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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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X_next (i : ι) : V | C.X (c.next i) | abbreviation | homological_complex.X_next | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Either `C.X j`, if there is some `j` with `c.rel i j`, or `C.X i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_next_iso {i j : ι} (r : c.rel i j) :
C.X_next i ≅ C.X j | eq_to_iso $ by rw ← c.next_eq' r | def | homological_complex.X_next_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_next i` is isomorphic to `C.X j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_next_iso_self {i : ι} (h : ¬c.rel i (c.next i)) :
C.X_next i ≅ C.X i | eq_to_iso $ congr_arg C.X begin
dsimp [complex_shape.next],
rw dif_neg, rintro ⟨j, hj⟩,
have : c.next i = j := c.next_eq' hj,
rw this at h, contradiction,
end | def | homological_complex.X_next_iso_self | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"complex_shape.next"
] | If there is no `j` so `c.rel i j`, then `C.X_next i` is isomorphic to `C.X i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_to (j : ι) : C.X_prev j ⟶ C.X j | C.d (c.prev j) j | abbreviation | homological_complex.d_to | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | The differential mapping into `C.X j`, or zero if there isn't one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_from (i : ι) : C.X i ⟶ C.X_next i | C.d i (c.next i) | abbreviation | homological_complex.d_from | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | The differential mapping out of `C.X i`, or zero if there isn't one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_to_eq {i j : ι} (r : c.rel i j) :
C.d_to j = (C.X_prev_iso r).hom ≫ C.d i j | begin
obtain rfl := c.prev_eq' r,
exact (category.id_comp _).symm,
end | lemma | homological_complex.d_to_eq | 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 | |
d_to_eq_zero {j : ι} (h : ¬c.rel (c.prev j) j) :
C.d_to j = 0 | C.shape _ _ h | lemma | homological_complex.d_to_eq_zero | 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 | |
d_from_eq {i j : ι} (r : c.rel i j) :
C.d_from i = C.d i j ≫ (C.X_next_iso r).inv | begin
obtain rfl := c.next_eq' r,
exact (category.comp_id _).symm,
end | lemma | homological_complex.d_from_eq | 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 | |
d_from_eq_zero {i : ι} (h : ¬c.rel i (c.next i)) :
C.d_from i = 0 | C.shape _ _ h | lemma | homological_complex.d_from_eq_zero | 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_iso_comp_d_to {i j : ι} (r : c.rel i j) :
(C.X_prev_iso r).inv ≫ C.d_to j = C.d i j | by simp [C.d_to_eq r] | lemma | homological_complex.X_prev_iso_comp_d_to | 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_iso_self_comp_d_to {j : ι} (h : ¬c.rel (c.prev j) j) :
(C.X_prev_iso_self h).inv ≫ C.d_to j = 0 | by simp [h] | lemma | homological_complex.X_prev_iso_self_comp_d_to | 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 | |
d_from_comp_X_next_iso {i j : ι} (r : c.rel i j) :
C.d_from i ≫ (C.X_next_iso r).hom = C.d i j | by simp [C.d_from_eq r] | lemma | homological_complex.d_from_comp_X_next_iso | 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 | |
d_from_comp_X_next_iso_self {i : ι} (h : ¬c.rel i (c.next i)) :
C.d_from i ≫ (C.X_next_iso_self h).hom = 0 | by simp [h] | lemma | homological_complex.d_from_comp_X_next_iso_self | 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 | |
d_to_comp_d_from (j : ι) : C.d_to j ≫ C.d_from j = 0 | C.d_comp_d _ _ _ | lemma | homological_complex.d_to_comp_d_from | 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 | |
kernel_from_eq_kernel [has_kernels V] {i j : ι} (r : c.rel i j) :
kernel_subobject (C.d_from i) = kernel_subobject (C.d i j) | begin
rw C.d_from_eq r,
apply kernel_subobject_comp_mono,
end | lemma | homological_complex.kernel_from_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_to_eq_image [has_images V] [has_equalizers V]
{i j : ι} (r : c.rel i j) :
image_subobject (C.d_to j) = image_subobject (C.d i j) | begin
rw C.d_to_eq r,
apply image_subobject_iso_comp,
end | lemma | homological_complex.image_to_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 | |
