statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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complex_shape (ι : Type*) | (rel : ι → ι → Prop)
(next_eq : ∀ {i j j'}, rel i j → rel i j' → j = j')
(prev_eq : ∀ {i i' j}, rel i j → rel i' j → i = i') | structure | complex_shape | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"rel"
] | A `c : complex_shape ι` describes the shape of a chain complex,
with chain groups indexed by `ι`.
Typically `ι` will be `ℕ`, `ℤ`, or `fin n`.
There is a relation `rel : ι → ι → Prop`,
and we will only allow a non-zero differential from `i` to `j` when `rel i j`.
There are axioms which imply `{ j // c.rel i j }` and `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (ι : Type*) : complex_shape ι | { rel := λ i j, i = j,
next_eq := λ i j j' w w', w.symm.trans w',
prev_eq := λ i i' j w w', w.trans w'.symm, } | def | complex_shape.refl | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape",
"rel"
] | The complex shape where only differentials from each `X.i` to itself are allowed.
This is mostly only useful so we can describe the relation of "related in `k` steps" below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (c : complex_shape ι) : complex_shape ι | { rel := λ i j, c.rel j i,
next_eq := λ i j j' w w', c.prev_eq w w',
prev_eq := λ i i' j w w', c.next_eq w w', } | def | complex_shape.symm | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape",
"rel"
] | The reverse of a `complex_shape`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (c : complex_shape ι) : c.symm.symm = c | by { ext, simp, } | lemma | complex_shape.symm_symm | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (c₁ c₂ : complex_shape ι) : complex_shape ι | { rel := relation.comp c₁.rel c₂.rel,
next_eq := λ i j j' w w',
begin
obtain ⟨k, w₁, w₂⟩ := w,
obtain ⟨k', w₁', w₂'⟩ := w',
rw c₁.next_eq w₁ w₁' at w₂,
exact c₂.next_eq w₂ w₂',
end,
prev_eq := λ i i' j w w',
begin
obtain ⟨k, w₁, w₂⟩ := w,
obtain ⟨k', w₁', w₂'⟩ := w',
rw c₂.prev_eq ... | def | complex_shape.trans | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape",
"rel",
"relation.comp"
] | The "composition" of two `complex_shape`s.
We need this to define "related in k steps" later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_next (c : complex_shape ι) (i : ι) :
subsingleton { j // c.rel i j } | begin
fsplit,
rintros ⟨j, rij⟩ ⟨k, rik⟩,
congr,
exact c.next_eq rij rik,
end | instance | complex_shape.subsingleton_next | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_prev (c : complex_shape ι) (j : ι) :
subsingleton { i // c.rel i j } | begin
fsplit,
rintros ⟨i, rik⟩ ⟨j, rjk⟩,
congr,
exact c.prev_eq rik rjk,
end | instance | complex_shape.subsingleton_prev | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next (c : complex_shape ι) (i : ι) : ι | if h : ∃ j, c.rel i j then h.some else i | def | complex_shape.next | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape"
] | An arbitary choice of index `j` such that `rel i j`, if such exists.
Returns `i` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prev (c : complex_shape ι) (j : ι) : ι | if h : ∃ i, c.rel i j then h.some else j | def | complex_shape.prev | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"complex_shape"
] | An arbitary choice of index `i` such that `rel i j`, if such exists.
Returns `j` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
next_eq' (c : complex_shape ι) {i j : ι} (h : c.rel i j) : c.next i = j | by { apply c.next_eq _ h, dsimp only [next], rw dif_pos, exact Exists.some_spec ⟨j, h⟩, } | lemma | complex_shape.next_eq' | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"Exists.some_spec",
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_eq' (c : complex_shape ι) {i j : ι} (h : c.rel i j) : c.prev j = i | by { apply c.prev_eq _ h, dsimp only [prev], rw dif_pos, exact Exists.some_spec ⟨i, h⟩, } | lemma | complex_shape.prev_eq' | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"Exists.some_spec",
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
up' {α : Type*} [add_right_cancel_semigroup α] (a : α) : complex_shape α | { rel := λ i j , i + a = j,
next_eq := λ i j k hi hj, hi.symm.trans hj,
prev_eq := λ i j k hi hj, add_right_cancel (hi.trans hj.symm), } | def | complex_shape.up' | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"complex_shape",
"rel"
] | The `complex_shape` allowing differentials from `X i` to `X (i+a)`.
(For example when `a = 1`, a cohomology theory indexed by `ℕ` or `ℤ`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
down' {α : Type*} [add_right_cancel_semigroup α] (a : α) : complex_shape α | { rel := λ i j , j + a = i,
next_eq := λ i j k hi hj, add_right_cancel (hi.trans (hj.symm)),
prev_eq := λ i j k hi hj, hi.symm.trans hj, } | def | complex_shape.down' | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"complex_shape",
"rel"
] | The `complex_shape` allowing differentials from `X (j+a)` to `X j`.
