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desc_f_one {Y Z : C}
(f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
J.cocomplex.X 1 ⟶ I.cocomplex.X 1 | exact.desc (desc_f_zero f I J ≫ I.cocomplex.d 0 1) (J.ι.f 0) (J.cocomplex.d 0 1)
(abelian.exact.op _ _ J.exact₀) (by simp [←category.assoc, desc_f_zero]) | def | category_theory.InjectiveResolution.desc_f_one | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | Auxiliary construction for `desc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_f_one_zero_comm {Y Z : C}
(f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
J.cocomplex.d 0 1 ≫ desc_f_one f I J = desc_f_zero f I J ≫ I.cocomplex.d 0 1 | by simp [desc_f_zero, desc_f_one] | lemma | category_theory.InjectiveResolution.desc_f_one_zero_comm | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc_f_succ {Y Z : C}
(I : InjectiveResolution Y) (J : InjectiveResolution Z)
(n : ℕ) (g : J.cocomplex.X n ⟶ I.cocomplex.X n) (g' : J.cocomplex.X (n+1) ⟶ I.cocomplex.X (n+1))
(w : J.cocomplex.d n (n+1) ≫ g' = g ≫ I.cocomplex.d n (n+1)) :
Σ' g'' : J.cocomplex.X (n+2) ⟶ I.cocomplex.X (n+2),
J.cocomplex.d (n+1... | ⟨@exact.desc C _ _ _ _ _ _ _ _ _
(g' ≫ I.cocomplex.d (n+1) (n+2))
(J.cocomplex.d n (n+1))
(J.cocomplex.d (n+1) (n+2)) (abelian.exact.op _ _ (J.exact _))
(by simp [←category.assoc, w]), (by simp)⟩ | def | category_theory.InjectiveResolution.desc_f_succ | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | Auxiliary construction for `desc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc {Y Z : C}
(f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
J.cocomplex ⟶ I.cocomplex | cochain_complex.mk_hom _ _ (desc_f_zero f _ _) (desc_f_one f _ _)
(desc_f_one_zero_comm f I J).symm
(λ n ⟨g, g', w⟩, ⟨(desc_f_succ I J n g g' w.symm).1, (desc_f_succ I J n g g' w.symm).2.symm⟩) | def | category_theory.InjectiveResolution.desc | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex.mk_hom"
] | A morphism in `C` descends to a chain map between injective resolutions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_commutes {Y Z : C}
(f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
J.ι ≫ desc f I J = (cochain_complex.single₀ C).map f ≫ I.ι | begin
ext n,
rcases n with (_|_|n);
{ dsimp [desc, desc_f_one, desc_f_zero], simp, },
end | lemma | category_theory.InjectiveResolution.desc_commutes | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex.single₀"
] | The resolution maps intertwine the descent of a morphism and that morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_homotopy_zero_zero {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}
(f : I.cocomplex ⟶ J.cocomplex)
(comm : I.ι ≫ f = 0) : I.cocomplex.X 1 ⟶ J.cocomplex.X 0 | exact.desc (f.f 0) (I.ι.f 0) (I.cocomplex.d 0 1) (abelian.exact.op _ _ I.exact₀)
(congr_fun (congr_arg homological_complex.hom.f comm) 0) | def | category_theory.InjectiveResolution.desc_homotopy_zero_zero | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"comm"
] | An auxiliary definition for `desc_homotopy_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_homotopy_zero_one {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}
(f : I.cocomplex ⟶ J.cocomplex)
(comm : I.ι ≫ f = (0 : _ ⟶ J.cocomplex)) : I.cocomplex.X 2 ⟶ J.cocomplex.X 1 | exact.desc (f.f 1 - desc_homotopy_zero_zero f comm ≫ J.cocomplex.d 0 1)
(I.cocomplex.d 0 1) (I.cocomplex.d 1 2) (abelian.exact.op _ _ (I.exact _))
(by simp [desc_homotopy_zero_zero, ←category.assoc]) | def | category_theory.InjectiveResolution.desc_homotopy_zero_one | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"comm"
] | An auxiliary definition for `desc_homotopy_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_homotopy_zero_succ {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}
(f : I.cocomplex ⟶ J.cocomplex) (n : ℕ)
(g : I.cocomplex.X (n + 1) ⟶ J.cocomplex.X n)
(g' : I.cocomplex.X (n + 2) ⟶ J.cocomplex.X (n + 1))
(w : f.f (n + 1) = I.cocomplex.d (n+1) (n+2) ≫ g' + g ≫ J.cocomplex.d n (n+1)) :
... | exact.desc (f.f (n+2) - g' ≫ J.cocomplex.d _ _) (I.cocomplex.d (n+1) (n+2))
(I.cocomplex.d (n+2) (n+3)) (abelian.exact.op _ _ (I.exact _))
(by simp [preadditive.comp_sub, ←category.assoc, preadditive.sub_comp,
show I.cocomplex.d (n+1) (n+2) ≫ g' = f.f (n + 1) - g ≫ J.cocomplex.d n (n+1),
by {rw w, s... | def | category_theory.InjectiveResolution.desc_homotopy_zero_succ | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | An auxiliary definition for `desc_homotopy_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_homotopy_zero {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}
(f : I.cocomplex ⟶ J.cocomplex)
(comm : I.ι ≫ f = 0) :
homotopy f 0 | homotopy.mk_coinductive _ (desc_homotopy_zero_zero f comm) (by simp [desc_homotopy_zero_zero])
(desc_homotopy_zero_one f comm) (by simp [desc_homotopy_zero_one])
(λ n ⟨g, g', w⟩, ⟨desc_homotopy_zero_succ f n g g' (by simp only [w, add_comm]),
by simp [desc_homotopy_zero_succ, w]⟩) | def | category_theory.InjectiveResolution.desc_homotopy_zero | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"comm",
"homotopy",
"homotopy.mk_coinductive"
] | Any descent of the zero morphism is homotopic to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_homotopy {Y Z : C} (f : Y ⟶ Z) {I : InjectiveResolution Y} {J : InjectiveResolution Z}