iso_app (f : C₁ ≅ C₂) (i : ι) : C₁.X i ≅ C₂.X i | (eval V c i).map_iso f | def | homological_complex.hom.iso_app | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | The `i`-th component of an isomorphism of chain complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_components (f : Π i, C₁.X i ≅ C₂.X i)
(hf : ∀ i j, c.rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom) :
C₁ ≅ C₂ | { hom := { f := λ i, (f i).hom, comm' := hf },
inv :=
{ f := λ i, (f i).inv,
comm' := λ i j hij,
calc (f i).inv ≫ C₁.d i j
= (f i).inv ≫ (C₁.d i j ≫ (f j).hom) ≫ (f j).inv : by simp
... = (f i).inv ≫ ((f i).hom ≫ C₂.d i j) ≫ (f j).inv : by rw hf i j hij
... = C₂.d i j ≫ (f j).inv : by simp ... | def | homological_complex.hom.iso_of_components | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Construct an isomorphism of chain complexes from isomorphism of the objects
which commute with the differentials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_components_app (f : Π i, C₁.X i ≅ C₂.X i)
(hf : ∀ i j, c.rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom) (i : ι) :
iso_app (iso_of_components f hf) i = f i | by { ext, simp, } | lemma | homological_complex.hom.iso_of_components_app | 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 | |
is_iso_of_components (f : C₁ ⟶ C₂) [∀ (n : ι), is_iso (f.f n)] : is_iso f | begin
convert is_iso.of_iso (homological_complex.hom.iso_of_components (λ n, as_iso (f.f n))
(by tidy)),
ext n,
refl,
end | lemma | homological_complex.hom.is_iso_of_components | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"homological_complex.hom.iso_of_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev (f : hom C₁ C₂) (j : ι) : C₁.X_prev j ⟶ C₂.X_prev j | f.f _ | abbreviation | homological_complex.hom.prev | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | `f.prev j` is `f.f i` if there is some `r i j`, and `f.f j` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prev_eq (f : hom C₁ C₂) {i j : ι} (w : c.rel i j) :
f.prev j = (C₁.X_prev_iso w).hom ≫ f.f i ≫ (C₂.X_prev_iso w).inv | begin
obtain rfl := c.prev_eq' w,
simp only [X_prev_iso, eq_to_iso_refl, iso.refl_hom, iso.refl_inv, id_comp, comp_id],
end | lemma | homological_complex.hom.prev_eq | 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 | |
next (f : hom C₁ C₂) (i : ι) : C₁.X_next i ⟶ C₂.X_next i | f.f _ | abbreviation | homological_complex.hom.next | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | `f.next i` is `f.f j` if there is some `r i j`, and `f.f j` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
next_eq (f : hom C₁ C₂) {i j : ι} (w : c.rel i j) :
f.next i = (C₁.X_next_iso w).hom ≫ f.f j ≫ (C₂.X_next_iso w).inv | begin
obtain rfl := c.next_eq' w,
simp only [X_next_iso, eq_to_iso_refl, iso.refl_hom, iso.refl_inv, id_comp, comp_id],
end | lemma | homological_complex.hom.next_eq | 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 | |
comm_from (f : hom C₁ C₂) (i : ι) :
f.f i ≫ C₂.d_from i = C₁.d_from i ≫ f.next i | f.comm _ _ | lemma | homological_complex.hom.comm_from | 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 | |
comm_to (f : hom C₁ C₂) (j : ι) :
f.prev j ≫ C₂.d_to j = C₁.d_to j ≫ f.f j | f.comm _ _ | lemma | homological_complex.hom.comm_to | 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 | |
sq_from (f : hom C₁ C₂) (i : ι) : arrow.mk (C₁.d_from i) ⟶ arrow.mk (C₂.d_from i) | arrow.hom_mk (f.comm_from i) | def | homological_complex.hom.sq_from | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | A morphism of chain complexes
induces a morphism of arrows of the differentials out of each object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_from_left (f : hom C₁ C₂) (i : ι) : (f.sq_from i).left = f.f i | rfl | lemma | homological_complex.hom.sq_from_left | 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 | |
sq_from_right (f : hom C₁ C₂) (i : ι) : (f.sq_from i).right = f.next i | rfl | lemma | homological_complex.hom.sq_from_right | 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 | |
sq_from_id (C₁ : homological_complex V c) (i : ι) : sq_from (𝟙 C₁) i = 𝟙 _ | rfl | lemma | homological_complex.hom.sq_from_id | 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 | |
