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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
mul_equiv.multiplicative_additive [mul_one_class H] : multiplicative (additive H) ≃* H | add_equiv.to_multiplicative'' (add_equiv.refl (additive H)) | def | mul_equiv.multiplicative_additive | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_equiv.to_multiplicative''",
"additive",
"mul_one_class",
"multiplicative"
] | `multiplicative (additive H)` is just `H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_units [group G] : G ≃* Gˣ | { to_fun := λ x, ⟨x, x⁻¹, mul_inv_self _, inv_mul_self _⟩,
inv_fun := coe,
left_inv := λ x, rfl,
right_inv := λ u, units.ext rfl,
map_mul' := λ x y, units.ext rfl } | def | to_units | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"group",
"inv_fun",
"inv_mul_self",
"mul_inv_self",
"units.ext"
] | A group is isomorphic to its group of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_units [group G] (g : G) :
(to_units g : G) = g | rfl | lemma | coe_to_units | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"group",
"to_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv (h : M ≃* N) : Mˣ ≃* Nˣ | { inv_fun := map h.symm.to_monoid_hom,
left_inv := λ u, ext $ h.left_inv u,
right_inv := λ u, ext $ h.right_inv u,
.. map h.to_monoid_hom } | def | units.map_equiv | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"inv_fun"
] | A multiplicative equivalence of monoids defines a multiplicative equivalence
of their groups of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_equiv_symm (h : M ≃* N) : (map_equiv h).symm = map_equiv h.symm | rfl | lemma | units.map_equiv_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map_equiv (h : M ≃* N) (x : Mˣ) : (map_equiv h x : N) = h x | rfl | lemma | units.coe_map_equiv | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left (u : Mˣ) : equiv.perm M | { to_fun := λx, u * x,
inv_fun := λx, ↑u⁻¹ * x,
left_inv := u.inv_mul_cancel_left,
right_inv := u.mul_inv_cancel_left } | def | units.mul_left | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.perm",
"inv_fun"
] | Left multiplication by a unit of a monoid is a permutation of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_symm (u : Mˣ) : u.mul_left.symm = u⁻¹.mul_left | equiv.ext $ λ x, rfl | lemma | units.mul_left_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_bijective (a : Mˣ) : function.bijective ((*) a : M → M) | (mul_left a).bijective | lemma | units.mul_left_bijective | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right (u : Mˣ) : equiv.perm M | { to_fun := λx, x * u,
inv_fun := λx, x * ↑u⁻¹,
left_inv := λ x, mul_inv_cancel_right x u,
right_inv := λ x, inv_mul_cancel_right x u } | def | units.mul_right | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.perm",
"inv_fun",
"inv_mul_cancel_right",
"mul_inv_cancel_right"
] | Right multiplication by a unit of a monoid is a permutation of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_symm (u : Mˣ) : u.mul_right.symm = u⁻¹.mul_right | equiv.ext $ λ x, rfl | lemma | units.mul_right_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_bijective (a : Mˣ) : function.bijective ((* a) : M → M) | (mul_right a).bijective | lemma | units.mul_right_bijective | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left (a : G) : perm G | (to_units a).mul_left | def | equiv.mul_left | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"to_units"
] | Left multiplication in a `group` is a permutation of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_left (a : G) : ⇑(equiv.mul_left a) = (*) a | rfl | lemma | equiv.coe_mul_left | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_symm_apply (a : G) : ((equiv.mul_left a).symm : G → G) = (*) a⁻¹ | rfl | lemma | equiv.mul_left_symm_apply | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_left"
] | Extra simp lemma that `dsimp` can use. `simp` will never use this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_symm (a : G) : (equiv.mul_left a).symm = equiv.mul_left a⁻¹ | ext $ λ x, rfl | lemma | equiv.mul_left_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.group.mul_left_bijective (a : G) : function.bijective ((*) a) | (equiv.mul_left a).bijective | lemma | group.mul_left_bijective | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right (a : G) : perm G | (to_units a).mul_right | def | equiv.mul_right | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"to_units"
] | Right multiplication in a `group` is a permutation of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_right (a : G) : ⇑(equiv.mul_right a) = λ x, x * a | rfl | lemma | equiv.coe_mul_right | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_symm (a : G) : (equiv.mul_right a).symm = equiv.mul_right a⁻¹ | ext $ λ x, rfl | lemma | equiv.mul_right_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_symm_apply (a : G) : ((equiv.mul_right a).symm : G → G) = λ x, x * a⁻¹ | rfl | lemma | equiv.mul_right_symm_apply | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_right"
