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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from_next (i : ι) : (Π i j, C.X i ⟶ D.X j) →+ (C.X_next i ⟶ D.X i) | add_monoid_hom.mk' (λ f, f (c.next i) i) $ λ f g, rfl | def | from_next | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | `f i' i` if `i'` comes after `i`, and 0 if there's no such `i'`.
Hopefully there won't be much need for this, except in `d_next_eq_d_from_from_next`
to see that `d_next` factors through `C.d_from i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_next_eq_d_from_from_next (f : Π i j, C.X i ⟶ D.X j) (i : ι) :
d_next i f = C.d_from i ≫ from_next i f | rfl | lemma | d_next_eq_d_from_from_next | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next",
"from_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_eq (f : Π i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.rel i i') :
d_next i f = C.d i i' ≫ f i' i | by { obtain rfl := c.next_eq' w, refl } | lemma | d_next_eq | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_comp_left (f : C ⟶ D) (g : Π i j, D.X i ⟶ E.X j) (i : ι) :
d_next i (λ i j, f.f i ≫ g i j) = f.f i ≫ d_next i g | (f.comm_assoc _ _ _).symm | lemma | d_next_comp_left | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_comp_right (f : Π i j, C.X i ⟶ D.X j) (g : D ⟶ E) (i : ι) :
d_next i (λ i j, f i j ≫ g.f j) = d_next i f ≫ g.f i | (category.assoc _ _ _).symm | lemma | d_next_comp_right | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d (j : ι) : (Π i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.X j) | add_monoid_hom.mk' (λ f, f j (c.prev j) ≫ D.d (c.prev j) j) $
λ f g, preadditive.add_comp _ _ _ _ _ _ | def | prev_d | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | The composition of `f j j' ≫ D.d j' j` if there is some `j'` coming before `j`,
and `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prev (j : ι) : (Π i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.X_prev j) | add_monoid_hom.mk' (λ f, f j (c.prev j)) $ λ f g, rfl | def | to_prev | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | `f j j'` if `j'` comes after `j`, and 0 if there's no such `j'`.
Hopefully there won't be much need for this, except in `d_next_eq_d_from_from_next`
to see that `d_next` factors through `C.d_from i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prev_d_eq_to_prev_d_to (f : Π i j, C.X i ⟶ D.X j) (j : ι) :
prev_d j f = to_prev j f ≫ D.d_to j | rfl | lemma | prev_d_eq_to_prev_d_to | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"prev_d",
"to_prev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_eq (f : Π i j, C.X i ⟶ D.X j) {j j' : ι} (w : c.rel j' j) :
prev_d j f = f j j' ≫ D.d j' j | by { obtain rfl := c.prev_eq' w, refl } | lemma | prev_d_eq | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_comp_left (f : C ⟶ D) (g : Π i j, D.X i ⟶ E.X j) (j : ι) :
prev_d j (λ i j, f.f i ≫ g i j) = f.f j ≫ prev_d j g | category.assoc _ _ _ | lemma | prev_d_comp_left | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_comp_right (f : Π i j, C.X i ⟶ D.X j) (g : D ⟶ E) (j : ι) :
prev_d j (λ i j, f i j ≫ g.f j) = prev_d j f ≫ g.f j | by { dsimp [prev_d], simp only [category.assoc, g.comm] } | lemma | prev_d_comp_right | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_nat (C D : chain_complex V ℕ) (i : ℕ) (f : Π i j, C.X i ⟶ D.X j) :
d_next i f = C.d i (i-1) ≫ f (i-1) i | begin
dsimp [d_next],
cases i,
{ simp only [shape, chain_complex.next_nat_zero, complex_shape.down_rel,
nat.one_ne_zero, not_false_iff, zero_comp], },
{ dsimp only [nat.succ_eq_add_one],
have : (complex_shape.down ℕ).next (i + 1) = i + 1 - 1,
{ rw chain_complex.next_nat_succ, refl },
congr' 2,... | lemma | d_next_nat | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"chain_complex",
"chain_complex.next_nat_succ",
"chain_complex.next_nat_zero",
"complex_shape.down",
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_nat (C D : cochain_complex V ℕ) (i : ℕ) (f : Π i j, C.X i ⟶ D.X j) :
prev_d i f = f i (i-1) ≫ D.d (i-1) i | begin
dsimp [prev_d],
cases i,
{ simp only [shape, cochain_complex.prev_nat_zero, complex_shape.up_rel,
nat.one_ne_zero, not_false_iff, comp_zero]},
{ dsimp only [nat.succ_eq_add_one],
have : (complex_shape.up ℕ).prev (i + 1) = i + 1 - 1,
{ rw cochain_complex.prev_nat_succ, refl },
congr' 2, }... | lemma | prev_d_nat | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"cochain_complex",
"cochain_complex.prev_nat_succ",
"cochain_complex.prev_nat_zero",
"complex_shape.up",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy (f g : C ⟶ D) | (hom : Π i j, C.X i ⟶ D.X j)
(zero' : ∀ i j, ¬ c.rel j i → hom i j = 0 . obviously)
(comm : ∀ i, f.f i = d_next i hom + prev_d i hom + g.f i . obviously') | structure | homotopy | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"d_next",
"obviously'",
"prev_d"
] | A homotopy `h` between chain maps `f` and `g` consists of components `h i j : C.X i ⟶ D.X j`
which are zero unless `c.rel j i`, satisfying the homotopy condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_sub_zero : homotopy f g ≃ homotopy (f - g) 0 | { to_fun := λ h,
{ hom := λ i j, h.hom i j,
zero' := λ i j w, h.zero _ _ w,
comm := λ i, by simp [h.comm] },
