generalize is_exact

algebra/ses.hlean

@@ -13,16 +13,12 @@ import algebra.group_theory hit.set_quotient types.sigma types.list types.sum .q
 open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function group trunc
      equiv is_equiv
 
-structure is_exact {A B C : AbGroup} (f : A →g B) (g : B →g C) :=
-  ( im_in_ker : Π(a:A), g (f a) = 1)
-  ( ker_in_im : Π(b:B), (g b = 1) → image_subgroup f b)
-
 structure SES (A B C : AbGroup) :=
   ( f : A →g B)
   ( g : B →g C)
   ( Hf : is_embedding f)
   ( Hg : is_surjective g)
-  ( ex : is_exact f g)
+  ( ex : is_exact_ag f g)
 
 definition SES_of_inclusion {A B : AbGroup} (f : A →g B) (Hf : is_embedding f) : SES A B (quotient_ab_group (image_subgroup f)) :=
   begin
@@ -37,7 +33,7 @@ definition SES_of_inclusion {A B : AbGroup} (f : A →g B) (Hf : is_embedding f)
     intro a,
     fapply qg_map_eq_one, fapply tr, fapply fiber.mk, exact a, reflexivity,
     intro b, intro p,
-    fapply rel_of_ab_qg_map_eq_one, assumption
+    exact rel_of_ab_qg_map_eq_one _ p
   end
 
 definition SES_of_subgroup {B : AbGroup} (S : subgroup_rel B) : SES (ab_subgroup S) B (quotient_ab_group S) :=

homotopy/cohomology.hlean

@@ -11,18 +11,6 @@ import .spectrum .EM ..algebra.arrow_group .fwedge ..choice .pushout ..move_to_l
 open eq spectrum int trunc pointed EM group algebra circle sphere nat EM.ops equiv susp is_trunc
      function fwedge cofiber bool lift sigma is_equiv choice pushout algebra unit pi
 
--- TODO: move
-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)
-
-definition is_exact_g {A B C : Group} (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_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

homotopy/pushout.hlean

@@ -464,10 +464,6 @@ namespace pushout
 
   /- universal property of cofiber -/
 
-  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)
-
   definition cofiber_exact_1 {X Y Z : Type*} (f : X →* Y) (g : pcofiber f →* Z) :
     (g ∘* pcod f) ∘* f ~* pconst X Z :=
   !passoc ⬝* pwhisker_left _ !pcod_pcompose ⬝* !pcompose_pconst

homotopy/spherical_fibrations.hlean

@@ -59,7 +59,7 @@ namespace spherical_fibrations
   begin
     intro X, cases X with X p,
     apply sigma.mk (psusp X), induction p with f, apply tr,
-    apply susp.psusp_equiv f
+    apply susp.psusp_pequiv f
   end
 
   definition BF_of_BG {n : ℕ} : BG n → BF n :=

move_to_lib.hlean

@@ -8,6 +8,24 @@ open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc
 definition add_comm_right {A : Type} [add_comm_semigroup A] (n m k : A) : n + m + k = n + k + m :=
 !add.assoc ⬝ ap (add n) !add.comm ⬝ !add.assoc⁻¹
 
+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)
+
+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
+
 namespace algebra
   definition inf_group_loopn (n : ℕ) (A : Type*) [H : is_succ n] : inf_group (Ω[n] A) :=
   by induction H; exact _

pointed.hlean

@@ -930,7 +930,7 @@ namespace pointed
             (ppcompose_left_loop_phomotopy' (pmap_of_map f x₀) !pconst) :=
   begin
     fapply phomotopy.mk,
-    { esimp, esimp [pmap_eq_equiv], intro p,
+    { esimp, intro p,
      refine _ ⬝ ap011 (λx y, phomotopy_of_eq (ap1_gen _ x y _))
                  proof !eq_of_phomotopy_refl⁻¹ qed proof !eq_of_phomotopy_refl⁻¹ qed,
      refine _ ⬝ ap phomotopy_of_eq !ap1_gen_idp_left⁻¹,