(g h : I.cocomplex ⟶ J.cocomplex)
(g_comm : I.ι ≫ g = (cochain_complex.single₀ C).map f ≫ J.ι)
(h_comm : I.ι ≫ h = (cochain_complex.single₀ C).map f ≫ J.ι) :
homotopy g h | homotopy.equiv_sub_zero.inv_fun (desc_homotopy_zero _ (by simp [g_comm, h_comm])) | def | category_theory.InjectiveResolution.desc_homotopy | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex.single₀",
"homotopy"
] | Two descents of the same morphism are homotopic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_id_homotopy (X : C) (I : InjectiveResolution X) :
homotopy (desc (𝟙 X) I I) (𝟙 I.cocomplex) | by apply desc_homotopy (𝟙 X); simp | def | category_theory.InjectiveResolution.desc_id_homotopy | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"homotopy"
] | The descent of the identity morphism is homotopic to the identity cochain map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_comp_homotopy {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)
(I : InjectiveResolution X) (J : InjectiveResolution Y) (K : InjectiveResolution Z) :
homotopy (desc (f ≫ g) K I) (desc f J I ≫ desc g K J) | by apply desc_homotopy (f ≫ g); simp | def | category_theory.InjectiveResolution.desc_comp_homotopy | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"homotopy"
] | The descent of a composition is homotopic to the composition of the descents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv {X : C} (I J : InjectiveResolution X) :
homotopy_equiv I.cocomplex J.cocomplex | { hom := desc (𝟙 X) J I,
inv := desc (𝟙 X) I J,
homotopy_hom_inv_id := (desc_comp_homotopy (𝟙 X) (𝟙 X) I J I).symm.trans $
by simpa [category.id_comp] using desc_id_homotopy _ _,
homotopy_inv_hom_id := (desc_comp_homotopy (𝟙 X) (𝟙 X) J I J).symm.trans $
by simpa [category.id_comp] using desc_id_homo... | def | category_theory.InjectiveResolution.homotopy_equiv | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"homotopy_equiv"
] | Any two injective resolutions are homotopy equivalent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv_hom_ι {X : C} (I J : InjectiveResolution X) :
I.ι ≫ (homotopy_equiv I J).hom = J.ι | by simp [homotopy_equiv] | lemma | category_theory.InjectiveResolution.homotopy_equiv_hom_ι | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"homotopy_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_equiv_inv_ι {X : C} (I J : InjectiveResolution X) :
J.ι ≫ (homotopy_equiv I J).inv = I.ι | by simp [homotopy_equiv] | lemma | category_theory.InjectiveResolution.homotopy_equiv_inv_ι | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"homotopy_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_resolution (Z : C) [has_injective_resolution Z] : cochain_complex C ℕ | (has_injective_resolution.out Z).some.cocomplex | abbreviation | category_theory.injective_resolution | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex"
] | An arbitrarily chosen injective resolution of an object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_resolution.ι (Z : C) [has_injective_resolution Z] :
(cochain_complex.single₀ C).obj Z ⟶ injective_resolution Z | (has_injective_resolution.out Z).some.ι | abbreviation | category_theory.injective_resolution.ι | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex.single₀"
] | The cochain map from cochain complex consisting of `Z` supported in degree `0`
back to the arbitrarily chosen injective resolution `injective_resolution Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_resolution.desc {X Y : C} (f : X ⟶ Y)
[has_injective_resolution X] [has_injective_resolution Y] :
injective_resolution X ⟶ injective_resolution Y | InjectiveResolution.desc f _ _ | abbreviation | category_theory.injective_resolution.desc | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | The descent of a morphism to a cochain map between the arbitrarily chosen injective resolutions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_resolutions : C ⥤ homotopy_category C (complex_shape.up ℕ) | { obj := λ X, (homotopy_category.quotient _ _).obj (injective_resolution X),
map := λ X Y f, (homotopy_category.quotient _ _).map (injective_resolution.desc f),
map_id' := λ X, begin
rw ←(homotopy_category.quotient _ _).map_id,
apply homotopy_category.eq_of_homotopy,
apply InjectiveResolution.desc_id_ho... | def | category_theory.injective_resolutions | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"complex_shape.up",
"homotopy_category",
"homotopy_category.eq_of_homotopy",
"homotopy_category.quotient",
"map_comp",
"map_id"
] | Taking injective resolutions is functorial,
if considered with target the homotopy category
(`ℕ`-indexed cochain complexes and chain maps up to homotopy). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_f_d {X Y : C} (f : X ⟶ Y) : exact f (d f) | (abelian.exact_iff _ _).2 $
⟨by simp, zero_of_comp_mono (ι _) $ by rw [category.assoc, kernel.condition]⟩ | lemma | category_theory.exact_f_d | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_cocomplex (Z : C) : cochain_complex C ℕ | cochain_complex.mk'
(injective.under Z) (injective.syzygies (injective.ι Z)) (injective.d (injective.ι Z))
(λ ⟨X, Y, f⟩, ⟨injective.syzygies f, injective.d f, (exact_f_d f).w⟩) | def | category_theory.InjectiveResolution.of_cocomplex | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex",