] | Extra simp lemma that `dsimp` can use. `simp` will never use this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.group.mul_right_bijective (a : G) : function.bijective (* a) | (equiv.mul_right a).bijective | lemma | group.mul_right_bijective | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"equiv.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left (a : G) : G ≃ G | { to_fun := λ b, a / b,
inv_fun := λ b, b⁻¹ * a,
left_inv := λ b, by simp [div_eq_mul_inv],
right_inv := λ b, by simp [div_eq_mul_inv] } | def | equiv.div_left | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"div_eq_mul_inv",
"inv_fun"
] | A version of `equiv.mul_left a b⁻¹` that is defeq to `a / b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_left_eq_inv_trans_mul_left (a : G) :
equiv.div_left a = (equiv.inv G).trans (equiv.mul_left a) | ext $ λ _, div_eq_mul_inv _ _ | lemma | equiv.div_left_eq_inv_trans_mul_left | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"div_eq_mul_inv",
"equiv.div_left",
"equiv.inv",
"equiv.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right (a : G) : G ≃ G | { to_fun := λ b, b / a,
inv_fun := λ b, b * a,
left_inv := λ b, by simp [div_eq_mul_inv],
right_inv := λ b, by simp [div_eq_mul_inv] } | def | equiv.div_right | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"div_eq_mul_inv",
"inv_fun"
] | A version of `equiv.mul_right a⁻¹ b` that is defeq to `b / a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_right_eq_mul_right_inv (a : G) : equiv.div_right a = equiv.mul_right a⁻¹ | ext $ λ _, div_eq_mul_inv _ _ | lemma | equiv.div_right_eq_mul_right_inv | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"div_eq_mul_inv",
"equiv.div_right",
"equiv.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.inv (G : Type*) [division_comm_monoid G] : G ≃* G | { to_fun := has_inv.inv,
inv_fun := has_inv.inv,
map_mul' := mul_inv,
..equiv.inv G } | def | mul_equiv.inv | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"division_comm_monoid",
"equiv.inv",
"inv_fun",
"mul_inv"
] | In a `division_comm_monoid`, `equiv.inv` is a `mul_equiv`. There is a variant of this
`mul_equiv.inv' G : G ≃* Gᵐᵒᵖ` for the non-commutative case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.inv_symm (G : Type*) [division_comm_monoid G] :
(mul_equiv.inv G).symm = mul_equiv.inv G | rfl | lemma | mul_equiv.inv_symm | algebra.hom.equiv.units | src/algebra/hom/equiv/units/basic.lean | [
"algebra.hom.equiv.basic",
"algebra.hom.units"
] | [
"division_comm_monoid",
"mul_equiv.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left₀ (a : G) (ha : a ≠ 0) : perm G | (units.mk0 a ha).mul_left | def | equiv.mul_left₀ | algebra.hom.equiv.units | src/algebra/hom/equiv/units/group_with_zero.lean | [
"algebra.hom.equiv.units.basic",
"algebra.group_with_zero.units.basic"
] | [
"units.mk0"
] | Left multiplication by a nonzero element in a `group_with_zero` is a permutation of the
underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.mul_left_bijective₀ (a : G) (ha : a ≠ 0) :
function.bijective ((*) a : G → G) | (equiv.mul_left₀ a ha).bijective | lemma | mul_left_bijective₀ | algebra.hom.equiv.units | src/algebra/hom/equiv/units/group_with_zero.lean | [
"algebra.hom.equiv.units.basic",
"algebra.group_with_zero.units.basic"
] | [
"equiv.mul_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right₀ (a : G) (ha : a ≠ 0) : perm G | (units.mk0 a ha).mul_right | def | equiv.mul_right₀ | algebra.hom.equiv.units | src/algebra/hom/equiv/units/group_with_zero.lean | [
"algebra.hom.equiv.units.basic",
"algebra.group_with_zero.units.basic"
] | [
"units.mk0"
] | Right multiplication by a nonzero element in a `group_with_zero` is a permutation of the
underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.mul_right_bijective₀ (a : G) (ha : a ≠ 0) :
function.bijective ((* a) : G → G) | (equiv.mul_right₀ a ha).bijective | lemma | mul_right_bijective₀ | algebra.hom.equiv.units | src/algebra/hom/equiv/units/group_with_zero.lean | [
"algebra.hom.equiv.units.basic",
"algebra.group_with_zero.units.basic"
] | [
"equiv.mul_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_scalar : has_smul ℕ (C ⟶ D) | ⟨λ n f,
{ f := λ i, n • f.f i,
comm' := λ i j h, by simp [preadditive.nsmul_comp, preadditive.comp_nsmul] }⟩ | instance | homological_complex.has_nat_scalar | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ (C ⟶ D) | ⟨λ n f,
{ f := λ i, n • f.f i,