sq_from_comp (f : C₁ ⟶ C₂) (g : C₂ ⟶ C₃) (i : ι) :
sq_from (f ≫ g) i = sq_from f i ≫ sq_from g i | rfl | lemma | homological_complex.hom.sq_from_comp | 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 | |
sq_to (f : hom C₁ C₂) (j : ι) : arrow.mk (C₁.d_to j) ⟶ arrow.mk (C₂.d_to j) | arrow.hom_mk (f.comm_to j) | def | homological_complex.hom.sq_to | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | A morphism of chain complexes
induces a morphism of arrows of the differentials into each object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_to_left (f : hom C₁ C₂) (j : ι) : (f.sq_to j).left = f.prev j | rfl | lemma | homological_complex.hom.sq_to_left | 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 | |
sq_to_right (f : hom C₁ C₂) (j : ι) : (f.sq_to j).right = f.f j | rfl | lemma | homological_complex.hom.sq_to_right | 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 | |
of (X : α → V) (d : Π n, X (n+1) ⟶ X n) (sq : ∀ n, d (n+1) ≫ d n = 0) : chain_complex V α | { X := X,
d := λ i j, if h : i = j + 1 then
eq_to_hom (by subst h) ≫ d j
else
0,
shape' := λ i j w, by rw dif_neg (ne.symm w),
d_comp_d' := λ i j k hij hjk,
begin
dsimp at hij hjk, substs hij hjk,
simp only [category.id_comp, dif_pos rfl, eq_to_hom_refl],
exact sq k,
end } | def | chain_complex.of | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"chain_complex"
] | Construct an `α`-indexed chain complex from a dependently-typed differential. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_X (n : α) : (of X d sq).X n = X n | rfl | lemma | chain_complex.of_X | 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 | |
of_d (j : α) : (of X d sq).d (j+1) j = d j | by { dsimp [of], rw [if_pos rfl, category.id_comp] } | lemma | chain_complex.of_d | 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 | |
of_d_ne {i j : α} (h : i ≠ j + 1) : (of X d sq).d i j = 0 | by { dsimp [of], rw [dif_neg h], } | lemma | chain_complex.of_d_ne | 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 | |
of_hom (f : Π i : α, X i ⟶ Y i) (comm : ∀ i : α, f (i+1) ≫ d_Y i = d_X i ≫ f i) :
of X d_X sq_X ⟶ of Y d_Y sq_Y | { f := f,
comm' := λ n m,
begin
by_cases h : n = m + 1,
{ subst h,
simpa using comm m, },
{ rw [of_d_ne X _ _ h, of_d_ne Y _ _ h], simp }
end } | def | chain_complex.of_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"comm"
] | A constructor for chain maps between `α`-indexed chain complexes built using `chain_complex.of`,
from a dependently typed collection of morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_struct | (X₀ X₁ X₂ : V)
(d₀ : X₁ ⟶ X₀)
(d₁ : X₂ ⟶ X₁)
(s : d₁ ≫ d₀ = 0) | structure | chain_complex.mk_struct | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Auxiliary structure for setting up the recursion in `mk`.
This is purely an implementation detail: for some reason just using the dependent 6-tuple directly
results in `mk_aux` taking much longer (well over the `-T100000` limit) to elaborate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_struct.flat (t : mk_struct V) :
Σ' (X₀ X₁ X₂ : V) (d₀ : X₁ ⟶ X₀) (d₁ : X₂ ⟶ X₁), d₁ ≫ d₀ = 0 | ⟨t.X₀, t.X₁, t.X₂, t.d₀, t.d₁, t.s⟩ | def | chain_complex.mk_struct.flat | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Flatten to a tuple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_aux :
Π n : ℕ, mk_struct V | | 0 := ⟨X₀, X₁, X₂, d₀, d₁, s⟩
| (n+1) :=
let p := mk_aux n in
⟨p.X₁, p.X₂, (succ p.flat).1, p.d₁, (succ p.flat).2.1, (succ p.flat).2.2⟩ | def | chain_complex.mk_aux | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Auxiliary definition for `mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : chain_complex V ℕ | of (λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).X₀) (λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).d₀)
(λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).s) | def | chain_complex.mk | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"chain_complex"
] | A inductive constructor for `ℕ`-indexed chain complexes.
You provide explicitly the first two differentials,
then a function which takes two differentials and the fact they compose to zero,
and returns the next object, its differential, and the fact it composes appropiately to zero.