"cochain_complex.mk'"
] | Auxiliary definition for `InjectiveResolution.of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (Z : C) : InjectiveResolution Z | { cocomplex := of_cocomplex Z,
ι := cochain_complex.mk_hom _ _ (injective.ι Z) 0
(by { simp only [of_cocomplex_d, eq_self_iff_true, eq_to_hom_refl, category.comp_id,
dite_eq_ite, if_true, comp_zero],
exact (exact_f_d (injective.ι Z)).w, } ) (λ n _, ⟨0, by ext⟩),
injective := by { rintros (_|_|_|n); ... | def | category_theory.InjectiveResolution.of | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex.mk_hom",
"dite_eq_ite"
] | In any abelian category with enough injectives,
`InjectiveResolution.of Z` constructs an injective resolution of the object `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homological_complex.hom.from_single₀_InjectiveResolution (X : cochain_complex C ℕ) (Y : C)
(f : (cochain_complex.single₀ C).obj Y ⟶ X) [quasi_iso f]
(H : ∀ n, injective (X.X n)) :
InjectiveResolution Y | { cocomplex := X,
ι := f,
injective := H,
exact₀ := f.from_single₀_exact_f_d_at_zero,
exact := f.from_single₀_exact_at_succ,
mono := f.from_single₀_mono_at_zero } | def | homological_complex.hom.homological_complex.hom.from_single₀_InjectiveResolution | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [
"cochain_complex",
"cochain_complex.single₀",
"quasi_iso"
] | If `X` is a cochain complex of injective objects and we have a quasi-isomorphism
`f : Y[0] ⟶ X`, then `X` is an injective resolution of `Y.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_exact_of_preserves_finite_colimits_of_epi [preserves_finite_colimits F] [epi g]
(ex : exact f g) : exact (F.map f) (F.map g) | abelian.exact_of_is_cokernel _ _ (by simp [← functor.map_comp, ex.w])
$ limits.is_colimit_cofork_map_of_is_colimit' _ ex.w (abelian.is_colimit_of_exact_of_epi _ _ ex) | lemma | category_theory.abelian.functor.preserves_exact_of_preserves_finite_colimits_of_epi | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [] | If `preserves_finite_colimits F` and `epi g`, then `exact (F.map f) (F.map g)` if
`exact f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_of_map_projective_resolution (P: ProjectiveResolution X) [preserves_finite_colimits F] :
exact (((F.map_homological_complex (complex_shape.down ℕ)).obj P.complex).d_to 0)
(F.map (P.π.f 0)) | preadditive.exact_of_iso_of_exact' (F.map (P.complex.d 1 0)) (F.map (P.π.f 0)) _ _
(homological_complex.X_prev_iso ((F.map_homological_complex _).obj P.complex) rfl).symm
(iso.refl _) (iso.refl _) (by simp) (by simp)
(preserves_exact_of_preserves_finite_colimits_of_epi _ (P.exact₀)) | lemma | category_theory.abelian.functor.exact_of_map_projective_resolution | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"complex_shape.down",
"homological_complex.X_prev_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_derived_zero_to_self_app [enough_projectives C] {X : C}
(P : ProjectiveResolution X) : (F.left_derived 0).obj X ⟶ F.obj X | (left_derived_obj_iso F 0 P).hom ≫ homology.desc' _ _ _ (kernel.ι _ ≫ (F.map (P.π.f 0)))
begin
rw [kernel.lift_ι_assoc, homological_complex.d_to_eq _ (by simp : (complex_shape.down ℕ).rel 1 0),
map_homological_complex_obj_d, category.assoc, ← functor.map_comp],
simp
end | def | category_theory.abelian.functor.left_derived_zero_to_self_app | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"complex_shape.down",
"homological_complex.d_to_eq",
"homology.desc'",
"rel"
] | Given `P : ProjectiveResolution X`, a morphism `(F.left_derived 0).obj X ⟶ F.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_derived_zero_to_self_app_inv [enough_projectives C] [preserves_finite_colimits F] {X : C}
(P : ProjectiveResolution X) : F.obj X ⟶ (F.left_derived 0).obj X | begin
refine ((as_iso (cokernel.desc _ _ (exact_of_map_projective_resolution F P).w)).inv) ≫ _ ≫
(homology_iso_cokernel_lift _ _ _).inv ≫ (left_derived_obj_iso F 0 P).inv,
exact cokernel.map _ _ (𝟙 _) (kernel.lift _ (𝟙 _) (by simp)) (by { ext, simp }),
end | def | category_theory.abelian.functor.left_derived_zero_to_self_app_inv | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"homology_iso_cokernel_lift"
] | Given `P : ProjectiveResolution X`, a morphism `F.obj X ⟶ (F.left_derived 0).obj X` given
`preserves_finite_colimits F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_derived_zero_to_self_app_comp_inv [enough_projectives C] [preserves_finite_colimits F]
{X : C} (P : ProjectiveResolution X) : left_derived_zero_to_self_app F P ≫
left_derived_zero_to_self_app_inv F P = 𝟙 _ | begin
dsimp [left_derived_zero_to_self_app, left_derived_zero_to_self_app_inv],
rw [← category.assoc, ← category.assoc, ← category.assoc, iso.comp_inv_eq, category.id_comp,
category.assoc, category.assoc, category.assoc],
convert category.comp_id _,
rw [← category.assoc, ← category.assoc, iso.comp_inv_eq, c... | lemma | category_theory.abelian.functor.left_derived_zero_to_self_app_comp_inv | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"homology.π'",
"homology.π'_desc'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_derived_zero_to_self_app_inv_comp [enough_projectives C] [preserves_finite_colimits F]
