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definition is_embedding_of_is_exact_g {A B C : Group} {g : B →g C} {f : A →g B}
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(gf : is_exact_g f g) [is_contr A] : is_embedding g :=
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begin
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apply to_is_embedding_homomorphism, intro a p,
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induction is_exact.ker_in_im gf a p with x q,
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exact q⁻¹ ⬝ ap f !is_prop.elim ⬝ to_respect_one f
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end
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definition map_left_of_is_exact {G₃' G₃ G₂ : Type} {G₁ : Type*}
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{g : G₃ → G₂} {g' : G₃' → G₂} {f : G₂ → G₁} (H1 : is_exact g f) (H2 : is_exact g' f)
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(Hg' : is_embedding g') : G₃ → G₃' :=
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begin
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intro a,
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have fiber g' (g a),
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begin
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have is_prop (fiber g' (g a)), from !is_prop_fiber_of_is_embedding,
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induction is_exact.ker_in_im H2 (g a) (is_exact.im_in_ker H1 a) with a' p,
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exact fiber.mk a' p
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end,
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exact point this
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end
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definition map_left_of_is_exact_compute {G₃' G₃ G₂ : Type} {G₁ : Type*}
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{g : G₃ → G₂} {g' : G₃' → G₂} {f : G₂ → G₁} (H1 : is_exact g f) (H2 : is_exact g' f)
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(Hg' : is_embedding g') (a : G₃) : g' (map_left_of_is_exact H1 H2 Hg' a) = g a :=
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@point_eq _ _ g' _ _
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definition map_left_of_is_exact_compose {G₃'' G₃' G₃ G₂ : Type} {G₁ : Type*}
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{g : G₃ → G₂} {g' : G₃' → G₂} {g'' : G₃'' → G₂} {f : G₂ → G₁}
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(H1 : is_exact g f) (H2 : is_exact g' f) (H3 : is_exact g'' f)
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(Hg' : is_embedding g') (Hg'' : is_embedding g'') (a : G₃) :
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map_left_of_is_exact H2 H3 Hg'' (map_left_of_is_exact H1 H2 Hg' a) =
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map_left_of_is_exact H1 H3 Hg'' a :=
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begin
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refine @is_injective_of_is_embedding _ _ g'' _ _ _ _,
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refine !map_left_of_is_exact_compute ⬝ _ ⬝ !map_left_of_is_exact_compute⁻¹,
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exact map_left_of_is_exact_compute H1 H2 Hg' a
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end
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definition map_left_of_is_exact_id {G₃ G₂ : Type} {G₁ : Type*}
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{g : G₃ → G₂} {f : G₂ → G₁} (H1 : is_exact g f) (Hg : is_embedding g) (a : G₃) :
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map_left_of_is_exact H1 H1 Hg a = a :=
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begin
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refine @is_injective_of_is_embedding _ _ g _ _ _ _,
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exact map_left_of_is_exact_compute H1 H1 Hg a
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end
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definition map_left_of_is_exact_homotopy {G₃' G₃ G₂ : Type} {G₁ : Type*}
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{g : G₃ → G₂} {g' g'' : G₃' → G₂} {f : G₂ → G₁} (H1 : is_exact g f) (H2 : is_exact g' f)
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(H3 : is_exact g'' f) (Hg' : is_embedding g') (Hg'' : is_embedding g'') (p : g' ~ g'') :
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map_left_of_is_exact H1 H2 Hg' ~ map_left_of_is_exact H1 H3 Hg'' :=
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begin
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intro a,
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refine @is_injective_of_is_embedding _ _ g' _ _ _ _,
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exact !map_left_of_is_exact_compute ⬝ (!p ⬝ !map_left_of_is_exact_compute)⁻¹,
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end
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definition homomorphism_left_of_is_exact_g {G₃' G₃ G₂ G₁ : Group}
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{g : G₃ →g G₂} {g' : G₃' →g G₂} {f : G₂ →g G₁} (H1 : is_exact_g g f) (H2 : is_exact_g g' f)
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(Hg' : is_embedding g') : G₃ →g G₃' :=
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begin
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apply homomorphism.mk (map_left_of_is_exact H1 H2 Hg'),
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{ intro a a', refine @is_injective_of_is_embedding _ _ g' _ _ _ _,
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exact !point_eq ⬝ to_respect_mul g a a' ⬝
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(to_respect_mul g' _ _ ⬝ ap011 mul !point_eq !point_eq)⁻¹ }
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end
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definition isomorphism_left_of_is_exact_g {G₃' G₃ G₂ G₁ : Group}
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{g : G₃ →g G₂} {g' : G₃' →g G₂} {f : G₂ →g G₁} (H1 : is_exact g f) (H2 : is_exact g' f)