See also `mk'`, which only sees th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_X_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).X 0 = X₀ | rfl | lemma | chain_complex.mk_X_0 | 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 | |
mk_X_1 : (mk X₀ X₁ X₂ d₀ d₁ s succ).X 1 = X₁ | rfl | lemma | chain_complex.mk_X_1 | 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 | |
mk_X_2 : (mk X₀ X₁ X₂ d₀ d₁ s succ).X 2 = X₂ | rfl | lemma | chain_complex.mk_X_2 | 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 | |
mk_d_1_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 1 0 = d₀ | by { change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀, rw [if_pos rfl, category.id_comp] } | lemma | chain_complex.mk_d_1_0 | 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 | |
mk_d_2_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 2 1 = d₁ | by { change ite (2 = 1 + 1) (𝟙 X₂ ≫ d₁) 0 = d₁, rw [if_pos rfl, category.id_comp] } | lemma | chain_complex.mk_d_2_0 | 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 | |
mk' (X₀ X₁ : V) (d : X₁ ⟶ X₀)
(succ' : Π (t : Σ (X₀ X₁ : V), X₁ ⟶ X₀), Σ' (X₂ : V) (d : X₂ ⟶ t.2.1), d ≫ t.2.2 = 0) :
chain_complex V ℕ | mk X₀ X₁ (succ' ⟨X₀, X₁, d⟩).1 d (succ' ⟨X₀, X₁, d⟩).2.1 (succ' ⟨X₀, X₁, d⟩).2.2
(λ t, succ' ⟨t.2.1, t.2.2.1, t.2.2.2.2.1⟩) | def | chain_complex.mk' | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"chain_complex",
"mk'"
] | A simpler inductive constructor for `ℕ`-indexed chain complexes.
You provide explicitly the first differential,
then a function which takes a differential,
and returns the next object, its differential, and the fact it composes appropriately to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_X_0 : (mk' X₀ X₁ d₀ succ').X 0 = X₀ | rfl | lemma | chain_complex.mk'_X_0 | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_X_1 : (mk' X₀ X₁ d₀ succ').X 1 = X₁ | rfl | lemma | chain_complex.mk'_X_1 | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_d_1_0 : (mk' X₀ X₁ d₀ succ').d 1 0 = d₀ | by { change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀, rw [if_pos rfl, category.id_comp] } | lemma | chain_complex.mk'_d_1_0 | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_hom_aux :
Π n, Σ' (f : P.X n ⟶ Q.X n) (f' : P.X (n+1) ⟶ Q.X (n+1)), f' ≫ Q.d (n+1) n = P.d (n+1) n ≫ f | | 0 := ⟨zero, one, one_zero_comm⟩
| (n+1) := ⟨(mk_hom_aux n).2.1,
(succ n (mk_hom_aux n)).1, (succ n (mk_hom_aux n)).2⟩ | def | chain_complex.mk_hom_aux | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | An auxiliary construction for `mk_hom`.
Here we build by induction a family of commutative squares,
but don't require at the type level that these successive commutative squares actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a chain map)
in `mk_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom : P ⟶ Q | { f := λ n, (mk_hom_aux P Q zero one one_zero_comm succ n).1,
comm' := λ n m,
begin
rintro (rfl : m + 1 = n),
exact (mk_hom_aux P Q zero one one_zero_comm succ m).2.2,
end } | def | chain_complex.mk_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | A constructor for chain maps between `ℕ`-indexed chain complexes,
working by induction on commutative squares.
You need to provide the components of the chain map in degrees 0 and 1,
show that these form a commutative square,
and then give a construction of each component,
and the fact that it forms a commutative squa... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom_f_0 : (mk_hom P Q zero one one_zero_comm succ).f 0 = zero | rfl | lemma | chain_complex.mk_hom_f_0 | 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 | |
mk_hom_f_1 : (mk_hom P Q zero one one_zero_comm succ).f 1 = one | rfl | lemma | chain_complex.mk_hom_f_1 | 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 | |
mk_hom_f_succ_succ (n : ℕ) :
(mk_hom P Q zero one one_zero_comm succ).f (n+2) =
(succ n ⟨(mk_hom P Q zero one one_zero_comm succ).f n,
(mk_hom P Q zero one one_zero_comm succ).f (n+1),
(mk_hom P Q zero one one_zero_comm succ).comm (n+1) n⟩).1 | begin
dsimp [mk_hom, mk_hom_aux],
induction n; congr,
end | lemma | chain_complex.mk_hom_f_succ_succ | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (X : α → V) (d : Π n, X n ⟶ X (n+1)) (sq : ∀ n, d n ≫ d (n+1) = 0) : cochain_complex V α | { X := X,
d := λ i j, if h : i + 1 = j then
d _ ≫ eq_to_hom (by subst h)
else
0,
shape' := λ i j w, by {rw dif_neg, exact w},
d_comp_d' := λ i j k,
begin
split_ifs with h h' h',
{ substs h h',
simp [sq] },
all_goals { simp },
end } | def | cochain_complex.of | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"cochain_complex"
] | Construct an `α`-indexed cochain complex from a dependently-typed differential. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_d (j : α) : (of X d sq).d j (j+1) = d j | by { dsimp [of], rw [if_pos rfl, category.comp_id] } | lemma | cochain_complex.of_d | 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 | |
of_d_ne {i j : α} (h : i + 1 ≠ j) : (of X d sq).d i j = 0 | by { dsimp [of], rw [dif_neg h] } | lemma | cochain_complex.of_d_ne | 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 | |
of_hom (f : Π i : α, X i ⟶ Y i) (comm : ∀ i : α, f i ≫ d_Y i = d_X i ≫ f (i+1)) :
of X d_X sq_X ⟶ of Y d_Y sq_Y | { f := f,
comm' := λ n m,
begin
by_cases h : n + 1 = m,
{ subst h,
simpa using comm n },
{ rw [of_d_ne X _ _ h, of_d_ne Y _ _ h], simp }
end } | def | cochain_complex.of_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"comm"
] | A constructor for chain maps between `α`-indexed cochain complexes built using `cochain_complex.of`,
from a dependently typed collection of morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_struct | (X₀ X₁ X₂ : V)
(d₀ : X₀ ⟶ X₁)
(d₁ : X₁ ⟶ X₂)
(s : d₀ ≫ d₁ = 0) | structure | cochain_complex.mk_struct | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Auxiliary structure for setting up the recursion in `mk`.