inv_fun := λ h,
{ hom := λ i j, h.hom i j,
zero' := λ i j w, h.zero _ _ w,
comm := λ i, by simpa [sub_eq_iff_eq_add] using h.comm i },
left_inv := by tidy,
right_inv := by tidy, } | def | homotopy.equiv_sub_zero | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homotopy",
"inv_fun"
] | `f` is homotopic to `g` iff `f - g` is homotopic to `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_eq (h : f = g) : homotopy f g | { hom := 0,
zero' := λ _ _ _, rfl,
comm := λ _, by simp only [add_monoid_hom.map_zero, zero_add, h] } | def | homotopy.of_eq | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homotopy",
"of_eq"
] | Equal chain maps are homotopic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : C ⟶ D) : homotopy f f | of_eq (rfl : f = f) | def | homotopy.refl | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homotopy",
"of_eq"
] | Every chain map is homotopic to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm {f g : C ⟶ D} (h : homotopy f g) : homotopy g f | { hom := -h.hom,
zero' := λ i j w, by rw [pi.neg_apply, pi.neg_apply, h.zero i j w, neg_zero],
comm := λ i, by rw [add_monoid_hom.map_neg, add_monoid_hom.map_neg, h.comm, ← neg_add,
← add_assoc, neg_add_self, zero_add] } | def | homotopy.symm | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homotopy"
] | `f` is homotopic to `g` iff `g` is homotopic to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans {e f g : C ⟶ D} (h : homotopy e f) (k : homotopy f g) : homotopy e g | { hom := h.hom + k.hom,
zero' := λ i j w, by rw [pi.add_apply, pi.add_apply, h.zero i j w, k.zero i j w, zero_add],
comm := λ i, by { rw [add_monoid_hom.map_add, add_monoid_hom.map_add, h.comm, k.comm], abel }, } | def | homotopy.trans | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homotopy"
] | homotopy is a transitive relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add {f₁ g₁ f₂ g₂ : C ⟶ D}
(h₁ : homotopy f₁ g₁) (h₂ : homotopy f₂ g₂) : homotopy (f₁+f₂) (g₁+g₂) | { hom := h₁.hom + h₂.hom,
zero' := λ i j hij, by
rw [pi.add_apply, pi.add_apply, h₁.zero' i j hij, h₂.zero' i j hij, add_zero],
comm := λ i, by
{ simp only [homological_complex.add_f_apply, h₁.comm, h₂.comm,
add_monoid_hom.map_add],
abel, }, } | def | homotopy.add | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homological_complex.add_f_apply",
"homotopy"
] | the sum of two homotopies is a homotopy between the sum of the respective morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right {e f : C ⟶ D} (h : homotopy e f) (g : D ⟶ E) : homotopy (e ≫ g) (f ≫ g) | { hom := λ i j, h.hom i j ≫ g.f j,
zero' := λ i j w, by rw [h.zero i j w, zero_comp],
comm := λ i, by simp only [h.comm i, d_next_comp_right, preadditive.add_comp,
prev_d_comp_right, comp_f], } | def | homotopy.comp_right | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"d_next_comp_right",
"homotopy",
"prev_d_comp_right"
] | homotopy is closed under composition (on the right) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_left {f g : D ⟶ E} (h : homotopy f g) (e : C ⟶ D) : homotopy (e ≫ f) (e ≫ g) | { hom := λ i j, e.f i ≫ h.hom i j,
zero' := λ i j w, by rw [h.zero i j w, comp_zero],
comm := λ i, by simp only [h.comm i, d_next_comp_left, preadditive.comp_add,
prev_d_comp_left, comp_f], } | def | homotopy.comp_left | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"d_next_comp_left",
"homotopy",
"prev_d_comp_left"
] | homotopy is closed under composition (on the left) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {C₁ C₂ C₃ : homological_complex V c} {f₁ g₁ : C₁ ⟶ C₂} {f₂ g₂ : C₂ ⟶ C₃}
(h₁ : homotopy f₁ g₁) (h₂ : homotopy f₂ g₂) : homotopy (f₁ ≫ f₂) (g₁ ≫ g₂) | (h₁.comp_right _).trans (h₂.comp_left _) | def | homotopy.comp | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homological_complex",
"homotopy"
] | homotopy is closed under composition | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right_id {f : C ⟶ C} (h : homotopy f (𝟙 C)) (g : C ⟶ D) : homotopy (f ≫ g) g | (h.comp_right g).trans (of_eq $ category.id_comp _) | def | homotopy.comp_right_id | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homotopy",
"of_eq"
] | a variant of `homotopy.comp_right` useful for dealing with homotopy equivalences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_left_id {f : D ⟶ D} (h : homotopy f (𝟙 D)) (g : C ⟶ D) : homotopy (g ≫ f) g | (h.comp_left g).trans (of_eq $ category.comp_id _) | def | homotopy.comp_left_id | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homotopy",
"of_eq"
] | a variant of `homotopy.comp_left` useful for dealing with homotopy equivalences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopic_map (hom : Π i j, C.X i ⟶ D.X j) : C ⟶ D | { f := λ i, d_next i hom + prev_d i hom,
comm' := λ i j hij,
begin
have eq1 : prev_d i hom ≫ D.d i j = 0,
{ simp only [prev_d, add_monoid_hom.mk'_apply, category.assoc, d_comp_d, comp_zero], },
have eq2 : C.d i j ≫ d_next j hom = 0,
{ simp only [d_next, add_monoid_hom.mk'_apply, d_comp_d_assoc... | def | homotopy.null_homotopic_map | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next",
"d_next_eq",
"prev_d",
"prev_d_eq"
] | The null homotopic map associated to a family `hom` of morphisms `C_i ⟶ D_j`.