{X : C} (P : ProjectiveResolution X) : left_derived_zero_to_self_app_inv F P ≫
left_derived_zero_to_self_app F P = 𝟙 _ | begin
dsimp [left_derived_zero_to_self_app, left_derived_zero_to_self_app_inv],
rw [category.assoc, category.assoc, category.assoc,
← category.assoc (F.left_derived_obj_iso 0 P).inv, iso.inv_hom_id, category.id_comp,
is_iso.inv_comp_eq, category.comp_id],
ext,
simp only [cokernel.π_desc_assoc, category.... | lemma | category_theory.abelian.functor.left_derived_zero_to_self_app_inv_comp | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"homology.desc'",
"homology_iso_cokernel_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_derived_zero_to_self_app_iso [enough_projectives C] [preserves_finite_colimits F]
{X : C} (P : ProjectiveResolution X) : (F.left_derived 0).obj X ≅ F.obj X | { hom := left_derived_zero_to_self_app _ P,
inv := left_derived_zero_to_self_app_inv _ P,
hom_inv_id' := left_derived_zero_to_self_app_comp_inv _ P,
inv_hom_id' := left_derived_zero_to_self_app_inv_comp _ P } | def | category_theory.abelian.functor.left_derived_zero_to_self_app_iso | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [] | Given `P : ProjectiveResolution X`, the isomorphism `(F.left_derived 0).obj X ≅ F.obj X` if
`preserves_finite_colimits F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_derived_zero_to_self_natural [enough_projectives C] {X : C} {Y : C} (f : X ⟶ Y)
(P : ProjectiveResolution X) (Q : ProjectiveResolution Y) :
(F.left_derived 0).map f ≫ left_derived_zero_to_self_app F Q =
left_derived_zero_to_self_app F P ≫ F.map f | begin
dsimp only [left_derived_zero_to_self_app],
rw [functor.left_derived_map_eq F 0 f (ProjectiveResolution.lift f P Q) (by simp),
category.assoc, category.assoc, ← category.assoc _ (F.left_derived_obj_iso 0 Q).hom,
iso.inv_hom_id, category.id_comp, category.assoc, whisker_eq],
dsimp only [homology_func... | lemma | category_theory.abelian.functor.left_derived_zero_to_self_natural | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [
"homological_complex.congr_hom",
"homological_complex.hom.sq_to_right",
"homology.π'_desc'",
"map_comp"
] | Given `P : ProjectiveResolution X` and `Q : ProjectiveResolution Y` and a morphism `f : X ⟶ Y`,
naturality of the square given by `left_derived_zero_to_self_obj_hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_derived_zero_iso_self [enough_projectives C] [preserves_finite_colimits F] :
(F.left_derived 0) ≅ F | nat_iso.of_components (λ X, left_derived_zero_to_self_app_iso _ (ProjectiveResolution.of X))
(λ X Y f, left_derived_zero_to_self_natural _ _ _ _) | def | category_theory.abelian.functor.left_derived_zero_iso_self | category_theory.abelian | src/category_theory/abelian/left_derived.lean | [
"category_theory.abelian.homology",
"category_theory.functor.left_derived",
"category_theory.abelian.projective",
"category_theory.limits.constructions.epi_mono"
] | [] | Given `preserves_finite_colimits F`, the natural isomorphism `(F.left_derived 0) ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_preadditive_abelian extends has_zero_morphisms C, normal_mono_category C,
normal_epi_category C | [has_zero_object : has_zero_object C]
[has_kernels : has_kernels C]
[has_cokernels : has_cokernels C]
[has_finite_products : has_finite_products C]
[has_finite_coproducts : has_finite_coproducts C] | class | category_theory.non_preadditive_abelian | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | We call a category `non_preadditive_abelian` if it has a zero object, kernels, cokernels, binary
products and coproducts, and every monomorphism and every epimorphism is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_factor_thru_image [mono f] : is_iso (abelian.factor_thru_image f) | is_iso_of_mono_of_epi _ | instance | category_theory.non_preadditive_abelian.is_iso_factor_thru_image | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_factor_thru_coimage [epi f] :
is_iso (abelian.factor_thru_coimage f) | is_iso_of_mono_of_epi _ | instance | category_theory.non_preadditive_abelian.is_iso_factor_thru_coimage | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_is_cokernel_of_kernel [epi f] (s : fork f 0) (h : is_limit s) :
is_colimit (cokernel_cofork.of_π f (kernel_fork.condition s)) | is_cokernel.cokernel_iso _ _
(cokernel.of_iso_comp _ _
(limits.is_limit.cone_point_unique_up_to_iso (limit.is_limit _) h)
(cone_morphism.w (limits.is_limit.unique_up_to_iso (limit.is_limit _) h).hom _))
(as_iso $ abelian.factor_thru_coimage f) (abelian.coimage.fac f) | def | category_theory.non_preadditive_abelian.epi_is_cokernel_of_kernel | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | In a `non_preadditive_abelian` category, an epi is the cokernel of its kernel. More precisely:
If `f` is an epimorphism and `s` is some limit kernel cone on `f`, then `f` is a cokernel
of `fork.ι s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_is_kernel_of_cokernel [mono f] (s : cofork f 0) (h : is_colimit s) :
is_limit (kernel_fork.of_ι f (cokernel_cofork.condition s)) | is_kernel.iso_kernel _ _
(kernel.of_comp_iso _ _
(limits.is_colimit.cocone_point_unique_up_to_iso h (colimit.is_colimit _))
(cocone_morphism.w (limits.is_colimit.unique_up_to_iso h $ colimit.is_colimit _).hom _))