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(Hg : is_embedding g) (Hg' : is_embedding g') : G₃ ≃g G₃' :=
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begin
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fapply isomorphism.mk, exact homomorphism_left_of_is_exact_g H1 H2 Hg',
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fapply adjointify, exact homomorphism_left_of_is_exact_g H2 H1 Hg,
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{ intro a, refine @is_injective_of_is_embedding _ _ g' _ _ _ _,
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refine map_left_of_is_exact_compute H1 H2 Hg' _ ⬝ map_left_of_is_exact_compute H2 H1 Hg a },
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{ intro a, refine @is_injective_of_is_embedding _ _ g _ _ _ _,
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refine map_left_of_is_exact_compute H2 H1 Hg _ ⬝ map_left_of_is_exact_compute H1 H2 Hg' a },
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end
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definition is_exact_incl_of_subgroup {G H : Group} (f : G →g H) :
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is_exact (incl_of_subgroup (kernel_subgroup f)) f :=
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begin
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apply is_exact.mk,
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{ intro x, cases x with x p, exact p },
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{ intro x p, exact image.mk ⟨x, p⟩ idp }
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end
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definition isomorphism_kernel_of_is_exact {G₄ G₃ G₂ G₁ : Group}
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{h : G₄ →g G₃} {g : G₃ →g G₂} {f : G₂ →g G₁} (H1 : is_exact h g) (H2 : is_exact g f)
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(HG : is_contr G₄) : G₃ ≃g kernel f :=
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isomorphism_left_of_is_exact_g H2 (is_exact_incl_of_subgroup f)
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(is_embedding_of_is_exact_g H1) (is_embedding_incl_of_subgroup _)
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section chain_complex
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open succ_str chain_complex
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definition is_exact_of_is_exact_at {N : succ_str} {A : chain_complex N} {n : N}
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@ -89,12 +181,10 @@ structure is_short_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C)
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lemma is_short_exact_of_is_exact {X A B C Y : Group}
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(k : X →g A) (f : A →g B) (g : B →g C) (l : C →g Y)
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(hX : is_contr X) (hY : is_contr Y)
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(kf : is_exact k f) (fg : is_exact f g) (gl : is_exact g l) : is_short_exact f g :=
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(kf : is_exact_g k f) (fg : is_exact_g f g) (gl : is_exact_g g l) : is_short_exact f g :=
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begin
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constructor,
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{ apply to_is_embedding_homomorphism, intro a p,
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induction is_exact.ker_in_im kf a p with x q,
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exact q⁻¹ ⬝ ap k !is_prop.elim ⬝ to_respect_one k },
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{ exact is_embedding_of_is_exact_g kf },
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{ exact is_exact.im_in_ker fg },
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{ exact is_exact.ker_in_im fg },
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{ intro c, exact is_exact.ker_in_im gl c !is_prop.elim },
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@ -129,4 +219,46 @@ begin
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{ exact is_short_exact.ker_in_im H }
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end
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/- TODO: move and remove other versions -/
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definition is_surjective_qg_map {A : Group} (N : normal_subgroup_rel A) :
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is_surjective (qg_map N) :=
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begin
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intro x, induction x,
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fapply image.mk,
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exact a, reflexivity,
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apply is_prop.elimo
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end
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definition is_surjective_ab_qg_map {A : AbGroup} (N : subgroup_rel A) :
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is_surjective (ab_qg_map N) :=
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is_surjective_qg_map _
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definition qg_map_eq_one {A : Group} {K : normal_subgroup_rel A} (g : A) (H : K g) :
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qg_map K g = 1 :=
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begin
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apply set_quotient.eq_of_rel,
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have e : g * 1⁻¹ = g,
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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exact transport (λx, K x) e⁻¹ H
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end
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definition ab_qg_map_eq_one {A : AbGroup} {K : subgroup_rel A} (g : A) (H : K g) :
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ab_qg_map K g = 1 :=
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qg_map_eq_one g H