This is the same datum as for the field `hom` in the structure `homotopy`. For
this definition, we do not need the field `zero` of that structure
as this definition uses only the maps `C_i ⟶ C_j` when `c.rel j i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopic_map' (h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) : C ⟶ D | null_homotopic_map (λ i j, dite (c.rel j i) (h i j) (λ _, 0)) | def | homotopy.null_homotopic_map' | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | Variant of `null_homotopic_map` where the input consists only of the
relevant maps `C_i ⟶ D_j` such that `c.rel j i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopic_map_comp (hom : Π i j, C.X i ⟶ D.X j) (g : D ⟶ E) :
null_homotopic_map hom ≫ g = null_homotopic_map (λ i j, hom i j ≫ g.f j) | begin
ext n,
dsimp [null_homotopic_map, from_next, to_prev, add_monoid_hom.mk'_apply],
simp only [preadditive.add_comp, category.assoc, g.comm],
end | lemma | homotopy.null_homotopic_map_comp | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"from_next",
"to_prev"
] | Compatibility of `null_homotopic_map` with the postcomposition by a morphism
of complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopic_map'_comp (hom : Π i j, c.rel j i → (C.X i ⟶ D.X j)) (g : D ⟶ E) :
null_homotopic_map' hom ≫ g = null_homotopic_map' (λ i j hij, hom i j hij ≫ g.f j) | begin
ext n,
erw null_homotopic_map_comp,
congr',
ext i j,
split_ifs,
{ refl, },
{ rw zero_comp, },
end | lemma | homotopy.null_homotopic_map'_comp | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | Compatibility of `null_homotopic_map'` with the postcomposition by a morphism
of complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_null_homotopic_map (f : C ⟶ D) (hom : Π i j, D.X i ⟶ E.X j) :
f ≫ null_homotopic_map hom = null_homotopic_map (λ i j, f.f i ≫ hom i j) | begin
ext n,
dsimp [null_homotopic_map, from_next, to_prev, add_monoid_hom.mk'_apply],
simp only [preadditive.comp_add, category.assoc, f.comm_assoc],
end | lemma | homotopy.comp_null_homotopic_map | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"from_next",
"to_prev"
] | Compatibility of `null_homotopic_map` with the precomposition by a morphism
of complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_null_homotopic_map' (f : C ⟶ D) (hom : Π i j, c.rel j i → (D.X i ⟶ E.X j)) :
f ≫ null_homotopic_map' hom = null_homotopic_map' (λ i j hij, f.f i ≫ hom i j hij) | begin
ext n,
erw comp_null_homotopic_map,
congr',
ext i j,
split_ifs,
{ refl, },
{ rw comp_zero, },
end | lemma | homotopy.comp_null_homotopic_map' | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | Compatibility of `null_homotopic_map'` with the precomposition by a morphism
of complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_null_homotopic_map {W : Type*} [category W] [preadditive W]
(G : V ⥤ W) [G.additive] (hom : Π i j, C.X i ⟶ D.X j) :
(G.map_homological_complex c).map (null_homotopic_map hom) =
null_homotopic_map (λ i j, G.map (hom i j)) | begin
ext i,
dsimp [null_homotopic_map, d_next, prev_d],
simp only [G.map_comp, functor.map_add],
end | lemma | homotopy.map_null_homotopic_map | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next",
"prev_d"
] | Compatibility of `null_homotopic_map` with the application of additive functors | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_null_homotopic_map' {W : Type*} [category W] [preadditive W]
(G : V ⥤ W) [G.additive] (hom : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
(G.map_homological_complex c).map (null_homotopic_map' hom) =
null_homotopic_map' (λ i j hij, G.map (hom i j hij)) | begin
ext n,
erw map_null_homotopic_map,
congr',
ext i j,
split_ifs,
{ refl, },
{ rw G.map_zero, }
end | lemma | homotopy.map_null_homotopic_map' | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | Compatibility of `null_homotopic_map'` with the application of additive functors | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopy (hom : Π i j, C.X i ⟶ D.X j) (zero' : ∀ i j, ¬ c.rel j i → hom i j = 0) :
homotopy (null_homotopic_map hom) 0 | { hom := hom,
zero' := zero',
comm := by { intro i, rw [homological_complex.zero_f_apply, add_zero], refl, }, } | def | homotopy.null_homotopy | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"homological_complex.zero_f_apply",
"homotopy"
] | Tautological construction of the `homotopy` to zero for maps constructed by
`null_homotopic_map`, at least when we have the `zero'` condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopy' (h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
homotopy (null_homotopic_map' h) 0 | begin
apply null_homotopy (λ i j, dite (c.rel j i) (h i j) (λ _, 0)),
intros i j hij,
dsimp,
rw [dite_eq_right_iff],
intro hij',
exfalso,
exact hij hij',
end | def | homotopy.null_homotopy' | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"dite_eq_right_iff",
"homotopy"
] | Homotopy to zero for maps constructed with `null_homotopic_map'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
null_homotopic_map_f {k₂ k₁ k₀ : ι} (r₂₁ : c.rel k₂ k₁) (r₁₀ : c.rel k₁ k₀)
(hom : Π i j, C.X i ⟶ D.X j) :
(null_homotopic_map hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁ | by { dsimp only [null_homotopic_map], rw [d_next_eq hom r₁₀, prev_d_eq hom r₂₁], } | lemma | homotopy.null_homotopic_map_f | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next_eq",
"prev_d_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map'_f {k₂ k₁ k₀ : ι} (r₂₁ : c.rel k₂ k₁) (r₁₀ : c.rel k₁ k₀)
(h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
(null_homotopic_map' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁ | begin
simp only [← null_homotopic_map'],
rw null_homotopic_map_f r₂₁ r₁₀ (λ i j, dite (c.rel j i) (h i j) (λ _, 0)),
dsimp,
split_ifs,
refl,
end | lemma | homotopy.null_homotopic_map'_f | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.rel k₀ l)
(hom : Π i j, C.X i ⟶ D.X j) :
(null_homotopic_map hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀ | begin
dsimp only [null_homotopic_map],