This is purely an implementation detail: for some reason just using the dependent 6-tuple directly
results in `mk_aux` taking much longer (well over the `-T100000` limit) to elaborate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_struct.flat (t : mk_struct V) :
Σ' (X₀ X₁ X₂ : V) (d₀ : X₀ ⟶ X₁) (d₁ : X₁ ⟶ X₂), d₀ ≫ d₁ = 0 | ⟨t.X₀, t.X₁, t.X₂, t.d₀, t.d₁, t.s⟩ | def | cochain_complex.mk_struct.flat | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | Flatten to a tuple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : cochain_complex V ℕ | of (λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).X₀) (λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).d₀)
(λ n, (mk_aux X₀ X₁ X₂ d₀ d₁ s succ n).s) | def | cochain_complex.mk | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"cochain_complex"
] | A inductive constructor for `ℕ`-indexed cochain complexes.
You provide explicitly the first two differentials,
then a function which takes two differentials and the fact they compose to zero,
and returns the next object, its differential, and the fact it composes appropiately to zero.
See also `mk'`, which only sees ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_d_1_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 0 1 = d₀ | by { change ite (1 = 0 + 1) (d₀ ≫ 𝟙 X₁) 0 = d₀, rw [if_pos rfl, category.comp_id] } | lemma | cochain_complex.mk_d_1_0 | 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 | |
mk_d_2_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 1 2 = d₁ | by { change ite (2 = 1 + 1) (d₁ ≫ 𝟙 X₂) 0 = d₁, rw [if_pos rfl, category.comp_id] } | lemma | cochain_complex.mk_d_2_0 | 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 | |
mk' (X₀ X₁ : V) (d : X₀ ⟶ X₁)
(succ' : Π (t : Σ (X₀ X₁ : V), X₀ ⟶ X₁), Σ' (X₂ : V) (d : t.2.1 ⟶ X₂), t.2.2 ≫ d = 0) :
cochain_complex V ℕ | mk X₀ X₁ (succ' ⟨X₀, X₁, d⟩).1 d (succ' ⟨X₀, X₁, d⟩).2.1 (succ' ⟨X₀, X₁, d⟩).2.2
(λ t, succ' ⟨t.2.1, t.2.2.1, t.2.2.2.2.1⟩) | def | cochain_complex.mk' | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"cochain_complex",
"mk'"
] | A simpler inductive constructor for `ℕ`-indexed cochain complexes.
You provide explicitly the first differential,
then a function which takes a differential,
and returns the next object, its differential, and the fact it composes appropriately to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_d_1_0 : (mk' X₀ X₁ d₀ succ').d 0 1 = d₀ | by { change ite (1 = 0 + 1) (d₀ ≫ 𝟙 X₁) 0 = d₀, rw [if_pos rfl, category.comp_id] } | lemma | cochain_complex.mk'_d_1_0 | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_hom_aux :
Π n, Σ' (f : P.X n ⟶ Q.X n) (f' : P.X (n+1) ⟶ Q.X (n+1)), f ≫ Q.d n (n+1) = P.d n (n+1) ≫ f' | | 0 := ⟨zero, one, one_zero_comm⟩
| (n+1) := ⟨(mk_hom_aux n).2.1,
(succ n (mk_hom_aux n)).1, (succ n (mk_hom_aux n)).2⟩ | def | cochain_complex.mk_hom_aux | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | An auxiliary construction for `mk_hom`.