comm' := λ i j h, by simp [preadditive.zsmul_comp, preadditive.comp_zsmul] }⟩ | instance | homological_complex.has_int_scalar | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_f_apply (i : ι) : (0 : C ⟶ D).f i = 0 | rfl | lemma | homological_complex.zero_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_f_apply (f g : C ⟶ D) (i : ι) : (f + g).f i = f.f i + g.f i | rfl | lemma | homological_complex.add_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_f_apply (f : C ⟶ D) (i : ι) : (-f).f i = -(f.f i) | rfl | lemma | homological_complex.neg_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_f_apply (f g : C ⟶ D) (i : ι) : (f - g).f i = f.f i - g.f i | rfl | lemma | homological_complex.sub_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_f_apply (n : ℕ) (f : C ⟶ D) (i : ι) : (n • f).f i = n • f.f i | rfl | lemma | homological_complex.nsmul_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_f_apply (n : ℤ) (f : C ⟶ D) (i : ι) : (n • f).f i = n • f.f i | rfl | lemma | homological_complex.zsmul_f_apply | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.f_add_monoid_hom {C₁ C₂ : homological_complex V c} (i : ι) :
(C₁ ⟶ C₂) →+ (C₁.X i ⟶ C₂.X i) | add_monoid_hom.mk' (λ f, hom.f f i) (λ _ _, rfl) | def | homological_complex.hom.f_add_monoid_hom | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"homological_complex"
] | The `i`-th component of a chain map, as an additive map from chain maps to morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_additive (i : ι) : (eval V c i).additive | {} | instance | homological_complex.eval_additive | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_additive [has_equalizers V] : (cycles_functor V c i).additive | {} | instance | homological_complex.cycles_additive | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"additive",
"cycles_functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundaries_additive : (boundaries_functor V c i).additive | {} | instance | homological_complex.boundaries_additive | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"additive",
"boundaries_functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_additive : (homology_functor V c i).additive | { map_add' := λ C D f g, begin
dsimp [homology_functor],
ext,
simp only [homology.π_map, preadditive.comp_add, ←preadditive.add_comp],
congr,
ext, simp,
end } | instance | homological_complex.homology_additive | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"additive",
"homology.π_map",
"homology_functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.map_homological_complex (F : V ⥤ W) [F.additive] (c : complex_shape ι) :
homological_complex V c ⥤ homological_complex W c | { obj := λ C,
{ X := λ i, F.obj (C.X i),
d := λ i j, F.map (C.d i j),
shape' := λ i j w, by rw [C.shape _ _ w, F.map_zero],
d_comp_d' := λ i j k _ _, by rw [←F.map_comp, C.d_comp_d, F.map_zero], },
map := λ C D f,
{ f := λ i, F.map (f.f i),
comm' := λ i j h, by { dsimp, rw [←F.map_comp, ←F.map_co... | def | category_theory.functor.map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape",
"homological_complex"
] | An additive functor induces a functor between homological complexes.
This is sometimes called the "prolongation". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.map_homological_complex_id_iso (c : complex_shape ι) :
(𝟭 V).map_homological_complex c ≅ 𝟭 _ | nat_iso.of_components (λ K, hom.iso_of_components (λ i, iso.refl _) (by tidy)) (by tidy) | def | category_theory.functor.map_homological_complex_id_iso | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | The functor on homological complexes induced by the identity functor is
isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.map_homogical_complex_additive
(F : V ⥤ W) [F.additive] (c : complex_shape ι) : (F.map_homological_complex c).additive | {} | instance | category_theory.functor.map_homogical_complex_additive | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"additive",
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.map_homological_complex_reflects_iso
(F : V ⥤ W) [F.additive] [reflects_isomorphisms F] (c : complex_shape ι) :
reflects_isomorphisms (F.map_homological_complex c) | ⟨λ X Y f, begin
introI,
haveI : ∀ (n : ι), is_iso (F.map (f.f n)) := λ n, is_iso.of_iso
((homological_complex.eval W c n).map_iso (as_iso ((F.map_homological_complex c).map f))),
haveI := λ n, is_iso_of_reflects_iso (f.f n) F,
exact homological_complex.hom.is_iso_of_components f,
end⟩ | instance | category_theory.functor.map_homological_complex_reflects_iso | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape",
"homological_complex.eval",
"homological_complex.hom.is_iso_of_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.map_homological_complex {F G : V ⥤ W} [F.additive] [G.additive]