rw [prev_d_eq hom r₁₀, d_next, add_monoid_hom.mk'_apply, C.shape, zero_comp, zero_add],
exact hk₀ _
end | lemma | homotopy.null_homotopic_map_f_of_not_rel_left | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next",
"prev_d_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map'_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.rel k₀ l)
(h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
(null_homotopic_map' h).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀ | begin
simp only [← null_homotopic_map'],
rw null_homotopic_map_f_of_not_rel_left r₁₀ hk₀ (λ i j, dite (c.rel j i) (h i j) (λ _, 0)),
dsimp,
split_ifs,
refl,
end | lemma | homotopy.null_homotopic_map'_f_of_not_rel_left | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.rel l k₁)
(hom : Π i j, C.X i ⟶ D.X j) :
(null_homotopic_map hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ | begin
dsimp only [null_homotopic_map],
rw [d_next_eq hom r₁₀, prev_d, add_monoid_hom.mk'_apply, D.shape, comp_zero, add_zero],
exact hk₁ _,
end | lemma | homotopy.null_homotopic_map_f_of_not_rel_right | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next_eq",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map'_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.rel l k₁)
(h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
(null_homotopic_map' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ | begin
simp only [← null_homotopic_map'],
rw null_homotopic_map_f_of_not_rel_right r₁₀ hk₁ (λ i j, dite (c.rel j i) (h i j) (λ _, 0)),
dsimp,
split_ifs,
refl,
end | lemma | homotopy.null_homotopic_map'_f_of_not_rel_right | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map_f_eq_zero {k₀ : ι}
(hk₀ : ∀ l : ι, ¬c.rel k₀ l) (hk₀' : ∀ l : ι, ¬c.rel l k₀)
(hom : Π i j, C.X i ⟶ D.X j) :
(null_homotopic_map hom).f k₀ = 0 | begin
dsimp [null_homotopic_map, d_next, prev_d],
rw [C.shape, D.shape, zero_comp, comp_zero, add_zero]; apply_assumption,
end | lemma | homotopy.null_homotopic_map_f_eq_zero | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"d_next",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
null_homotopic_map'_f_eq_zero {k₀ : ι}
(hk₀ : ∀ l : ι, ¬c.rel k₀ l) (hk₀' : ∀ l : ι, ¬c.rel l k₀)
(h : Π i j, c.rel j i → (C.X i ⟶ D.X j)) :
(null_homotopic_map' h).f k₀ = 0 | begin
simp only [← null_homotopic_map'],
exact null_homotopic_map_f_eq_zero hk₀ hk₀'
(λ i j, dite (c.rel j i) (h i j) (λ _, 0)),
end | lemma | homotopy.null_homotopic_map'_f_eq_zero | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_chain_complex (f : Π i j, P.X i ⟶ Q.X j) (j : ℕ) :
prev_d j f = f j (j+1) ≫ Q.d _ _ | begin
dsimp [prev_d],
have : (complex_shape.down ℕ).prev j = j + 1 := chain_complex.prev ℕ j,
congr' 2,
end | lemma | homotopy.prev_d_chain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"chain_complex.prev",
"complex_shape.down",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_succ_chain_complex (f : Π i j, P.X i ⟶ Q.X j) (i : ℕ) :
d_next (i+1) f = P.d _ _ ≫ f i (i+1) | begin
dsimp [d_next],
have : (complex_shape.down ℕ).next (i + 1) = i := chain_complex.next_nat_succ _,
congr' 2,
end | lemma | homotopy.d_next_succ_chain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"chain_complex.next_nat_succ",
"complex_shape.down",
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_next_zero_chain_complex (f : Π i j, P.X i ⟶ Q.X j) :
d_next 0 f = 0 | begin
dsimp [d_next],
rw [P.shape, zero_comp],
rw chain_complex.next_nat_zero, dsimp, dec_trivial,
end | lemma | homotopy.d_next_zero_chain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"chain_complex.next_nat_zero",
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_inductive_aux₁ :
Π n, Σ' (f : P.X n ⟶ Q.X (n+1)) (f' : P.X (n+1) ⟶ Q.X (n+2)),
e.f (n+1) = P.d (n+1) n ≫ f + f' ≫ Q.d (n+2) (n+1) | | 0 := ⟨zero, one, comm_one⟩
| 1 := ⟨one, (succ 0 ⟨zero, one, comm_one⟩).1, (succ 0 ⟨zero, one, comm_one⟩).2⟩
| (n+2) :=
⟨(mk_inductive_aux₁ (n+1)).2.1,
(succ (n+1) (mk_inductive_aux₁ (n+1))).1,
(succ (n+1) (mk_inductive_aux₁ (n+1))).2⟩ | def | homotopy.mk_inductive_aux₁ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | An auxiliary construction for `mk_inductive`.
Here we build by induction a family of diagrams,
but don't require at the type level that these successive diagrams actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a homotopy)
in `mk_inductive`.
At this stage, we don... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_inductive_aux₂ :
Π n, Σ' (f : P.X_next n ⟶ Q.X n) (f' : P.X n ⟶ Q.X_prev n), e.f n = P.d_from n ≫ f + f' ≫ Q.d_to n | | 0 := ⟨0, zero ≫ (Q.X_prev_iso rfl).inv, by simpa using comm_zero⟩
| (n+1) := let I := mk_inductive_aux₁ e zero comm_zero one comm_one succ n in
⟨(P.X_next_iso rfl).hom ≫ I.1, I.2.1 ≫ (Q.X_prev_iso rfl).inv, by simpa using I.2.2⟩ | def | homotopy.mk_inductive_aux₂ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | An auxiliary construction for `mk_inductive`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_inductive_aux₃ (i j : ℕ) (h : i+1 = j) :
(mk_inductive_aux₂ e zero comm_zero one comm_one succ i).2.1 ≫ (Q.X_prev_iso h).hom
= (P.X_next_iso h).inv ≫ (mk_inductive_aux₂ e zero comm_zero one comm_one succ j).1 | by subst j; rcases i with (_|_|i); { dsimp, simp, } | lemma | homotopy.mk_inductive_aux₃ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_inductive : homotopy e 0 | { hom := λ i j, if h : i + 1 = j then
(mk_inductive_aux₂ e zero comm_zero one comm_one succ i).2.1 ≫ (Q.X_prev_iso h).hom
else
0,
zero' := λ i j w, by rwa dif_neg,
comm := λ i, begin
dsimp, simp only [add_zero],
convert (mk_inductive_aux₂ e zero comm_zero one comm_one succ i).2.2,
{ cases i,
... | def | homotopy.mk_inductive | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"aux",
"chain_complex.next_nat_succ",
"chain_complex.next_nat_zero",
"chain_complex.prev",
"comm",
"complex_shape.down",
"from_next",
"homotopy",
"to_prev"
] | A constructor for a `homotopy e 0`, for `e` a chain map between `ℕ`-indexed chain complexes,
working by induction.
