57 lines
2.6 KiB
Text
57 lines
2.6 KiB
Text
/-
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Copyright (c) 2017 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad
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Short exact sequences
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-/
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import .quotient_group
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open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
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is_trunc function sphere unit sum prod
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structure is_short_exact {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
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(is_emb : is_embedding f)
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(im_in_ker : Π(a:A), g (f a) = pt)
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(ker_in_im : Π(b:B), (g b = pt) → image f b)
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(is_surj : is_surjective g)
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structure is_short_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
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(is_emb : is_embedding f)
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(im_in_ker : Π(a:A), g (f a) = pt)
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(ker_in_im : Π(b:B), (g b = pt) → fiber f b)
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(is_surj : is_split_surjective g)
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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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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_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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end
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lemma is_short_exact_equiv {A B A' B' : Type} {C C' : Type*}
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{f' : A' → B'} {g' : B' → C'} (f : A → B) (g : B → C)
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(eA : A ≃ A') (eB : B ≃ B') (eC : C ≃* C')
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(h₁ : hsquare f f' eA eB) (h₂ : hsquare g g' eB eC)
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(H : is_short_exact f' g') : is_short_exact f g :=
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begin
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constructor,
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{ apply is_embedding_homotopy_closed_rev (homotopy_top_of_hsquare h₁),
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apply is_embedding_compose, apply is_embedding_of_is_equiv,
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apply is_embedding_compose, apply is_short_exact.is_emb H, apply is_embedding_of_is_equiv },
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{ intro a, refine homotopy_top_of_hsquare' (hhconcat h₁ h₂) a ⬝ _,
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refine ap eC⁻¹ _ ⬝ respect_pt eC⁻¹ᵉ*, exact is_short_exact.im_in_ker H (eA a) },
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{ intro b p, note q := eq_of_inv_eq ((homotopy_top_of_hsquare' h₂ b)⁻¹ ⬝ p) ⬝ respect_pt eC,
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induction is_short_exact.ker_in_im H (eB b) q with a' r,
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apply image.mk (eA⁻¹ a'),
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exact eq_of_fn_eq_fn eB ((homotopy_top_of_hsquare h₁⁻¹ʰᵗʸᵛ a')⁻¹ ⬝ r) },
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{ apply is_surjective_homotopy_closed_rev (homotopy_top_of_hsquare' h₂),
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apply is_surjective_compose, apply is_surjective_of_is_equiv,
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apply is_surjective_compose, apply is_short_exact.is_surj H, apply is_surjective_of_is_equiv }
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end
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