Here we build by induction a family of commutative squares,
but don't require at the type level that these successive commutative squares actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a chain map)
in `mk_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom : P ⟶ Q | { f := λ n, (mk_hom_aux P Q zero one one_zero_comm succ n).1,
comm' := λ n m,
begin
rintro (rfl : n + 1 = m),
exact (mk_hom_aux P Q zero one one_zero_comm succ n).2.2,
end } | def | cochain_complex.mk_hom | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [] | A constructor for chain maps between `ℕ`-indexed cochain complexes,
working by induction on commutative squares.
You need to provide the components of the chain map in degrees 0 and 1,
show that these form a commutative square,
and then give a construction of each component,
and the fact that it forms a commutative sq... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_hom_f_succ_succ (n : ℕ) :
(mk_hom P Q zero one one_zero_comm succ).f (n+2) =
(succ n ⟨(mk_hom P Q zero one one_zero_comm succ).f n,
(mk_hom P Q zero one one_zero_comm succ).f (n+1),
(mk_hom P Q zero one one_zero_comm succ).comm n (n+1)⟩).1 | begin
dsimp [mk_hom, mk_hom_aux],
induction n; congr,
end | lemma | cochain_complex.mk_hom_f_succ_succ | algebra.homology | src/algebra/homology/homological_complex.lean | [
"algebra.homology.complex_shape",
"category_theory.subobject.limits",
"category_theory.graded_object"
] | [
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles (i : ι) : subobject (C.X i) | kernel_subobject (C.d_from i) | abbreviation | homological_complex.cycles | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | The cycles at index `i`, as a subobject. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cycles_eq_kernel_subobject {i j : ι} (r : c.rel i j) :
C.cycles i = kernel_subobject (C.d i j) | C.kernel_from_eq_kernel r | lemma | homological_complex.cycles_eq_kernel_subobject | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_iso_kernel {i j : ι} (r : c.rel i j) :
(C.cycles i : V) ≅ kernel (C.d i j) | subobject.iso_of_eq _ _ (C.cycles_eq_kernel_subobject r) ≪≫
kernel_subobject_iso (C.d i j) | def | homological_complex.cycles_iso_kernel | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | The underlying object of `C.cycles i` is isomorphic to `kernel (C.d i j)`,
for any `j` such that `rel i j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cycles_eq_top {i} (h : ¬c.rel i (c.next i)) : C.cycles i = ⊤ | begin
rw eq_top_iff,
apply le_kernel_subobject,
rw [C.d_from_eq_zero h, comp_zero],
end | lemma | homological_complex.cycles_eq_top | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries (C : homological_complex V c) (j : ι) : subobject (C.X j) | image_subobject (C.d_to j) | abbreviation | homological_complex.boundaries | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex"
] | The boundaries at index `i`, as a subobject. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundaries_eq_image_subobject [has_equalizers V] {i j : ι} (r : c.rel i j) :
C.boundaries j = image_subobject (C.d i j) | C.image_to_eq_image r | lemma | homological_complex.boundaries_eq_image_subobject | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries_iso_image [has_equalizers V] {i j : ι} (r : c.rel i j) :
(C.boundaries j : V) ≅ image (C.d i j) | subobject.iso_of_eq _ _ (C.boundaries_eq_image_subobject r) ≪≫
image_subobject_iso (C.d i j) | def | homological_complex.boundaries_iso_image | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | The underlying object of `C.boundaries j` is isomorphic to `image (C.d i j)`,
for any `i` such that `rel i j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundaries_eq_bot [has_zero_object V] {j} (h : ¬c.rel (c.prev j) j) :
C.boundaries j = ⊥ | begin
rw eq_bot_iff,
refine image_subobject_le _ 0 _,
rw [C.d_to_eq_zero h, zero_comp],
end | lemma | homological_complex.boundaries_eq_bot | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries_le_cycles (C : homological_complex V c) (i : ι) :
C.boundaries i ≤ C.cycles i | image_le_kernel _ _ (C.d_to_comp_d_from i) | lemma | homological_complex.boundaries_le_cycles | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"image_le_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries_to_cycles (C : homological_complex V c) (i : ι) :
(C.boundaries i : V) ⟶ (C.cycles i : V) | image_to_kernel _ _ (C.d_to_comp_d_from i) | abbreviation | homological_complex.boundaries_to_cycles | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"image_to_kernel"