You need to provide the components of the homotopy in degrees 0 and 1,
show that these satisfy the homotopy condition,
and then give a construction of each component,
and the fact that it satisfies the ho... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_next_cochain_complex (f : Π i j, P.X i ⟶ Q.X j) (j : ℕ) :
d_next j f = P.d _ _ ≫ f (j+1) j | begin
dsimp [d_next],
have : (complex_shape.up ℕ).next j = j + 1 := cochain_complex.next ℕ j,
congr' 2,
end | lemma | homotopy.d_next_cochain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"cochain_complex.next",
"complex_shape.up",
"d_next"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_succ_cochain_complex (f : Π i j, P.X i ⟶ Q.X j) (i : ℕ) :
prev_d (i+1) f = f (i+1) _ ≫ Q.d i (i+1) | begin
dsimp [prev_d],
have : (complex_shape.up ℕ).prev (i+1) = i := cochain_complex.prev_nat_succ i,
congr' 2,
end | lemma | homotopy.prev_d_succ_cochain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"cochain_complex.prev_nat_succ",
"complex_shape.up",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prev_d_zero_cochain_complex (f : Π i j, P.X i ⟶ Q.X j) :
prev_d 0 f = 0 | begin
dsimp [prev_d],
rw [Q.shape, comp_zero],
rw [cochain_complex.prev_nat_zero], dsimp, dec_trivial,
end | lemma | homotopy.prev_d_zero_cochain_complex | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"cochain_complex.prev_nat_zero",
"prev_d"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coinductive_aux₁ :
Π n, Σ' (f : P.X (n+1) ⟶ Q.X n) (f' : P.X (n+2) ⟶ Q.X (n+1)),
e.f (n+1) = f ≫ Q.d n (n+1) + P.d (n+1) (n+2) ≫ f' | | 0 := ⟨zero, one, comm_one⟩
| 1 := ⟨one, (succ 0 ⟨zero, one, comm_one⟩).1, (succ 0 ⟨zero, one, comm_one⟩).2⟩
| (n+2) :=
⟨(mk_coinductive_aux₁ (n+1)).2.1,
(succ (n+1) (mk_coinductive_aux₁ (n+1))).1,
(succ (n+1) (mk_coinductive_aux₁ (n+1))).2⟩ | def | homotopy.mk_coinductive_aux₁ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | An auxiliary construction for `mk_coinductive`.
Here we build by induction a family of diagrams,
but don't require at the type level that these successive diagrams actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a homotopy)
in `mk_coinductive`.
At this stage, we... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_coinductive_aux₂ :
Π n, Σ' (f : P.X n ⟶ Q.X_prev n) (f' : P.X_next n ⟶ Q.X n),
e.f n = f ≫ Q.d_to n + P.d_from n ≫ f' | | 0 := ⟨0, (P.X_next_iso rfl).hom ≫ zero, by simpa using comm_zero⟩
| (n+1) := let I := mk_coinductive_aux₁ e zero comm_zero one comm_one succ n in
⟨I.1 ≫ (Q.X_prev_iso rfl).inv, (P.X_next_iso rfl).hom ≫ I.2.1, by simpa using I.2.2⟩ | def | homotopy.mk_coinductive_aux₂ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | An auxiliary construction for `mk_inductive`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_coinductive_aux₃ (i j : ℕ) (h : i + 1 = j) :
(P.X_next_iso h).inv ≫ (mk_coinductive_aux₂ e zero comm_zero one comm_one succ i).2.1
= (mk_coinductive_aux₂ e zero comm_zero one comm_one succ j).1 ≫ (Q.X_prev_iso h).hom | by subst j; rcases i with (_|_|i); { dsimp, simp, } | lemma | homotopy.mk_coinductive_aux₃ | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coinductive : homotopy e 0 | { hom := λ i j, if h : j + 1 = i then
(P.X_next_iso h).inv ≫ (mk_coinductive_aux₂ e zero comm_zero one comm_one succ j).2.1
else
0,
zero' := λ i j w, by rwa dif_neg,
comm := λ i, begin
dsimp,
rw [add_zero, add_comm],
convert (mk_coinductive_aux₂ e zero comm_zero one comm_one succ i).2.2 using ... | def | homotopy.mk_coinductive | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"aux",
"cochain_complex.next",
"cochain_complex.prev_nat_succ",
"cochain_complex.prev_nat_zero",
"comm",
"complex_shape.up",
"from_next",
"homotopy",
"to_prev"
] | A constructor for a `homotopy e 0`, for `e` a chain map between `ℕ`-indexed cochain complexes,
working by induction.
You need to provide the components of the homotopy in degrees 0 and 1,
show that these satisfy the homotopy condition,
and then give a construction of each component,
and the fact that it satisfies the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv (C D : homological_complex V c) | (hom : C ⟶ D)
(inv : D ⟶ C)
(homotopy_hom_inv_id : homotopy (hom ≫ inv) (𝟙 C))
(homotopy_inv_hom_id : homotopy (inv ≫ hom) (𝟙 D)) | structure | homotopy_equiv | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homological_complex",
"homotopy"
] | A homotopy equivalence between two chain complexes consists of a chain map each way,
and homotopies from the compositions to the identity chain maps.