(For example when `a = 1`, a homology theory indexed by `ℕ` or `ℤ`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
down'_mk {α : Type*} [add_right_cancel_semigroup α] (a : α)
(i j : α) (h : j + a = i) : (down' a).rel i j | h | lemma | complex_shape.down'_mk | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
up (α : Type*) [add_right_cancel_semigroup α] [has_one α] : complex_shape α | up' 1 | def | complex_shape.up | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"complex_shape"
] | The `complex_shape` appropriate for cohomology, so `d : X i ⟶ X j` only when `j = i + 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
down (α : Type*) [add_right_cancel_semigroup α] [has_one α] : complex_shape α | down' 1 | def | complex_shape.down | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"complex_shape"
] | The `complex_shape` appropriate for homology, so `d : X i ⟶ X j` only when `i = j + 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
down_mk {α : Type*} [add_right_cancel_semigroup α] [has_one α]
(i j : α) (h : j + 1 = i) : (down α).rel i j | down'_mk (1 : α) i j h | lemma | complex_shape.down_mk | algebra.homology | src/algebra/homology/complex_shape.lean | [
"algebra.group.defs",
"logic.relation"
] | [
"add_right_cancel_semigroup",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.differential_object.X_eq_to_hom
(X : differential_object (graded_object_with_shift b V))
{i j : β} (h : i = j) : X.X i ⟶ X.X j | eq_to_hom (congr_arg X.X h) | abbreviation | category_theory.differential_object.X_eq_to_hom | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [] | Since `eq_to_hom` only preserves the fact that `X.X i = X.X j` but not `i = j`, this definition
is used to aid the simplifier. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.category_theory.differential_object.X_eq_to_hom_refl
(X : differential_object (graded_object_with_shift b V)) (i : β) :
X.X_eq_to_hom (refl i) = 𝟙 _ | rfl | lemma | category_theory.differential_object.X_eq_to_hom_refl | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_d (X : differential_object (graded_object_with_shift b V))
{x y : β} (h : x = y) :
X.X_eq_to_hom h ≫ X.d y = X.d x ≫ X.X_eq_to_hom (by { cases h, refl }) | by { cases h, dsimp, simp } | lemma | homological_complex.eq_to_hom_d | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_eq_to_hom (X : homological_complex V (complex_shape.up' b))
{x y z : β} (h : y = z) :
X.d x y ≫ eq_to_hom (congr_arg X.X h) = X.d x z | by { cases h, simp } | lemma | homological_complex.d_eq_to_hom | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [
"complex_shape.up'",
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_f' {X Y : differential_object (graded_object_with_shift b V)}
(f : X ⟶ Y) {x y : β} (h : x = y) :
X.X_eq_to_hom h ≫ f.f y = f.f x ≫ Y.X_eq_to_hom h | by { cases h, simp } | lemma | homological_complex.eq_to_hom_f' | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dgo_to_homological_complex :
differential_object (graded_object_with_shift b V) ⥤
homological_complex V (complex_shape.up' b) | { obj := λ X,
{ X := λ i, X.X i,
d := λ i j, if h : i + b = j then
X.d i ≫ X.X_eq_to_hom (show i + (1 : ℤ) • b = j, by simp [h]) else 0,
shape' := λ i j w, by { dsimp at w, convert dif_neg w },
d_comp_d' := λ i j k hij hjk, begin
dsimp at hij hjk, substs hij hjk,
have : X.d i ≫ X.d _ = _... | def | homological_complex.dgo_to_homological_complex | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [
"complex_shape.up'",
"homological_complex"
] | The functor from differential graded objects to homological complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homological_complex_to_dgo :
homological_complex V (complex_shape.up' b) ⥤
differential_object (graded_object_with_shift b V) | { obj := λ X,
{ X := λ i, X.X i,
d := λ i, X.d i (i + 1 • b),
d_squared' := by { ext i, dsimp, simp, } },
map := λ X Y f,
{ f := f.f,
comm' := by { ext i, dsimp, simp, }, } } | def | homological_complex.homological_complex_to_dgo | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [
"complex_shape.up'",
"homological_complex"
] | The functor from homological complexes to differential graded objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dgo_equiv_homological_complex_unit_iso :
𝟭 (differential_object (graded_object_with_shift b V)) ≅
dgo_to_homological_complex b V ⋙ homological_complex_to_dgo b V | nat_iso.of_components (λ X,