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definition is_short_exact_normal_subgroup {G : Group} (S : normal_subgroup_rel G) :
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is_short_exact (incl_of_subgroup S) (qg_map S) :=
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begin
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fconstructor,
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{ exact is_embedding_incl_of_subgroup S },
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{ intro a, fapply qg_map_eq_one, induction a with b p, exact p },
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{ intro b p, fapply image.mk,
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{ apply sigma.mk b, fapply rel_of_qg_map_eq_one, exact p },
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reflexivity },
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{ exact is_surjective_qg_map S },
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end
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end algebra
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@ -1,7 +1,7 @@
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-- Authors: Floris van Doorn
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import homotopy.EM algebra.category.functor.equivalence types.pointed2 ..pointed_pi ..pointed
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..move_to_lib .susp ..algebra.quotient_group
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..move_to_lib .susp ..algebra.exactness
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open eq equiv is_equiv algebra group nat pointed EM.ops is_trunc trunc susp function is_conn
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@ -620,31 +620,32 @@ namespace EM
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apply is_conn_fiber, apply is_conn_of_is_conn_succ
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end
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section --move
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open chain_complex succ_str
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-- definition isomorphism_kernel_of_trivial {N : succ_str} (X : chain_complex N) {n : N}
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-- (H1 : is_exact_at X n) (H2 : is_exact_at X (S n))
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-- (HX1 : is_contr (X n)) (HG2 : pgroup (X (S n)))
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-- : Group_of_pgroup (X (S n)) ≃g kernel (homomorphism.mk (cc_to_fn X _) _) :=
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-- _
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section
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open chain_complex prod fin
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/- TODO: other cases -/
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definition LES_isomorphism_kernel_of_trivial.{u}
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{X Y : pType.{u}} (f : X →* Y) (n : ℕ) [H : is_succ n]
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(H1 : is_contr (πg[n+1] Y)) : πg[n] (pfiber f) ≃g kernel (π→g[n] f) :=
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begin
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induction H with n,
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have H2 : is_exact (π→g[n+1] (ppoint f)) (π→g[n+1] f),
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from is_exact_of_is_exact_at (is_exact_LES_of_homotopy_groups f (n+1, 0)),
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have H3 : is_exact (π→g[n+1] (boundary_map f) ∘g ghomotopy_group_succ_in Y n)
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(π→g[n+1] (ppoint f)),
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from is_exact_of_is_exact_at (is_exact_LES_of_homotopy_groups f (n+1, 1)),
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exact isomorphism_kernel_of_is_exact H3 H2 H1
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end
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end
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-- definition is_equiv_of_trivial (X : chain_complex N) {n : N}
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-- (H1 : is_exact_at X n) (H2 : is_exact_at X (S n))
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-- [HX1 : is_contr (X n)] [HX2 : is_contr (X (S (S (S n))))]
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-- [pgroup (X (S n))] [pgroup (X (S (S n)))] [is_mul_hom (cc_to_fn X (S n))]
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-- : is_equiv (cc_to_fn X (S n)) :=
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-- begin
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-- apply is_equiv_of_is_surjective_of_is_embedding,
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-- { apply is_embedding_of_trivial X, apply H2},
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-- { apply is_surjective_of_trivial X, apply H1},
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-- end
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open group algebra
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definition homotopy_group_fiber_EM1_functor {G H : Group} (φ : G →g H) :
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open group algebra is_trunc
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definition homotopy_group_fiber_EM1_functor.{u} {G H : Group.{u}} (φ : G →g H) :
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π₁ (pfiber (EM1_functor φ)) ≃g kernel φ :=
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have H1 : is_trunc 1 (EM1 H), from sorry,
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have H2 : 1 <[ℕ] 1 + 1, from sorry,
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LES_isomorphism_kernel_of_trivial (EM1_functor φ) 1
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(@trivial_homotopy_group_of_is_trunc _ 1 2 H1 H2) ⬝g
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sorry
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definition homotopy_group_fiber_EMadd1_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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