Note that this contains data;
arguably it might be more useful for many applications if we truncated it to a Prop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (C : homological_complex V c) : homotopy_equiv C C | { hom := 𝟙 C,
inv := 𝟙 C,
homotopy_hom_inv_id := by simp,
homotopy_inv_hom_id := by simp, } | def | homotopy_equiv.refl | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homological_complex",
"homotopy_equiv"
] | Any complex is homotopy equivalent to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm
{C D : homological_complex V c} (f : homotopy_equiv C D) :
homotopy_equiv D C | { hom := f.inv,
inv := f.hom,
homotopy_hom_inv_id := f.homotopy_inv_hom_id,
homotopy_inv_hom_id := f.homotopy_hom_inv_id, } | def | homotopy_equiv.symm | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homological_complex",
"homotopy_equiv"
] | Being homotopy equivalent is a symmetric relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans
{C D E : homological_complex V c} (f : homotopy_equiv C D) (g : homotopy_equiv D E) :
homotopy_equiv C E | { hom := f.hom ≫ g.hom,
inv := g.inv ≫ f.inv,
homotopy_hom_inv_id := by simpa using
((g.homotopy_hom_inv_id.comp_right_id f.inv).comp_left f.hom).trans f.homotopy_hom_inv_id,
homotopy_inv_hom_id := by simpa using
((f.homotopy_inv_hom_id.comp_right_id g.hom).comp_left g.inv).trans g.homotopy_inv_hom_id, } | def | homotopy_equiv.trans | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homological_complex",
"homotopy_equiv"
] | Homotopy equivalence is a transitive relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso {ι : Type*} {V : Type u} [category.{v} V] [preadditive V]
{c : complex_shape ι} {C D : homological_complex V c} (f : C ≅ D) :
homotopy_equiv C D | ⟨f.hom, f.inv, homotopy.of_eq f.3, homotopy.of_eq f.4⟩ | def | homotopy_equiv.of_iso | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"complex_shape",
"homological_complex",
"homotopy.of_eq",
"homotopy_equiv"
] | An isomorphism of complexes induces a homotopy equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_map_eq_of_homotopy (h : homotopy f g) (i : ι) :
(homology_functor V c i).map f = (homology_functor V c i).map g | begin
dsimp [homology_functor],
apply eq_of_sub_eq_zero,
ext,
simp only [homology.π_map, comp_zero, preadditive.comp_sub],
dsimp [kernel_subobject_map],
simp_rw [h.comm i],
simp only [zero_add, zero_comp, d_next_eq_d_from_from_next, kernel_subobject_arrow_comp_assoc,
preadditive.comp_add],
rw [←prea... | theorem | homology_map_eq_of_homotopy | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"category_theory.subobject.factor_thru_add_sub_factor_thru_right",
"d_next_eq_d_from_from_next",
"homology.π_map",
"homology_functor",
"homotopy",
"prev_d_eq_to_prev_d_to"
] | Homotopic maps induce the same map on homology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_obj_iso_of_homotopy_equiv (f : homotopy_equiv C D) (i : ι) :
(homology_functor V c i).obj C ≅ (homology_functor V c i).obj D | { hom := (homology_functor V c i).map f.hom,
inv := (homology_functor V c i).map f.inv,
hom_inv_id' := begin
rw [←functor.map_comp, homology_map_eq_of_homotopy f.homotopy_hom_inv_id,
category_theory.functor.map_id],
end,
inv_hom_id' := begin
rw [←functor.map_comp, homology_map_eq_of_homotopy f.hom... | def | homology_obj_iso_of_homotopy_equiv | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homology_functor",
"homology_map_eq_of_homotopy",
"homotopy_equiv"
] | Homotopy equivalent complexes have isomorphic homologies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.map_homotopy (F : V ⥤ W) [F.additive] {f g : C ⟶ D} (h : homotopy f g) :
homotopy ((F.map_homological_complex c).map f) ((F.map_homological_complex c).map g) | { hom := λ i j, F.map (h.hom i j),
zero' := λ i j w, by { rw [h.zero i j w, F.map_zero], },
comm := λ i, begin
dsimp [d_next, prev_d] at *,
rw h.comm i,
simp only [F.map_add, ← F.map_comp],
refl
end, } | def | category_theory.functor.map_homotopy | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"comm",
"d_next",
"homotopy",
"prev_d"
] | An additive functor takes homotopies to homotopies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.map_homotopy_equiv (F : V ⥤ W) [F.additive] (h : homotopy_equiv C D) :
homotopy_equiv ((F.map_homological_complex c).obj C) ((F.map_homological_complex c).obj D) | { hom := (F.map_homological_complex c).map h.hom,
inv := (F.map_homological_complex c).map h.inv,
homotopy_hom_inv_id := begin
rw [←(F.map_homological_complex c).map_comp, ←(F.map_homological_complex c).map_id],
exact F.map_homotopy h.homotopy_hom_inv_id,
end,
homotopy_inv_hom_id := begin
rw [←(F.ma... | def | category_theory.functor.map_homotopy_equiv | algebra.homology | src/algebra/homology/homotopy.lean | [
"algebra.homology.additive",
"tactic.abel"
] | [
"homotopy_equiv",
"map_comp",
"map_id"
] | An additive functor preserves homotopy equivalences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopic : hom_rel (homological_complex V c) | λ C D f g, nonempty (homotopy f g) | def | homotopic | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"hom_rel",
"homological_complex",
"homotopy"
] | The congruence on `homological_complex V c` given by the existence of a homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_congruence : congruence (homotopic V c) | { is_equiv := λ C D,
{ refl := λ C, ⟨homotopy.refl C⟩,
symm := λ f g ⟨w⟩, ⟨w.symm⟩,
trans := λ f g h ⟨w₁⟩ ⟨w₂⟩, ⟨w₁.trans w₂⟩, },
comp_left := λ E F G m₁ m₂ g ⟨i⟩, ⟨i.comp_left _⟩,
comp_right := λ E F G f m₁ m₂ ⟨i⟩, ⟨i.comp_right _⟩, } | instance | homotopy_congruence | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_category | category_theory.quotient (homotopic V c) | def | homotopy_category | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient",
"homotopic"
] | `homotopy_category V c` is the category of chain complexes of shape `c` in `V`,