(as_iso $ abelian.factor_thru_image f) (abelian.image.fac f) | def | category_theory.non_preadditive_abelian.mono_is_kernel_of_cokernel | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | In a `non_preadditive_abelian` category, a mono is the kernel of its cokernel. More precisely:
If `f` is a monomorphism and `s` is some colimit cokernel cocone on `f`, then `f` is a kernel
of `cofork.π s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
r (A : C) : A ⟶ cokernel (diag A) | prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) | abbreviation | category_theory.non_preadditive_abelian.r | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | The composite `A ⟶ A ⨯ A ⟶ cokernel (Δ A)`, where the first map is `(𝟙 A, 0)` and the second map
is the canonical projection into the cokernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_Δ {A : C} : mono (diag A) | mono_of_mono_fac $ prod.lift_fst _ _ | instance | category_theory.non_preadditive_abelian.mono_Δ | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_r {A : C} : mono (r A) | begin
let hl : is_limit (kernel_fork.of_ι (diag A) (cokernel.condition (diag A))),
{ exact mono_is_kernel_of_cokernel _ (colimit.is_colimit _) },
apply normal_epi_category.mono_of_cancel_zero,
intros Z x hx,
have hxx : (x ≫ prod.lift (𝟙 A) (0 : A ⟶ A)) ≫ cokernel.π (diag A) = 0,
{ rw [category.assoc, hx] }... | instance | category_theory.non_preadditive_abelian.mono_r | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_r {A : C} : epi (r A) | begin
have hlp : prod.lift (𝟙 A) (0 : A ⟶ A) ≫ limits.prod.snd = 0 := prod.lift_snd _ _,
let hp1 : is_limit (kernel_fork.of_ι (prod.lift (𝟙 A) (0 : A ⟶ A)) hlp),
{ refine fork.is_limit.mk _ (λ s, fork.ι s ≫ limits.prod.fst) _ _,
{ intro s,
ext; simp, erw category.comp_id },
{ intros s m h,
h... | instance | category_theory.non_preadditive_abelian.epi_r | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_r {A : C} : is_iso (r A) | is_iso_of_mono_of_epi _ | instance | category_theory.non_preadditive_abelian.is_iso_r | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ {A : C} : A ⨯ A ⟶ A | cokernel.π (diag A) ≫ inv (r A) | abbreviation | category_theory.non_preadditive_abelian.σ | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | The composite `A ⨯ A ⟶ cokernel (diag A) ⟶ A` given by the natural projection into the cokernel
followed by the inverse of `r`. In the category of modules, using the normal kernels and
cokernels, this map is equal to the map `(a, b) ↦ a - b`, hence the name `σ` for
"subtraction". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diag_σ {X : C} : diag X ≫ σ = 0 | by rw [cokernel.condition_assoc, zero_comp] | lemma | category_theory.non_preadditive_abelian.diag_σ | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_σ {X : C} : prod.lift (𝟙 X) 0 ≫ σ = 𝟙 X | by rw [←category.assoc, is_iso.hom_inv_id] | lemma | category_theory.non_preadditive_abelian.lift_σ | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_map {X Y : C} (f : X ⟶ Y) :
prod.lift (𝟙 X) 0 ≫ limits.prod.map f f = f ≫ prod.lift (𝟙 Y) 0 | by simp | lemma | category_theory.non_preadditive_abelian.lift_map | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_σ {X : C} : is_colimit (cokernel_cofork.of_π σ diag_σ) | cokernel.cokernel_iso _ σ (as_iso (r X)).symm (by rw [iso.symm_hom, as_iso_inv]) | def | category_theory.non_preadditive_abelian.is_colimit_σ | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | σ is a cokernel of Δ X. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
σ_comp {X Y : C} (f : X ⟶ Y) : σ ≫ f = limits.prod.map f f ≫ σ | begin
obtain ⟨g, hg⟩ :=
cokernel_cofork.is_colimit.desc' is_colimit_σ (limits.prod.map f f ≫ σ) (by simp),
suffices hfg : f = g,
{ rw [←hg, cofork.π_of_π, hfg] },
calc f = f ≫ prod.lift (𝟙 Y) 0 ≫ σ : by rw [lift_σ, category.comp_id]
... = prod.lift (𝟙 X) 0 ≫ limits.prod.map f f ≫ σ : by rw lift_map_as... | lemma | category_theory.non_preadditive_abelian.σ_comp | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | This is the key identity satisfied by `σ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sub {X Y : C} : has_sub (X ⟶ Y) | ⟨λ f g, prod.lift f g ≫ σ⟩ | def | category_theory.non_preadditive_abelian.has_sub | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | Subtraction of morphisms in a `non_preadditive_abelian` category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_neg {X Y : C} : has_neg (X ⟶ Y) | ⟨λ f, 0 - f⟩ | def | category_theory.non_preadditive_abelian.has_neg | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | Negation of morphisms in a `non_preadditive_abelian` category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_add {X Y : C} : has_add (X ⟶ Y) | ⟨λ f g, f - (-g)⟩ | def | category_theory.non_preadditive_abelian.has_add | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | Addition of morphisms in a `non_preadditive_abelian` category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_def {X Y : C} (a b : X ⟶ Y) : a - b = prod.lift a b ≫ σ | rfl | lemma | category_theory.non_preadditive_abelian.sub_def | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_def {X Y : C} (a b : X ⟶ Y) : a + b = a - (-b) | rfl | lemma | category_theory.non_preadditive_abelian.add_def | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_def {X Y : C} (a : X ⟶ Y) : -a = 0 - a | rfl | lemma | category_theory.non_preadditive_abelian.neg_def | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_zero {X Y : C} (a : X ⟶ Y) : a - 0 = a | begin