] | The canonical map from `boundaries i` to `cycles i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_to_kernel_as_boundaries_to_cycles (C : homological_complex V c) (i : ι) (h) :
(C.boundaries i).of_le (C.cycles i) h = C.boundaries_to_cycles i | rfl | lemma | homological_complex.image_to_kernel_as_boundaries_to_cycles | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex"
] | Prefer `boundaries_to_cycles`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology (C : homological_complex V c) (i : ι) : V | homology (C.d_to i) (C.d_from i) (C.d_to_comp_d_from i) | abbreviation | homological_complex.homology | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"homology"
] | The homology of a complex at index `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_iso (C : homological_complex V c) {i j k : ι} (hij : c.rel i j) (hjk : c.rel j k) :
C.homology j ≅ _root_.homology (C.d i j) (C.d j k) (C.d_comp_d i j k) | homology.map_iso _ _ (arrow.iso_mk (C.X_prev_iso hij) (iso.refl _) $ by dsimp;
rw [C.d_to_eq hij, category.comp_id])
(arrow.iso_mk (iso.refl _) (C.X_next_iso hjk) $ by dsimp; rw [C.d_from_comp_X_next_iso hjk,
category.id_comp]) rfl | def | homological_complex.homology_iso | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"homology.map_iso"
] | The `j`th homology of a homological complex (as kernel of 'the differential from `Cⱼ`' modulo
the image of 'the differential to `Cⱼ`') is isomorphic to the kernel of `d : Cⱼ → Cₖ` modulo
the image of `d : Cᵢ → Cⱼ` when `rel i j` and `rel j k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
chain_complex.homology_zero_iso [has_kernels V] [has_images V] [has_cokernels V]
(C : chain_complex V ℕ) [epi (factor_thru_image (C.d 1 0))] :
C.homology 0 ≅ cokernel (C.d 1 0) | (homology.map_iso _ _ (arrow.iso_mk (C.X_prev_iso rfl) (iso.refl _) $
by rw C.d_to_eq rfl; exact (category.comp_id _).symm : arrow.mk (C.d_to 0) ≅ arrow.mk (C.d 1 0))
(arrow.iso_mk (iso.refl _) (iso.refl _) $
by simp [C.d_from_eq_zero (λ (h : _ = _), one_ne_zero $ by
rwa chain_complex.next_nat_zero at h)] : arrow.m... | def | chain_complex.homology_zero_iso | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"chain_complex",
"chain_complex.next_nat_zero",
"homology.map_iso",
"homology_of_zero_right",
"one_ne_zero"
] | The 0th homology of a chain complex is isomorphic to the cokernel of `d : C₁ ⟶ C₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cochain_complex.homology_zero_iso [has_zero_object V]
[has_kernels V] [has_images V] [has_cokernels V] (C : cochain_complex V ℕ) :
C.homology 0 ≅ kernel (C.d 0 1) | (homology.map_iso _ _ (arrow.iso_mk (C.X_prev_iso_self (by rw cochain_complex.prev_nat_zero;
exact one_ne_zero)) (iso.refl _) (by simp) : arrow.mk (C.d_to 0) ≅ arrow.mk 0)
(arrow.iso_mk (iso.refl _) (C.X_next_iso rfl)
(by simp) : arrow.mk (C.d_from 0) ≅ arrow.mk (C.d 0 1)) $ by simpa).trans $
homology_of_zero_lef... | def | cochain_complex.homology_zero_iso | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cochain_complex",
"cochain_complex.prev_nat_zero",
"homology.map_iso",
"homology_of_zero_left",
"one_ne_zero"
] | The 0th cohomology of a cochain complex is isomorphic to the kernel of `d : C₀ → C₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
chain_complex.homology_succ_iso [has_kernels V] [has_images V] [has_cokernels V]
(C : chain_complex V ℕ) (n : ℕ) :
C.homology (n + 1) ≅ homology (C.d (n + 2) (n + 1)) (C.d (n + 1) n) (C.d_comp_d _ _ _) | C.homology_iso rfl rfl | def | chain_complex.homology_succ_iso | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"chain_complex",
"homology"
] | The `n + 1`th homology of a chain complex (as kernel of 'the differential from `Cₙ₊₁`' modulo
the image of 'the differential to `Cₙ₊₁`') is isomorphic to the kernel of `d : Cₙ₊₁ → Cₙ` modulo
the image of `d : Cₙ₊₂ → Cₙ₊₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cochain_complex.homology_succ_iso [has_kernels V] [has_images V] [has_cokernels V]
(C : cochain_complex V ℕ) (n : ℕ) :
C.homology (n + 1) ≅ homology (C.d n (n + 1)) (C.d (n + 1) (n + 2)) (C.d_comp_d _ _ _) | C.homology_iso rfl rfl | def | cochain_complex.homology_succ_iso | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cochain_complex",
"homology"
] | The `n + 1`th cohomology of a cochain complex (as kernel of 'the differential from `Cₙ₊₁`'
modulo the image of 'the differential to `Cₙ₊₁`') is isomorphic to the kernel of `d : Cₙ₊₁ → Cₙ₊₂`