with chain maps identified when they are homotopic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient : homological_complex V c ⥤ homotopy_category V c | category_theory.quotient.functor _ | def | homotopy_category.quotient | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient.functor",
"homological_complex",
"homotopy_category"
] | The quotient functor from complexes to the homotopy category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_obj_as (C : homological_complex V c) :
((quotient V c).obj C).as = C | rfl | lemma | homotopy_category.quotient_obj_as | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_out {C D : homotopy_category V c} (f : C ⟶ D) :
(quotient V c).map f.out = f | quot.out_eq _ | lemma | homotopy_category.quotient_map_out | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homotopy_category",
"quot.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_homotopy {C D : homological_complex V c} (f g : C ⟶ D) (h : homotopy f g) :
(quotient V c).map f = (quotient V c).map g | category_theory.quotient.sound _ ⟨h⟩ | lemma | homotopy_category.eq_of_homotopy | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient.sound",
"homological_complex",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_of_eq {C D : homological_complex V c} (f g : C ⟶ D)
(w : (quotient V c).map f = (quotient V c).map g) : homotopy f g | ((quotient.functor_map_eq_iff _ _ _).mp w).some | def | homotopy_category.homotopy_of_eq | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex",
"homotopy"
] | If two chain maps become equal in the homotopy category, then they are homotopic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_out_map {C D : homological_complex V c} (f : C ⟶ D) :
homotopy ((quotient V c).map f).out f | begin
apply homotopy_of_eq,
simp,
end | def | homotopy_category.homotopy_out_map | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex",
"homotopy"
] | An arbitrarily chosen representation of the image of a chain map in the homotopy category
is homotopic to the original chain map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_out_comp_out {C D E : homotopy_category V c} (f : C ⟶ D) (g : D ⟶ E) :
(quotient V c).map (quot.out f ≫ quot.out g) = f ≫ g | by conv_rhs { erw [←quotient_map_out f, ←quotient_map_out g, ←(quotient V c).map_comp], } | lemma | homotopy_category.quotient_map_out_comp_out | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homotopy_category",
"map_comp",
"quot.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_homotopy_equiv {C D : homological_complex V c} (f : homotopy_equiv C D) :
(quotient V c).obj C ≅ (quotient V c).obj D | { hom := (quotient V c).map f.hom,
inv := (quotient V c).map f.inv,
hom_inv_id' := begin
rw [←(quotient V c).map_comp, ←(quotient V c).map_id],
exact eq_of_homotopy _ _ f.homotopy_hom_inv_id,
end,
inv_hom_id' := begin
rw [←(quotient V c).map_comp, ←(quotient V c).map_id],
exact eq_of_homotopy _ ... | def | homotopy_category.iso_of_homotopy_equiv | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex",
"homotopy_equiv",
"map_comp",
"map_id"
] | Homotopy equivalent complexes become isomorphic in the homotopy category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_equiv_of_iso
{C D : homological_complex V c} (i : (quotient V c).obj C ≅ (quotient V c).obj D) :
homotopy_equiv C D | { hom := quot.out i.hom,
inv := quot.out i.inv,
homotopy_hom_inv_id := homotopy_of_eq _ _ (by { simp, refl, }),
homotopy_inv_hom_id := homotopy_of_eq _ _ (by { simp, refl, }), } | def | homotopy_category.homotopy_equiv_of_iso | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex",
"homotopy_equiv",
"quot.out"
] | If two complexes become isomorphic in the homotopy category,
then they were homotopy equivalent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_functor (i : ι) : homotopy_category V c ⥤ V | category_theory.quotient.lift _ (homology_functor V c i)
(λ C D f g ⟨h⟩, homology_map_eq_of_homotopy h i) | def | homotopy_category.homology_functor | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient.lift",
"homology_functor",
"homology_map_eq_of_homotopy",
"homotopy_category"
] | The `i`-th homology, as a functor from the homotopy category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_factors (i : ι) :
quotient V c ⋙ homology_functor V c i ≅ _root_.homology_functor V c i | category_theory.quotient.lift.is_lift _ _ _ | def | homotopy_category.homology_factors | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient.lift.is_lift",
"homology_functor"
] | The homology functor on the homotopy category is just the usual homology functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_factors_hom_app (i : ι) (C : homological_complex V c) :
(homology_factors V c i).hom.app C = 𝟙 _ | rfl | lemma | homotopy_category.homology_factors_hom_app | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_factors_inv_app (i : ι) (C : homological_complex V c) :
(homology_factors V c i).inv.app C = 𝟙 _ | rfl | lemma | homotopy_category.homology_factors_inv_app | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_functor_map_factors (i : ι) {C D : homological_complex V c} (f : C ⟶ D) :
(_root_.homology_functor V c i).map f =
((homology_functor V c i).map ((quotient V c).map f) : _) | (category_theory.quotient.lift_map_functor_map _ (_root_.homology_functor V c i) _ f).symm | lemma | homotopy_category.homology_functor_map_factors | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"category_theory.quotient.lift_map_functor_map",
"homological_complex",
"homology_functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.map_homotopy_category (c : complex_shape ι) (F : V ⥤ W) [F.additive] :
homotopy_category V c ⥤ homotopy_category W c | { obj := λ C, (homotopy_category.quotient W c).obj ((F.map_homological_complex c).obj C.as),
map := λ C D f,
(homotopy_category.quotient W c).map ((F.map_homological_complex c).map (quot.out f)),
map_id' := λ C, begin
