124 lines
4.9 KiB
Text
124 lines
4.9 KiB
Text
/-
|
|
Copyright (c) 2017 Jeremy Avigad. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Authors: Jeremy Avigad
|
|
|
|
Short exact sequences
|
|
-/
|
|
import homotopy.chain_complex eq2
|
|
open pointed is_trunc equiv is_equiv eq algebra group trunc function
|
|
|
|
structure is_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
|
|
( im_in_ker : Π(a:A), g (f a) = pt)
|
|
( ker_in_im : Π(b:B), (g b = pt) → fiber f b)
|
|
|
|
structure is_exact {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
|
|
( im_in_ker : Π(a:A), g (f a) = pt)
|
|
( ker_in_im : Π(b:B), (g b = pt) → image f b)
|
|
|
|
namespace algebra
|
|
|
|
definition is_exact_g {A B C : Group} (f : A →g B) (g : B →g C) :=
|
|
is_exact f g
|
|
|
|
definition is_exact_ag {A B C : AbGroup} (f : A →g B) (g : B →g C) :=
|
|
is_exact f g
|
|
|
|
definition is_exact_g.mk {A B C : Group} {f : A →g B} {g : B →g C}
|
|
(H₁ : Πa, g (f a) = 1) (H₂ : Πb, g b = 1 → image f b) : is_exact_g f g :=
|
|
is_exact.mk H₁ H₂
|
|
|
|
definition is_exact.im_in_ker2 {A B : Type} {C : Set*} {f : A → B} {g : B → C} (H : is_exact f g)
|
|
{b : B} (h : image f b) : g b = pt :=
|
|
begin
|
|
induction h with a p, exact ap g p⁻¹ ⬝ is_exact.im_in_ker H a
|
|
end
|
|
|
|
-- TO DO: give less univalency proof
|
|
definition is_exact_homotopy {A B : Type} {C : Type*} {f f' : A → B} {g g' : B → C}
|
|
(p : f ~ f') (q : g ~ g') (H : is_exact f g) : is_exact f' g' :=
|
|
begin
|
|
induction p using homotopy.rec_on_idp,
|
|
induction q using homotopy.rec_on_idp,
|
|
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
|
|
apply is_contr.mk (f pt),
|
|
intro b,
|
|
induction is_exact.ker_in_im H b !is_prop.elim,
|
|
exact ap f !is_prop.elim ⬝ p
|
|
end
|
|
|
|
definition is_surjective_of_is_exact_of_is_contr {A B : Type} {C : Type*} {f : A → B} {g : B → C}
|
|
(H : is_exact f g) [is_contr C] : is_surjective f :=
|
|
λb, is_exact.ker_in_im H b !is_prop.elim
|
|
|
|
section chain_complex
|
|
open succ_str chain_complex
|
|
definition is_exact_of_is_exact_at {N : succ_str} {A : chain_complex N} {n : N}
|
|
(H : is_exact_at A n) : is_exact (cc_to_fn A (S n)) (cc_to_fn A n) :=
|
|
is_exact.mk (cc_is_chain_complex A n) H
|
|
end chain_complex
|
|
|
|
structure is_short_exact {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
|
|
(is_emb : is_embedding f)
|
|
(im_in_ker : Π(a:A), g (f a) = pt)
|
|
(ker_in_im : Π(b:B), (g b = pt) → image f b)
|
|
(is_surj : is_surjective g)
|
|
|
|
structure is_short_exact_t {A B : Type} {C : Type*} (f : A → B) (g : B → C) :=
|
|
(is_emb : is_embedding f)
|
|
(im_in_ker : Π(a:A), g (f a) = pt)
|
|
(ker_in_im : Π(b:B), (g b = pt) → fiber f b)
|
|
(is_surj : is_split_surjective g)
|
|
|
|
lemma is_short_exact_of_is_exact {X A B C Y : Group}
|
|
(k : X →g A) (f : A →g B) (g : B →g C) (l : C →g Y)
|
|
(hX : is_contr X) (hY : is_contr Y)
|
|
(kf : is_exact k f) (fg : is_exact f g) (gl : is_exact g l) : is_short_exact f g :=
|
|
begin
|
|
constructor,
|
|
{ apply to_is_embedding_homomorphism, intro a p,
|
|
induction is_exact.ker_in_im kf a p with x q,
|
|
exact q⁻¹ ⬝ ap k !is_prop.elim ⬝ to_respect_one k },
|
|
{ exact is_exact.im_in_ker fg },
|
|
{ exact is_exact.ker_in_im fg },
|
|
{ intro c, exact is_exact.ker_in_im gl c !is_prop.elim },
|
|
end
|
|
|
|
lemma is_short_exact_equiv {A B A' B' : Type} {C C' : Type*}
|
|
{f' : A' → B'} {g' : B' → C'} (f : A → B) (g : B → C)
|
|
(eA : A ≃ A') (eB : B ≃ B') (eC : C ≃* C')
|
|
(h₁ : hsquare f f' eA eB) (h₂ : hsquare g g' eB eC)
|
|
(H : is_short_exact f' g') : is_short_exact f g :=
|
|
begin
|
|
constructor,
|
|
{ apply is_embedding_homotopy_closed_rev (homotopy_top_of_hsquare h₁),
|
|
apply is_embedding_compose, apply is_embedding_of_is_equiv,
|
|
apply is_embedding_compose, apply is_short_exact.is_emb H, apply is_embedding_of_is_equiv },
|
|
{ intro a, refine homotopy_top_of_hsquare' (hhconcat h₁ h₂) a ⬝ _,
|
|
refine ap eC⁻¹ _ ⬝ respect_pt eC⁻¹ᵉ*, exact is_short_exact.im_in_ker H (eA a) },
|
|
{ intro b p, note q := eq_of_inv_eq ((homotopy_top_of_hsquare' h₂ b)⁻¹ ⬝ p) ⬝ respect_pt eC,
|
|
induction is_short_exact.ker_in_im H (eB b) q with a' r,
|
|
apply image.mk (eA⁻¹ a'),
|
|
exact eq_of_fn_eq_fn eB ((homotopy_top_of_hsquare h₁⁻¹ʰᵗʸᵛ a')⁻¹ ⬝ r) },
|
|
{ apply is_surjective_homotopy_closed_rev (homotopy_top_of_hsquare' h₂),
|
|
apply is_surjective_compose, apply is_surjective_of_is_equiv,
|
|
apply is_surjective_compose, apply is_short_exact.is_surj H, apply is_surjective_of_is_equiv }
|
|
end
|
|
|
|
end algebra
|