rw sub_def,
conv_lhs { congr, congr, rw ←category.comp_id a, skip, rw (show 0 = a ≫ (0 : Y ⟶ Y), by simp)},
rw [← prod.comp_lift, category.assoc, lift_σ, category.comp_id]
end | lemma | category_theory.non_preadditive_abelian.sub_zero | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self {X Y : C} (a : X ⟶ Y) : a - a = 0 | by rw [sub_def, ←category.comp_id a, ← prod.comp_lift, category.assoc, diag_σ, comp_zero] | lemma | category_theory.non_preadditive_abelian.sub_self | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_sub_lift {X Y : C} (a b c d : X ⟶ Y) :
prod.lift a b - prod.lift c d = prod.lift (a - c) (b - d) | begin
simp only [sub_def],
ext,
{ rw [category.assoc, σ_comp, prod.lift_map_assoc, prod.lift_fst, prod.lift_fst, prod.lift_fst] },
{ rw [category.assoc, σ_comp, prod.lift_map_assoc, prod.lift_snd, prod.lift_snd, prod.lift_snd] }
end | lemma | category_theory.non_preadditive_abelian.lift_sub_lift | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_sub_sub {X Y : C} (a b c d : X ⟶ Y) : (a - c) - (b - d) = (a - b) - (c - d) | begin
rw [sub_def, ←lift_sub_lift, sub_def, category.assoc, σ_comp, prod.lift_map_assoc], refl
end | lemma | category_theory.non_preadditive_abelian.sub_sub_sub | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_sub {X Y : C} (a b : X ⟶ Y) : (-a) - b = (-b) - a | by conv_lhs { rw [neg_def, ←sub_zero b, sub_sub_sub, sub_zero, ←neg_def] } | lemma | category_theory.non_preadditive_abelian.neg_sub | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_neg {X Y : C} (a : X ⟶ Y) : -(-a) = a | begin
rw [neg_def, neg_def],
conv_lhs { congr, rw ←sub_self a },
rw [sub_sub_sub, sub_zero, sub_self, sub_zero]
end | lemma | category_theory.non_preadditive_abelian.neg_neg | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm {X Y : C} (a b : X ⟶ Y) : a + b = b + a | begin
rw [add_def],
conv_lhs { rw ←neg_neg a },
rw [neg_def, neg_def, neg_def, sub_sub_sub],
conv_lhs {congr, skip, rw [←neg_def, neg_sub] },
rw [sub_sub_sub, add_def, ←neg_def, neg_neg b, neg_def]
end | lemma | category_theory.non_preadditive_abelian.add_comm | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_neg {X Y : C} (a b : X ⟶ Y) : a + (-b) = a - b | by rw [add_def, neg_neg] | lemma | category_theory.non_preadditive_abelian.add_neg | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_neg_self {X Y : C} (a : X ⟶ Y) : a + (-a) = 0 | by rw [add_neg, sub_self] | lemma | category_theory.non_preadditive_abelian.add_neg_self | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_add_self {X Y : C} (a : X ⟶ Y) : (-a) + a = 0 | by rw [add_comm, add_neg_self] | lemma | category_theory.non_preadditive_abelian.neg_add_self | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_sub' {X Y : C} (a b : X ⟶ Y) : -(a - b) = (-a) + b | begin
rw [neg_def, neg_def],
conv_lhs { rw ←sub_self (0 : X ⟶ Y) },
rw [sub_sub_sub, add_def, neg_def]
end | lemma | category_theory.non_preadditive_abelian.neg_sub' | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_add {X Y : C} (a b : X ⟶ Y) : -(a + b) = (-a) - b | by rw [add_def, neg_sub', add_neg] | lemma | category_theory.non_preadditive_abelian.neg_add | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_add {X Y : C} (a b c : X ⟶ Y) : (a - b) + c = a - (b - c) | by rw [add_def, neg_def, sub_sub_sub, sub_zero] | lemma | category_theory.non_preadditive_abelian.sub_add | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_assoc {X Y : C} (a b c : X ⟶ Y) : (a + b) + c = a + (b + c) | begin
conv_lhs { congr, rw add_def },
rw [sub_add, ←add_neg, neg_sub', neg_neg]
end | lemma | category_theory.non_preadditive_abelian.add_assoc | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_zero {X Y : C} (a : X ⟶ Y) : a + 0 = a | by rw [add_def, neg_def, sub_self, sub_zero] | lemma | category_theory.non_preadditive_abelian.add_zero | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_sub {X Y Z : C} (f : X ⟶ Y) (g h : Y ⟶ Z) : f ≫ (g - h) = f ≫ g - f ≫ h | by rw [sub_def, ←category.assoc, prod.comp_lift, sub_def] | lemma | category_theory.non_preadditive_abelian.comp_sub | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_comp {X Y Z : C} (f g : X ⟶ Y) (h : Y ⟶ Z) : (f - g) ≫ h = f ≫ h - g ≫ h | by rw [sub_def, category.assoc, σ_comp, ←category.assoc, prod.lift_map, sub_def] | lemma | category_theory.non_preadditive_abelian.sub_comp | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_add (X Y Z : C) (f : X ⟶ Y) (g h : Y ⟶ Z) : f ≫ (g + h) = f ≫ g + f ≫ h | by rw [add_def, comp_sub, neg_def, comp_sub, comp_zero, add_def, neg_def] | lemma | category_theory.non_preadditive_abelian.comp_add | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp (X Y Z : C) (f g : X ⟶ Y) (h : Y ⟶ Z) : (f + g) ≫ h = f ≫ h + g ≫ h | by rw [add_def, sub_comp, neg_def, sub_comp, zero_comp, add_def, neg_def] | lemma | category_theory.non_preadditive_abelian.add_comp | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preadditive : preadditive C | { hom_group := λ X Y,
{ add := (+),
add_assoc := add_assoc,
zero := 0,