{ hom := { f := λ i, 𝟙 (X.X i), },
inv := { f := λ i, 𝟙 (X.X i), }, }) (by tidy) | def | homological_complex.dgo_equiv_homological_complex_unit_iso | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [] | The unit isomorphism for `dgo_equiv_homological_complex`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dgo_equiv_homological_complex_counit_iso :
homological_complex_to_dgo b V ⋙ dgo_to_homological_complex b V ≅
𝟭 (homological_complex V (complex_shape.up' b)) | nat_iso.of_components (λ X,
{ hom :=
{ f := λ i, 𝟙 (X.X i),
comm' := λ i j h, begin
dsimp at h ⊢, subst h,
delta homological_complex_to_dgo,
simp,
end },
inv :=
{ f := λ i, 𝟙 (X.X i),
comm' := λ i j h, begin
dsimp at h ⊢, subst h,
delta homologic... | def | homological_complex.dgo_equiv_homological_complex_counit_iso | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [
"complex_shape.up'",
"homological_complex"
] | The counit isomorphism for `dgo_equiv_homological_complex`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dgo_equiv_homological_complex :
differential_object (graded_object_with_shift b V) ≌
homological_complex V (complex_shape.up' b) | { functor := dgo_to_homological_complex b V,
inverse := homological_complex_to_dgo b V,
unit_iso := dgo_equiv_homological_complex_unit_iso b V,
counit_iso := dgo_equiv_homological_complex_counit_iso b V, } | def | homological_complex.dgo_equiv_homological_complex | algebra.homology | src/algebra/homology/differential_object.lean | [
"algebra.homology.homological_complex",
"category_theory.differential_object"
] | [
"complex_shape.up'",
"homological_complex"
] | The category of differential graded objects in `V` is equivalent
to the category of homological complexes in `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact [has_zero_morphisms V] [has_kernels V] {A B C : V} (f : A ⟶ B) (g : B ⟶ C) : Prop | (w : f ≫ g = 0)
(epi : epi (image_to_kernel f g w)) | structure | category_theory.exact | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preadditive.exact_iff_homology_zero {A B C : V} (f : A ⟶ B) (g : B ⟶ C) :
exact f g ↔ ∃ w : f ≫ g = 0, nonempty (homology f g w ≅ 0) | ⟨λ h, ⟨h.w, ⟨cokernel.of_epi _⟩⟩,
λ h, begin
obtain ⟨w, ⟨i⟩⟩ := h,
exact ⟨w, preadditive.epi_of_cokernel_zero ((cancel_mono i.hom).mp (by ext))⟩,
end⟩ | lemma | category_theory.preadditive.exact_iff_homology_zero | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"homology"
] | In any preadditive category,
composable morphisms `f g` are exact iff they compose to zero and the homology vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preadditive.exact_of_iso_of_exact {A₁ B₁ C₁ A₂ B₂ C₂ : V}
(f₁ : A₁ ⟶ B₁) (g₁ : B₁ ⟶ C₁) (f₂ : A₂ ⟶ B₂) (g₂ : B₂ ⟶ C₂)
(α : arrow.mk f₁ ≅ arrow.mk f₂) (β : arrow.mk g₁ ≅ arrow.mk g₂) (p : α.hom.right = β.hom.left)
(h : exact f₁ g₁) :
exact f₂ g₂ | begin
rw preadditive.exact_iff_homology_zero at h ⊢,
rcases h with ⟨w₁, ⟨i⟩⟩,
suffices w₂ : f₂ ≫ g₂ = 0, from ⟨w₂, ⟨(homology.map_iso w₁ w₂ α β p).symm.trans i⟩⟩,
rw [← cancel_epi α.hom.left, ← cancel_mono β.inv.right, comp_zero, zero_comp, ← w₁],
simp only [← arrow.mk_hom f₁, ← arrow.left_hom_inv_right α.hom... | lemma | category_theory.preadditive.exact_of_iso_of_exact | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"homology.map_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preadditive.exact_of_iso_of_exact' {A₁ B₁ C₁ A₂ B₂ C₂ : V}
(f₁ : A₁ ⟶ B₁) (g₁ : B₁ ⟶ C₁) (f₂ : A₂ ⟶ B₂) (g₂ : B₂ ⟶ C₂)
(α : A₁ ≅ A₂) (β : B₁ ≅ B₂) (γ : C₁ ≅ C₂) (hsq₁ : α.hom ≫ f₂ = f₁ ≫ β.hom)
(hsq₂ : β.hom ≫ g₂ = g₁ ≫ γ.hom)
(h : exact f₁ g₁) :
exact f₂ g₂ | preadditive.exact_of_iso_of_exact f₁ g₁ f₂ g₂ (arrow.iso_mk α β hsq₁) (arrow.iso_mk β γ hsq₂) rfl h | lemma | category_theory.preadditive.exact_of_iso_of_exact' | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | A reformulation of `preadditive.exact_of_iso_of_exact` that does not involve the arrow
category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preadditive.exact_iff_exact_of_iso {A₁ B₁ C₁ A₂ B₂ C₂ : V}
(f₁ : A₁ ⟶ B₁) (g₁ : B₁ ⟶ C₁) (f₂ : A₂ ⟶ B₂) (g₂ : B₂ ⟶ C₂)
(α : arrow.mk f₁ ≅ arrow.mk f₂) (β : arrow.mk g₁ ≅ arrow.mk g₂) (p : α.hom.right = β.hom.left) :
exact f₁ g₁ ↔ exact f₂ g₂ | ⟨preadditive.exact_of_iso_of_exact _ _ _ _ _ _ p,
preadditive.exact_of_iso_of_exact _ _ _ _ α.symm β.symm
begin
rw ← cancel_mono α.hom.right,
simp only [iso.symm_hom, ← comma.comp_right, α.inv_hom_id],