modulo the image of `d : Cₙ → Cₙ₊₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cycles_map (f : C₁ ⟶ C₂) (i : ι) : (C₁.cycles i : V) ⟶ (C₂.cycles i : V) | subobject.factor_thru _ ((C₁.cycles i).arrow ≫ f.f i) (kernel_subobject_factors _ _ (by simp)) | abbreviation | cycles_map | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | The morphism between cycles induced by a chain map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cycles_map_arrow (f : C₁ ⟶ C₂) (i : ι) :
(cycles_map f i) ≫ (C₂.cycles i).arrow = (C₁.cycles i).arrow ≫ f.f i | by { simp, } | lemma | cycles_map_arrow | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cycles_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_map_id (i : ι) : cycles_map (𝟙 C₁) i = 𝟙 _ | by { dunfold cycles_map, simp, } | lemma | cycles_map_id | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cycles_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_map_comp (f : C₁ ⟶ C₂) (g : C₂ ⟶ C₃) (i : ι) :
cycles_map (f ≫ g) i = cycles_map f i ≫ cycles_map g i | by { dunfold cycles_map, simp [subobject.factor_thru_right], } | lemma | cycles_map_comp | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cycles_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_functor (i : ι) : homological_complex V c ⥤ V | { obj := λ C, C.cycles i,
map := λ C₁ C₂ f, cycles_map f i, } | def | cycles_functor | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"cycles_map",
"homological_complex"
] | Cycles as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundaries_map (f : C₁ ⟶ C₂) (i : ι) : (C₁.boundaries i : V) ⟶ (C₂.boundaries i : V) | image_subobject_map (f.sq_to i) | abbreviation | boundaries_map | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [] | The morphism between boundaries induced by a chain map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundaries_functor (i : ι) : homological_complex V c ⥤ V | { obj := λ C, C.boundaries i,
map := λ C₁ C₂ f, image_subobject_map (f.sq_to i), } | def | boundaries_functor | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex"
] | Boundaries as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundaries_to_cycles_naturality (i : ι) :
boundaries_map f i ≫ C₂.boundaries_to_cycles i = C₁.boundaries_to_cycles i ≫ cycles_map f i | by { ext, simp, } | lemma | boundaries_to_cycles_naturality | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"boundaries_map",
"cycles_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries_to_cycles_nat_trans (i : ι) :
boundaries_functor V c i ⟶ cycles_functor V c i | { app := λ C, C.boundaries_to_cycles i,
naturality' := λ C₁ C₂ f, boundaries_to_cycles_naturality f i, } | def | boundaries_to_cycles_nat_trans | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"boundaries_functor",
"boundaries_to_cycles_naturality",
"cycles_functor"
] | The natural transformation from the boundaries functor to the cycles functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_functor [has_cokernels V] (i : ι) :
homological_complex V c ⥤ V | -- It would be nice if we could just write
-- `cokernel (boundaries_to_cycles_nat_trans V c i)`
-- here, but universe implementation details get in the way...
{ obj := λ C, C.homology i,
map := λ C₁ C₂ f, _root_.homology.map _ _ (f.sq_to i) (f.sq_from i) rfl,
map_id' :=
begin
intros, ext1,
simp only [homo... | def | homology_functor | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"homology.π_map"
] | The `i`-th homology, as a functor to `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_homology_functor [has_cokernels V] :
homological_complex V c ⥤ graded_object ι V | { obj := λ C i, C.homology i,
map := λ C C' f i, (homology_functor V c i).map f,
map_id' :=
begin
intros, ext,
simp only [pi.id_apply, homology.π_map, homology_functor_map, kernel_subobject_map_id,
hom.sq_from_id, category.id_comp, category.comp_id]
end,
map_comp' :=
begin
intros, ext,
... | def | graded_homology_functor | algebra.homology | src/algebra/homology/homology.lean | [
"algebra.homology.image_to_kernel",
"algebra.homology.homological_complex",
"category_theory.graded_object"
] | [
"homological_complex",
"homology.π_map",
"homology_functor"
] | The homology functor from `ι`-indexed complexes to `ι`-graded objects in `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_next (i : ι) : (Π i j, C.X i ⟶ D.X j) →+ (C.X i ⟶ D.X i) | add_monoid_hom.mk' (λ f, C.d i (c.next i) ≫ f (c.next i) i) $
λ f g, preadditive.comp_add _ _ _ _ _ _ | def | d_next | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | The composition of `C.d i i' ≫ f i' i` if there is some `i'` coming after `i`,
and `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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