zero_add := neg_neg,
add_zero := add_zero,
neg := λ f, -f,
add_left_neg := neg_add_self,
add_comm := add_comm },
add_comp' := add_comp,
comp_add' := comp_add } | def | category_theory.non_preadditive_abelian.preadditive | category_theory.abelian | src/category_theory/abelian/non_preadditive.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.kernels",
"category_theory.limits.shapes.normal_mono.equalizers",
"category_theory.abelian.images",
"category_theory.preadditive.basic"
] | [] | Every `non_preadditive_abelian` category is preadditive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_op_unop : (kernel f.op).unop ≅ cokernel f | { hom := (kernel.lift f.op (cokernel.π f).op $ by simp [← op_comp]).unop,
inv := cokernel.desc f (kernel.ι f.op).unop $
by { rw [← f.unop_op, ← unop_comp, f.unop_op], simp },
hom_inv_id' := begin
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp],
congr' 1,
dsimp,
ext,
simp [← op_c... | def | category_theory.kernel_op_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The kernel of `f.op` is the opposite of `cokernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_op_unop : (cokernel f.op).unop ≅ kernel f | { hom := kernel.lift f (cokernel.π f.op).unop $
by { rw [← f.unop_op, ← unop_comp, f.unop_op], simp },
inv := (cokernel.desc f.op (kernel.ι f).op $ by simp [← op_comp]).unop,
hom_inv_id' := begin
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp],
congr' 1,
dsimp,
ext,
simp [← op_com... | def | category_theory.cokernel_op_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The cokernel of `f.op` is the opposite of `kernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_unop_op : opposite.op (kernel g.unop) ≅ cokernel g | (cokernel_op_unop g.unop).op | def | category_theory.kernel_unop_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The kernel of `g.unop` is the opposite of `cokernel g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_unop_op : opposite.op (cokernel g.unop) ≅ kernel g | (kernel_op_unop g.unop).op | def | category_theory.cokernel_unop_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The cokernel of `g.unop` is the opposite of `kernel g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π_op : (cokernel.π f.op).unop =
(cokernel_op_unop f).hom ≫ kernel.ι f ≫ eq_to_hom (opposite.unop_op _).symm | by simp [cokernel_op_unop] | lemma | category_theory.cokernel.π_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.ι_op : (kernel.ι f.op).unop =
eq_to_hom (opposite.unop_op _) ≫ cokernel.π f ≫ (kernel_op_unop f).inv | by simp [kernel_op_unop] | lemma | category_theory.kernel.ι_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_op_op : kernel f.op ≅ opposite.op (cokernel f) | (kernel_op_unop f).op.symm | def | category_theory.kernel_op_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The kernel of `f.op` is the opposite of `cokernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_op_op : cokernel f.op ≅ opposite.op (kernel f) | (cokernel_op_unop f).op.symm | def | category_theory.cokernel_op_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The cokernel of `f.op` is the opposite of `kernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_unop_unop : kernel g.unop ≅ (cokernel g).unop | (kernel_unop_op g).unop.symm | def | category_theory.kernel_unop_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The kernel of `g.unop` is the opposite of `cokernel g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.ι_unop : (kernel.ι g.unop).op =
eq_to_hom (opposite.op_unop _) ≫ cokernel.π g ≫ (kernel_unop_op g).inv | by simp | lemma | category_theory.kernel.ι_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.π_unop : (cokernel.π g.unop).op =
(cokernel_unop_op g).hom ≫ kernel.ι g ≫ eq_to_hom (opposite.op_unop _).symm | by simp | lemma | category_theory.cokernel.π_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_unop_unop : cokernel g.unop ≅ (kernel g).unop | (cokernel_unop_op g).unop.symm | def | category_theory.cokernel_unop_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The cokernel of `g.unop` is the opposite of `kernel g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_unop_op : opposite.op (image g.unop) ≅ image g | (abelian.image_iso_image _).op ≪≫ (cokernel_op_op _).symm ≪≫
cokernel_iso_of_eq (cokernel.π_unop _) ≪≫ (cokernel_epi_comp _ _)
≪≫ (cokernel_comp_is_iso _ _) ≪≫ (abelian.coimage_iso_image' _) | def | category_theory.image_unop_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The opposite of the image of `g.unop` is the image of `g.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_op_op : opposite.op (image f) ≅ image f.op | image_unop_op f.op | def | category_theory.image_op_op | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"opposite.op"
] | The opposite of the image of `f` is the image of `f.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_op_unop : (image f.op).unop ≅ image f | (image_unop_op f.op).unop | def | category_theory.image_op_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The image of `f.op` is the opposite of the image of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_unop_unop : (image g).unop ≅ image g.unop | (image_unop_op g).unop | def | category_theory.image_unop_unop | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | The image of `g` is the opposite of the image of `g.unop.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_ι_op_comp_image_unop_op_hom :