simp only [p, ←comma.comp_left, arrow.id_right, arrow.id_left, iso.inv_hom_id],
refl
end⟩ | lemma | category_theory.preadditive.exact_iff_exact_of_iso | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_zero_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) : f ≫ g = 0 | begin
rw [←image_subobject_arrow_comp f, category.assoc],
convert comp_zero,
rw p,
simp,
end | lemma | category_theory.comp_eq_zero_of_image_eq_kernel | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_is_iso_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) :
is_iso (image_to_kernel f g (comp_eq_zero_of_image_eq_kernel f g p)) | begin
refine ⟨⟨subobject.of_le _ _ p.ge, _⟩⟩,
dsimp [image_to_kernel],
simp only [subobject.of_le_comp_of_le, subobject.of_le_refl],
simp,
end | lemma | category_theory.image_to_kernel_is_iso_of_image_eq_kernel | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) : exact f g | { w := comp_eq_zero_of_image_eq_kernel f g p,
epi := begin
haveI := image_to_kernel_is_iso_of_image_eq_kernel f g p,
apply_instance,
end } | lemma | category_theory.exact_of_image_eq_kernel | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_comp_hom_inv_comp (i : B ≅ D) (h : exact f g) : exact (f ≫ i.hom) (i.inv ≫ g) | begin
refine ⟨by simp [h.w], _⟩,
rw image_to_kernel_comp_hom_inv_comp,
haveI := h.epi,
apply_instance,
end | lemma | category_theory.exact_comp_hom_inv_comp | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_comp_hom_inv_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_comp_inv_hom_comp (i : D ≅ B) (h : exact f g) : exact (f ≫ i.inv) (i.hom ≫ g) | exact_comp_hom_inv_comp i.symm h | lemma | category_theory.exact_comp_inv_hom_comp | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_comp_hom_inv_comp_iff (i : B ≅ D) : exact (f ≫ i.hom) (i.inv ≫ g) ↔ exact f g | ⟨λ h, by simpa using exact_comp_inv_hom_comp i h, exact_comp_hom_inv_comp i⟩ | lemma | category_theory.exact_comp_hom_inv_comp_iff | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_epi_comp (hgh : exact g h) [epi f] : exact (f ≫ g) h | begin
refine ⟨by simp [hgh.w], _⟩,
rw image_to_kernel_comp_left,
apply_instance,
end | lemma | category_theory.exact_epi_comp | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_comp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_iso_comp [is_iso f] : exact (f ≫ g) h ↔ exact g h | ⟨λ w, by { rw ←is_iso.inv_hom_id_assoc f g, exact exact_epi_comp w }, λ w, exact_epi_comp w⟩ | lemma | category_theory.exact_iso_comp | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_comp_mono (hfg : exact f g) [mono h] : exact f (g ≫ h) | begin
refine ⟨by simp [hfg.w_assoc], _⟩,
rw image_to_kernel_comp_right f g h hfg.w,
apply_instance,
end | lemma | category_theory.exact_comp_mono | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_comp_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_comp_mono_iff [mono h] : exact f (g ≫ h) ↔ exact f g | begin
refine ⟨λ hfg, ⟨zero_of_comp_mono h (by rw [category.assoc, hfg.1]), _⟩, λ h, exact_comp_mono h⟩,
rw ← (iso.eq_comp_inv _).1 (image_to_kernel_comp_mono _ _ h hfg.1),
haveI := hfg.2, apply_instance
end | lemma | category_theory.exact_comp_mono_iff | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_comp_mono"
] | The dual of this lemma is only true when `V` is abelian, see `abelian.exact_epi_comp_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_comp_iso [is_iso h] : exact f (g ≫ h) ↔ exact f g | exact_comp_mono_iff | lemma | category_theory.exact_comp_iso | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_kernel_subobject_arrow : exact (kernel_subobject f).arrow f | begin
refine ⟨by simp, _⟩,
apply @is_iso.epi_of_iso _ _ _ _ _ _,
exact ⟨⟨factor_thru_image_subobject _, by { ext, simp, }, by { ext, simp, }⟩⟩,
end | lemma | category_theory.exact_kernel_subobject_arrow | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_kernel_ι : exact (kernel.ι f) f | by { rw [←kernel_subobject_arrow', exact_iso_comp], exact exact_kernel_subobject_arrow } | lemma | category_theory.exact_kernel_ι | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_subobject_arrow_eq_zero_of_exact_zero_left (h : exact (0 : A ⟶ B) g) :
(kernel_subobject g).arrow = 0 | begin
rw [←cancel_epi (image_to_kernel (0 : A ⟶ B) g h.w),
←cancel_epi (factor_thru_image_subobject (0 : A ⟶ B))],
simp
end | lemma | category_theory.kernel_subobject_arrow_eq_zero_of_exact_zero_left | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_ι_eq_zero_of_exact_zero_left (h : exact (0 : A ⟶ B) g) :