rw ←(homotopy_category.quotient W c).map_id,
apply homotopy_category.eq_of_homotopy,
... | def | category_theory.functor.map_homotopy_category | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"complex_shape",
"homotopy_category",
"homotopy_category.eq_of_homotopy",
"homotopy_category.homotopy_of_eq",
"homotopy_category.quotient",
"homotopy_category.quotient_map_out_comp_out",
"map_comp",
"map_id",
"quot.out",
"quot.out_eq"
] | An additive functor induces a functor between homotopy categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.map_homotopy_category {F G : V ⥤ W} [F.additive] [G.additive]
(α : F ⟶ G) (c : complex_shape ι) : F.map_homotopy_category c ⟶ G.map_homotopy_category c | { app := λ C,
(homotopy_category.quotient W c).map ((nat_trans.map_homological_complex α c).app C.as),
naturality' := λ C D f,
begin
dsimp,
simp only [←functor.map_comp],
congr' 1,
ext,
dsimp,
simp,
end } | def | category_theory.nat_trans.map_homotopy_category | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"complex_shape",
"homotopy_category.quotient"
] | A natural transformation induces a natural transformation between
the induced functors on the homotopy category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.map_homotopy_category_id (c : complex_shape ι) (F : V ⥤ W) [F.additive] :
nat_trans.map_homotopy_category (𝟙 F) c = 𝟙 (F.map_homotopy_category c) | by tidy | lemma | category_theory.nat_trans.map_homotopy_category_id | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.map_homotopy_category_comp (c : complex_shape ι)
{F G H : V ⥤ W} [F.additive] [G.additive] [H.additive]
(α : F ⟶ G) (β : G ⟶ H):
nat_trans.map_homotopy_category (α ≫ β) c =
nat_trans.map_homotopy_category α c ≫ nat_trans.map_homotopy_category β c | by tidy | lemma | category_theory.nat_trans.map_homotopy_category_comp | algebra.homology | src/algebra/homology/homotopy_category.lean | [
"algebra.homology.homotopy",
"category_theory.quotient"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_le_kernel (w : f ≫ g = 0) :
image_subobject f ≤ kernel_subobject g | image_subobject_le_mk _ _ (kernel.lift _ _ w) (by simp) | lemma | image_le_kernel | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel (w : f ≫ g = 0) :
(image_subobject f : V) ⟶ (kernel_subobject g : V) | (subobject.of_le _ _ (image_le_kernel _ _ w)) | def | image_to_kernel | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_le_kernel"
] | The canonical morphism `image_subobject f ⟶ kernel_subobject g` when `f ≫ g = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subobject_of_le_as_image_to_kernel (w : f ≫ g = 0) (h) :
subobject.of_le (image_subobject f) (kernel_subobject g) h = image_to_kernel f g w | rfl | lemma | subobject_of_le_as_image_to_kernel | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | Prefer `image_to_kernel`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_to_kernel_arrow (w : f ≫ g = 0) :
image_to_kernel f g w ≫ (kernel_subobject g).arrow = (image_subobject f).arrow | by simp [image_to_kernel] | lemma | image_to_kernel_arrow | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_thru_image_subobject_comp_image_to_kernel (w : f ≫ g = 0) :
factor_thru_image_subobject f ≫ image_to_kernel f g w = factor_thru_kernel_subobject g f w | by { ext, simp, } | lemma | factor_thru_image_subobject_comp_image_to_kernel | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_zero_left [has_kernels V] [has_zero_object V] {w} :
image_to_kernel (0 : A ⟶ B) g w = 0 | by { ext, simp, } | lemma | image_to_kernel_zero_left | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_zero_right [has_images V] {w} :
image_to_kernel f (0 : B ⟶ C) w =
(image_subobject f).arrow ≫ inv (kernel_subobject (0 : B ⟶ C)).arrow | by { ext, simp } | lemma | image_to_kernel_zero_right | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_comp_right {D : V} (h : C ⟶ D) (w : f ≫ g = 0) :
image_to_kernel f (g ≫ h) (by simp [reassoc_of w]) =
image_to_kernel f g w ≫ subobject.of_le _ _ (kernel_subobject_comp_le g h) | by { ext, simp } | lemma | image_to_kernel_comp_right | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_comp_left {Z : V} (h : Z ⟶ A) (w : f ≫ g = 0) :
image_to_kernel (h ≫ f) g (by simp [w]) =
subobject.of_le _ _ (image_subobject_comp_le h f) ≫ image_to_kernel f g w | by { ext, simp } | lemma | image_to_kernel_comp_left | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_comp_mono {D : V} (h : C ⟶ D) [mono h] (w) :
image_to_kernel f (g ≫ h) w =
image_to_kernel f g ((cancel_mono h).mp (by simpa using w : (f ≫ g) ≫ h = 0 ≫ h)) ≫
(subobject.iso_of_eq _ _ (kernel_subobject_comp_mono g h)).inv | by { ext, simp, } | lemma | image_to_kernel_comp_mono | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_epi_comp {Z : V} (h : Z ⟶ A) [epi h] (w) :
image_to_kernel (h ≫ f) g w =
subobject.of_le _ _ (image_subobject_comp_le h f) ≫
image_to_kernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) | by { ext, simp, } | lemma | image_to_kernel_epi_comp | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_comp_hom_inv_comp [has_equalizers V] [has_images V] {Z : V} {i : B ≅ Z} (w) :
image_to_kernel (f ≫ i.hom) (i.inv ≫ g) w =
(image_subobject_comp_iso _ _).hom ≫ image_to_kernel f g (by simpa using w) ≫
(kernel_subobject_iso_comp i.inv g).inv | by { ext, simp, } | lemma | image_to_kernel_comp_hom_inv_comp | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_epi_of_zero_of_mono [has_kernels V] [has_zero_object V] [mono g] :
epi (image_to_kernel (0 : A ⟶ B) g (by simp)) | epi_of_target_iso_zero _ (kernel_subobject_iso g ≪≫ kernel.of_mono g) | instance | image_to_kernel_epi_of_zero_of_mono | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | `image_to_kernel` for `A --0--> B --g--> C`, where `g` is a mono is itself an epi
(i.e. the sequence is exact at `B`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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