(image.ι g.unop).op ≫ (image_unop_op g).hom = factor_thru_image g | begin
dunfold image_unop_op,
simp only [←category.assoc, ←op_comp, iso.trans_hom, iso.symm_hom, iso.op_hom, cokernel_op_op_inv,
cokernel_comp_is_iso_hom, cokernel_epi_comp_hom, cokernel_iso_of_eq_hom_comp_desc_assoc,
abelian.coimage_iso_image'_hom, eq_to_hom_refl, is_iso.inv_id,
category.id_comp (cokern... | lemma | category_theory.image_ι_op_comp_image_unop_op_hom | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_unop_op_hom_comp_image_ι :
(image_unop_op g).hom ≫ image.ι g = (factor_thru_image g.unop).op | by simp only [←cancel_epi (image.ι g.unop).op, ←category.assoc, image_ι_op_comp_image_unop_op_hom,
←op_comp, image.fac, quiver.hom.op_unop] | lemma | category_theory.image_unop_op_hom_comp_image_ι | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_thru_image_comp_image_unop_op_inv :
factor_thru_image g ≫ (image_unop_op g).inv = (image.ι g.unop).op | by rw [iso.comp_inv_eq, image_ι_op_comp_image_unop_op_hom] | lemma | category_theory.factor_thru_image_comp_image_unop_op_inv | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_unop_op_inv_comp_op_factor_thru_image :
(image_unop_op g).inv ≫ (factor_thru_image g.unop).op = image.ι g | by rw [iso.inv_comp_eq, image_unop_op_hom_comp_image_ι] | lemma | category_theory.image_unop_op_inv_comp_op_factor_thru_image | category_theory.abelian | src/category_theory/abelian/opposite.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.opposite",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_d_f [enough_projectives C] {X Y : C} (f : X ⟶ Y) : exact (d f) f | (abelian.exact_iff _ _).2 $
⟨by simp, zero_of_epi_comp (π _) $ by rw [←category.assoc, cokernel.condition]⟩ | lemma | category_theory.exact_d_f | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [] | When `C` is abelian, `projective.d f` and `f` are exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_preadditive_coyoneda_obj_of_projective (P : C)
[hP : projective P] : preserves_finite_colimits (preadditive_coyoneda_obj (op P)) | begin
letI := (projective_iff_preserves_epimorphisms_preadditive_coyoneda_obj' P).mp hP,
apply functor.preserves_finite_colimits_of_preserves_epis_and_kernels,
end | def | category_theory.preserves_finite_colimits_preadditive_coyoneda_obj_of_projective | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [] | The preadditive Co-Yoneda functor on `P` preserves colimits if `P` is projective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_of_preserves_finite_colimits_preadditive_coyoneda_obj (P : C)
[hP : preserves_finite_colimits (preadditive_coyoneda_obj (op P))] : projective P | begin
rw projective_iff_preserves_epimorphisms_preadditive_coyoneda_obj',
apply_instance
end | lemma | category_theory.projective_of_preserves_finite_colimits_preadditive_coyoneda_obj | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [] | An object is projective if its preadditive Co-Yoneda functor preserves finite colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_complex (Z : C) : chain_complex C ℕ | chain_complex.mk'
(projective.over Z) (projective.syzygies (projective.π Z)) (projective.d (projective.π Z))
(λ ⟨X, Y, f⟩, ⟨projective.syzygies f, projective.d f, (exact_d_f f).w⟩) | def | category_theory.ProjectiveResolution.of_complex | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [
"chain_complex",
"chain_complex.mk'"
] | Auxiliary definition for `ProjectiveResolution.of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (Z : C) : ProjectiveResolution Z | { complex := of_complex Z,
π := chain_complex.mk_hom _ _ (projective.π Z) 0
(by { simp, exact (exact_d_f (projective.π Z)).w.symm, })
(λ n _, ⟨0, by ext⟩),
projective := by { rintros (_|_|_|n); apply projective.projective_over, },
exact₀ := by simpa using exact_d_f (projective.π Z),
exact := by { rintro... | def | category_theory.ProjectiveResolution.of | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [
"chain_complex.mk_hom",
"complex"
] | In any abelian category with enough projectives,
`ProjectiveResolution.of Z` constructs a projective resolution of the object `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_single₀_ProjectiveResolution {X : chain_complex C ℕ} {Y : C}
(f : X ⟶ (chain_complex.single₀ C).obj Y) [quasi_iso f]
(H : ∀ n, projective (X.X n)) :
ProjectiveResolution Y | { complex := X,
π := f,
projective := H,
exact₀ := f.to_single₀_exact_d_f_at_zero,
exact := f.to_single₀_exact_at_succ,
epi := f.to_single₀_epi_at_zero } | def | homological_complex.hom.to_single₀_ProjectiveResolution | category_theory.abelian | src/category_theory/abelian/projective.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.projective_resolution",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.projective"
] | [
"chain_complex",
"chain_complex.single₀",
"complex",
"quasi_iso"
] | If `X` is a chain complex of projective objects and we have a quasi-isomorphism `f : X ⟶ Y[0]`,
then `X` is a projective resolution of `Y.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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