kernel.ι g = 0 | by { rw ←kernel_subobject_arrow', simp [kernel_subobject_arrow_eq_zero_of_exact_zero_left A h], } | lemma | category_theory.kernel_ι_eq_zero_of_exact_zero_left | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_zero_left_of_mono [has_zero_object V] [mono g] : exact (0 : A ⟶ B) g | ⟨by simp, image_to_kernel_epi_of_zero_of_mono _⟩ | lemma | category_theory.exact_zero_left_of_mono | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_epi_of_zero_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_comp_cokernel (h : exact f g) : kernel.ι g ≫ cokernel.π f = 0 | begin
rw [←kernel_subobject_arrow', category.assoc],
convert comp_zero,
apply zero_of_epi_comp (image_to_kernel f g h.w) _,
rw [image_to_kernel_arrow_assoc, ←image_subobject_arrow, category.assoc, ←iso.eq_inv_comp],
ext,
simp,
end | lemma | category_theory.kernel_comp_cokernel | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_zero_of_exact (h : exact f g) {X Y : V} {ι : X ⟶ B} (hι : ι ≫ g = 0) {π : B ⟶ Y}
(hπ : f ≫ π = 0) : ι ≫ π = 0 | by rw [←kernel.lift_ι _ _ hι, ←cokernel.π_desc _ _ hπ, category.assoc,
kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero] | lemma | category_theory.comp_eq_zero_of_exact | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fork_ι_comp_cofork_π (h : exact f g) (s : kernel_fork g)
(t : cokernel_cofork f) : fork.ι s ≫ cofork.π t = 0 | comp_eq_zero_of_exact f g h (kernel_fork.condition s) (cokernel_cofork.condition t) | lemma | category_theory.fork_ι_comp_cofork_π | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_of_zero {A C : V} (f : A ⟶ 0) (g : 0 ⟶ C) : exact f g | begin
obtain rfl : f = 0 := by ext,
obtain rfl : g = 0 := by ext,
fsplit,
{ simp, },
{ exact image_to_kernel_epi_of_zero_of_mono 0, },
end | lemma | category_theory.exact_of_zero | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_epi_of_zero_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_zero_mono {B C : V} (f : B ⟶ C) [mono f] : exact (0 : (0 ⟶ B)) f | ⟨by simp, infer_instance⟩ | lemma | category_theory.exact_zero_mono | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_epi_zero {A B : V} (f : A ⟶ B) [epi f] : exact f (0 : (B ⟶ 0)) | ⟨by simp, infer_instance⟩ | lemma | category_theory.exact_epi_zero | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_exact_zero_left [has_kernels V] {B C : V} (f : B ⟶ C) :
mono f ↔ exact (0 : (0 ⟶ B)) f | ⟨λ h, by exactI exact_zero_mono _,
λ h, preadditive.mono_of_kernel_iso_zero
((kernel_subobject_iso f).symm ≪≫ iso_zero_of_epi_zero (by simpa using h.epi))⟩ | lemma | category_theory.mono_iff_exact_zero_left | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_exact_zero_right [has_equalizers V] {A B : V} (f : A ⟶ B) :
epi f ↔ exact f (0 : (B ⟶ 0)) | ⟨λ h, by exactI exact_epi_zero _,
λ h, begin
have e₁ := h.epi,
rw image_to_kernel_zero_right at e₁,
have e₂ : epi (((image_subobject f).arrow ≫ inv (kernel_subobject 0).arrow) ≫
(kernel_subobject 0).arrow) := @epi_comp _ _ _ _ _ _ e₁ _ _,
rw [category.assoc, is_iso.inv_hom_id, category.comp_id] ... | lemma | category_theory.epi_iff_exact_zero_right | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [
"image_to_kernel_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_exact_sequences (F : V ⥤ W) | (reflects : ∀ {A B C : V} (f : A ⟶ B) (g : B ⟶ C), exact (F.map f) (F.map g) → exact f g) | class | category_theory.functor.reflects_exact_sequences | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | A functor reflects exact sequences if any composable pair of morphisms that is mapped to an
exact pair is itself exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_of_exact_map (F : V ⥤ W) [reflects_exact_sequences F] {A B C : V} {f : A ⟶ B}
{g : B ⟶ C} (hfg : exact (F.map f) (F.map g)) : exact f g | reflects_exact_sequences.reflects f g hfg | lemma | category_theory.functor.exact_of_exact_map | algebra.homology | src/algebra/homology/exact.lean | [
"algebra.homology.image_to_kernel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_obj (C : homological_complex (homological_complex V c) c') :
homological_complex (homological_complex V c') c | { X := λ i,
{ X := λ j, (C.X j).X i,
d := λ j j', (C.d j j').f i,
shape' := λ j j' w, by { rw C.shape j j' w, simp, },
d_comp_d' := λ j₁ j₂ j₃ _ _, congr_hom (C.d_comp_d j₁ j₂ j₃) i, },
d := λ i i',
{ f := λ j, (C.X j).d i i',
comm' := λ j j' h, ((C.d j j').comm i i').symm, },
shape' := λ i i' w... | def | homological_complex.flip_obj | algebra.homology | src/algebra/homology/flip.lean | [
"algebra.homology.homological_complex"
] | [
"comm",
"homological_complex"
] | Flip a complex of complexes over the diagonal,
exchanging the horizontal and vertical directions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip : homological_complex (homological_complex V c) c' ⥤
homological_complex (homological_complex V c') c | { obj := λ C, flip_obj C,
map := λ C D f,
{ f := λ i,
{ f := λ j, (f.f j).f i,
comm' := λ j j' h, congr_hom (f.comm j j') i, }, }, }. | def | homological_complex.flip | algebra.homology | src/algebra/homology/flip.lean | [
"algebra.homology.homological_complex"
] | [
"homological_complex"
] | Flipping a complex of complexes over the diagonal, as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_equivalence_unit_iso :
𝟭 (homological_complex (homological_complex V c) c') ≅ flip V c c' ⋙ flip V c' c | nat_iso.of_components
(λ C,
{ hom :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, },
inv :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, catego... | def | homological_complex.flip_equivalence_unit_iso | algebra.homology | src/algebra/homology/flip.lean | [
"algebra.homology.homological_complex"
] | [
"homological_complex"
] | Auxiliary definition for `homological_complex.flip_equivalence` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_equivalence_counit_iso :
flip V c' c ⋙ flip V c c' ≅ 𝟭 (homological_complex (homological_complex V c') c) | nat_iso.of_components
(λ C,
{ hom :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, category.comp_id] }, },
inv :=
{ f := λ i, { f := λ j, 𝟙 ((C.X i).X j), },
comm' := λ i j h, by { ext, dsimp, simp only [category.id_comp, catego... | def | homological_complex.flip_equivalence_counit_iso | algebra.homology | src/algebra/homology/flip.lean | [
"algebra.homology.homological_complex"
] | [
"homological_complex"
] | Auxiliary definition for `homological_complex.flip_equivalence` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_equivalence :
homological_complex (homological_complex V c) c' ≌
homological_complex (homological_complex V c') c | { functor := flip V c c',
inverse := flip V c' c,
unit_iso := flip_equivalence_unit_iso V c c',
counit_iso := flip_equivalence_counit_iso V c c', } | def | homological_complex.flip_equivalence | algebra.homology | src/algebra/homology/flip.lean | [
"algebra.homology.homological_complex"
] | [
"homological_complex"
] | Flipping a complex of complexes over the diagonal, as an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_functor {T : Type*} [category T]
(C : homological_complex (T ⥤ V) c) :
T ⥤ homological_complex V c | { obj := λ t,
{ X := λ i, (C.X i).obj t,
d := λ i j, (C.d i j).app t,
d_comp_d' := λ i j k hij hjk, begin
have := C.d_comp_d i j k,
rw [nat_trans.ext_iff, function.funext_iff] at this,
exact this t
end,
shape' := λ i j h, begin
have := C.shape _ _ h,
rw [nat_trans.ext_iff... | def | homological_complex.as_functor | algebra.homology | src/algebra/homology/functor.lean | [
"algebra.homology.homological_complex"
] | [
"function.funext_iff",
"homological_complex",
"map_id"
] | A complex of functors gives a functor to complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex_of_functors_to_functor_to_complex {T : Type*} [category T] :
(homological_complex (T ⥤ V) c) ⥤ (T ⥤ homological_complex V c) | { obj := λ C, C.as_functor,
map := λ C D f,
{ app := λ t,
{ f := λ i, (f.f i).app t,
comm' := λ i j w, nat_trans.congr_app (f.comm i j) t, },
naturality' := λ t t' g, by { ext i, exact (f.f i).naturality g, }, } } | def | homological_complex.complex_of_functors_to_functor_to_complex | algebra.homology | src/algebra/homology/functor.lean | [
"algebra.homology.homological_complex"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homological_complex (c : complex_shape ι) | (X : ι → V)
(d : Π i j, X i ⟶ X j)
(shape' : ∀ i j, ¬ c.rel i j → d i j = 0 . obviously)
(d_comp_d' : ∀ i j k, c.rel i j → c.rel j k → d i j ≫ d j k = 0 . obviously) | structure | homological_complex | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape"
] | A `homological_complex V c` with a "shape" controlled by `c : complex_shape ι`
has chain groups `X i` (objects in `V`) indexed by `i : ι`,
and a differential `d i j` whenever `c.rel i j`.
We in fact ask for differentials `d i j` for all `i j : ι`,
but have a field `shape'` requiring that these are zero when not allowe... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_comp_d (C : homological_complex V c) (i j k : ι) :
C.d i j ≫ C.d j k = 0 | begin
by_cases hij : c.rel i j,
{ by_cases hjk : c.rel j k,
{ exact C.d_comp_d' i j k hij hjk },
{ rw [C.shape j k hjk, comp_zero] } },
{ rw [C.shape i j hij, zero_comp] }
end | lemma | homological_complex.d_comp_d | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {C₁ C₂ : homological_complex V c} (h_X : C₁.X = C₂.X)
(h_d : ∀ (i j : ι), c.rel i j → C₁.d i j ≫ eq_to_hom (congr_fun h_X j) =
eq_to_hom (congr_fun h_X i) ≫ C₂.d i j) : C₁ = C₂ | begin
cases C₁,
cases C₂,
dsimp at h_X,
subst h_X,
simp only [true_and, eq_self_iff_true, heq_iff_eq],
ext i j,
by_cases hij : c.rel i j,
{ simpa only [id_comp, eq_to_hom_refl, comp_id] using h_d i j hij, },
{ rw [C₁_shape' i j hij, C₂_shape' i j hij], }
end | lemma | homological_complex.ext | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"heq_iff_eq",
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
chain_complex (α : Type*) [add_right_cancel_semigroup α] [has_one α] : Type* | homological_complex V (complex_shape.down α) | abbreviation | chain_complex | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_right_cancel_semigroup",
"complex_shape.down",
"homological_complex"
] | An `α`-indexed chain complex is a `homological_complex`
in which `d i j ≠ 0` only if `j + 1 = i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cochain_complex (α : Type*) [add_right_cancel_semigroup α] [has_one α] : Type* | homological_complex V (complex_shape.up α) | abbreviation | cochain_complex | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_right_cancel_semigroup",
"complex_shape.up",
"homological_complex"
] | An `α`-indexed cochain complex is a `homological_complex`
in which `d i j ≠ 0` only if `i + 1 = j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prev (α : Type*) [add_right_cancel_semigroup α] [has_one α] (i : α) :
(complex_shape.down α).prev i = i+1 | (complex_shape.down α).prev_eq' rfl | lemma | chain_complex.prev | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_right_cancel_semigroup",
"complex_shape.down"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next (α : Type*) [add_group α] [has_one α] (i : α) :
(complex_shape.down α).next i = i-1 | (complex_shape.down α).next_eq' $ sub_add_cancel _ _ | lemma | chain_complex.next | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_group",
"complex_shape.down"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next_nat_zero :
(complex_shape.down ℕ).next 0 = 0 | by { classical, refine dif_neg _, push_neg, intro, apply nat.no_confusion } | lemma | chain_complex.next_nat_zero | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.down"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next_nat_succ (i : ℕ) :
(complex_shape.down ℕ).next (i+1) = i | (complex_shape.down ℕ).next_eq' rfl | lemma | chain_complex.next_nat_succ | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.down"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev (α : Type*) [add_group α] [has_one α] (i : α) :
(complex_shape.up α).prev i = i-1 | (complex_shape.up α).prev_eq' $ sub_add_cancel _ _ | lemma | cochain_complex.prev | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_group",
"complex_shape.up"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next (α : Type*) [add_right_cancel_semigroup α] [has_one α] (i : α) :
(complex_shape.up α).next i = i+1 | (complex_shape.up α).next_eq' rfl | lemma | cochain_complex.next | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"add_right_cancel_semigroup",
"complex_shape.up"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_nat_zero :
(complex_shape.up ℕ).prev 0 = 0 | by { classical, refine dif_neg _, push_neg, intro, apply nat.no_confusion } | lemma | cochain_complex.prev_nat_zero | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.up"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_nat_succ (i : ℕ) :
(complex_shape.up ℕ).prev (i+1) = i | (complex_shape.up ℕ).prev_eq' rfl | lemma | cochain_complex.prev_nat_succ | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.up"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom (A B : homological_complex V c) | (f : ∀ i, A.X i ⟶ B.X i)
(comm' : ∀ i j, c.rel i j → f i ≫ B.d i j = A.d i j ≫ f j . obviously) | structure | homological_complex.hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | A morphism of homological complexes consists of maps between the chain groups,
commuting with the differentials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.comm {A B : homological_complex V c} (f : A.hom B) (i j : ι) :
f.f i ≫ B.d i j = A.d i j ≫ f.f j | begin
by_cases hij : c.rel i j,
{ exact f.comm' i j hij },
rw [A.shape i j hij, B.shape i j hij, comp_zero, zero_comp],
end | lemma | homological_complex.hom.comm | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (A : homological_complex V c) : hom A A | { f := λ _, 𝟙 _ } | def | homological_complex.id | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | Identity chain map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (A B C : homological_complex V c) (φ : hom A B) (ψ : hom B C) : hom A C | { f := λ i, φ.f i ≫ ψ.f i } | def | homological_complex.comp | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | Composition of chain maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_f (C : homological_complex V c) (i : ι) : hom.f (𝟙 C) i = 𝟙 (C.X i) | rfl | lemma | homological_complex.id_f | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_f {C₁ C₂ C₃ : homological_complex V c} (f : C₁ ⟶ C₂) (g : C₂ ⟶ C₃) (i : ι) :
(f ≫ g).f i = f.f i ≫ g.f i | rfl | lemma | homological_complex.comp_f | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_f {C₁ C₂ : homological_complex V c} (h : C₁ = C₂) (n : ι) :
homological_complex.hom.f (eq_to_hom h) n =
eq_to_hom (congr_fun (congr_arg homological_complex.X h) n) | by { subst h, refl, } | lemma | homological_complex.eq_to_hom_f | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_f_injective {C₁ C₂ : homological_complex V c} :
function.injective (λ f : hom C₁ C₂, f.f) | by tidy | lemma | homological_complex.hom_f_injective | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (C D : homological_complex V c) (i : ι) :
(0 : C ⟶ D).f i = 0 | rfl | lemma | homological_complex.zero_apply | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero [has_zero_object V] : homological_complex V c | { X := λ i, 0, d := λ i j, 0 } | def | homological_complex.zero | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | The zero complex | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_zero_zero [has_zero_object V] : is_zero (zero : homological_complex V c) | by { refine ⟨λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩; ext, } | lemma | homological_complex.is_zero_zero | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_hom {C D : homological_complex V c} {f g : C ⟶ D} (w : f = g) (i : ι) : f.f i = g.f i | congr_fun (congr_arg hom.f w) i | lemma | homological_complex.congr_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval (i : ι) : homological_complex V c ⥤ V | { obj := λ C, C.X i,
map := λ C D f, f.f i, } | def | homological_complex.eval | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | The functor picking out the `i`-th object of a complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : homological_complex V c ⥤ graded_object ι V | { obj := λ C, C.X,
map := λ _ _ f, f.f } | def | homological_complex.forget | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex"
] | The functor forgetting the differential in a complex, obtaining a graded object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_eval (i : ι) : forget V c ⋙ graded_object.eval i ≅ eval V c i | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | homological_complex.forget_eval | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Forgetting the differentials than picking out the `i`-th object is the same as
just picking out the `i`-th object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_comp_eq_to_hom {i j j' : ι} (rij : c.rel i j) (rij' : c.rel i j') :
C.d i j' ≫ eq_to_hom (congr_arg C.X (c.next_eq rij' rij)) = C.d i j | begin
have P : ∀ h : j' = j, C.d i j' ≫ eq_to_hom (congr_arg C.X h) = C.d i j,
{ rintro rfl, simp },
apply P,
end | lemma | homological_complex.d_comp_eq_to_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | If `C.d i j` and `C.d i j'` are both allowed, then we must have `j = j'`,
and so the differentials only differ by an `eq_to_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom_comp_d {i i' j : ι} (rij : c.rel i j) (rij' : c.rel i' j) :
eq_to_hom (congr_arg C.X (c.prev_eq rij rij')) ≫ C.d i' j = C.d i j | begin
have P : ∀ h : i = i', eq_to_hom (congr_arg C.X h) ≫ C.d i' j = C.d i j,
{ rintro rfl, simp },
apply P,
end | lemma | homological_complex.eq_to_hom_comp_d | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | If `C.d i j` and `C.d i' j` are both allowed, then we must have `i = i'`,
and so the differentials only differ by an `eq_to_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_eq_kernel [has_kernels V] {i j j' : ι} (r : c.rel i j) (r' : c.rel i j') :
kernel_subobject (C.d i j) = kernel_subobject (C.d i j') | begin
rw ←d_comp_eq_to_hom C r r',
apply kernel_subobject_comp_mono,
end | lemma | homological_complex.kernel_eq_kernel | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_image [has_images V] [has_equalizers V]
{i i' j : ι} (r : c.rel i j) (r' : c.rel i' j) :
image_subobject (C.d i j) = image_subobject (C.d i' j) | begin
rw ←eq_to_hom_comp_d C r r',
apply image_subobject_iso_comp,
end | lemma | homological_complex.image_eq_image | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_prev (j : ι) : V | C.X (c.prev j) | abbreviation | homological_complex.X_prev | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Either `C.X i`, if there is some `i` with `c.rel i j`, or `C.X j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_prev_iso {i j : ι} (r : c.rel i j) :
C.X_prev j ≅ C.X i | eq_to_iso $ by rw ← c.prev_eq' r | def | homological_complex.X_prev_iso | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | If `c.rel i j`, then `C.X_prev j` is isomorphic to `C.X i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_prev_iso_self {j : ι} (h : ¬c.rel (c.prev j) j) :
C.X_prev j ≅ C.X j | eq_to_iso $ congr_arg C.X begin
dsimp [complex_shape.prev],
rw dif_neg, push_neg, intros i hi,
have : c.prev j = i := c.prev_eq' hi,
rw this at h, contradiction,
end | def | homological_complex.X_prev_iso_self | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.prev"
] | If there is no `i` so `c.rel i j`, then `C.X_prev j` is isomorphic to `C.X j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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