(α : F ⟶ G) (c : complex_shape ι) : F.map_homological_complex c ⟶ G.map_homological_complex c | { app := λ C, { f := λ i, α.app _, }, } | def | category_theory.nat_trans.map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | A natural transformation between functors induces a natural transformation
between those functors applied to homological complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.map_homological_complex_id (c : complex_shape ι) (F : V ⥤ W) [F.additive] :
nat_trans.map_homological_complex (𝟙 F) c = 𝟙 (F.map_homological_complex c) | by tidy | lemma | category_theory.nat_trans.map_homological_complex_id | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.map_homological_complex_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_homological_complex (α ≫ β) c =
nat_trans.map_homological_complex α c ≫ nat_trans.map_homological_complex β c | by tidy | lemma | category_theory.nat_trans.map_homological_complex_comp | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.map_homological_complex_naturality {c : complex_shape ι}
{F G : V ⥤ W} [F.additive] [G.additive] (α : F ⟶ G) {C D : homological_complex V c} (f : C ⟶ D) :
(F.map_homological_complex c).map f ≫ (nat_trans.map_homological_complex α c).app D =
(nat_trans.map_homological_complex α c).app C ≫ (G.map_homolo... | by tidy | lemma | category_theory.nat_trans.map_homological_complex_naturality | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape",
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso.map_homological_complex {F G : V ⥤ W} [F.additive] [G.additive]
(α : F ≅ G) (c : complex_shape ι) : F.map_homological_complex c ≅ G.map_homological_complex c | { hom := α.hom.map_homological_complex c,
inv := α.inv.map_homological_complex c,
hom_inv_id' := by simpa only [← nat_trans.map_homological_complex_comp, α.hom_inv_id],
inv_hom_id' := by simpa only [← nat_trans.map_homological_complex_comp, α.inv_hom_id], } | def | category_theory.nat_iso.map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | A natural isomorphism between functors induces a natural isomorphism
between those functors applied to homological complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence.map_homological_complex (e : V ≌ W) [e.functor.additive] (c : complex_shape ι):
homological_complex V c ≌ homological_complex W c | { functor := e.functor.map_homological_complex c,
inverse := e.inverse.map_homological_complex c,
unit_iso := (functor.map_homological_complex_id_iso V c).symm ≪≫
nat_iso.map_homological_complex e.unit_iso c,
counit_iso := nat_iso.map_homological_complex e.counit_iso c ≪≫
functor.map_homological_complex_i... | def | category_theory.equivalence.map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape",
"homological_complex"
] | An equivalence of categories induces an equivalences between the respective categories
of homological complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_chain_complex_of (F : V ⥤ W) [F.additive] (X : α → V) (d : Π n, X (n+1) ⟶ X n)
(sq : ∀ n, d (n+1) ≫ d n = 0) :
(F.map_homological_complex _).obj (chain_complex.of X d sq) =
chain_complex.of (λ n, F.obj (X n))
(λ n, F.map (d n)) (λ n, by rw [ ← F.map_comp, sq n, functor.map_zero]) | begin
refine homological_complex.ext rfl _,
rintro i j (rfl : j + 1 = i),
simp only [category_theory.functor.map_homological_complex_obj_d, of_d,
eq_to_hom_refl, comp_id, id_comp],
end | lemma | chain_complex.map_chain_complex_of | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"chain_complex.of",
"homological_complex.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_map_homological_complex (F : V ⥤ W) [F.additive] (c : complex_shape ι) (j : ι):
single V c j ⋙ F.map_homological_complex _ ≅ F ⋙ single W c j | nat_iso.of_components (λ X,
{ hom := { f := λ i, if h : i = j then
eq_to_hom (by simp [h])
else
0, },
inv := { f := λ i, if h : i = j then
eq_to_hom (by simp [h])
else
0, },
hom_inv_id' := begin
ext i,
dsimp,
split_ifs with h,
{ simp [h] },
{ rw [zero_comp, if_neg h],
e... | def | homological_complex.single_map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [
"complex_shape"
] | Turning an object into a complex supported at `j` then applying a functor is
the same as applying the functor then forming the complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_map_homological_complex_hom_app_self (j : ι) (X : V) :
((single_map_homological_complex F c j).hom.app X).f j = eq_to_hom (by simp) | by simp [single_map_homological_complex] | lemma | homological_complex.single_map_homological_complex_hom_app_self | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_map_homological_complex_hom_app_ne
{i j : ι} (h : i ≠ j) (X : V) :
((single_map_homological_complex F c j).hom.app X).f i = 0 | by simp [single_map_homological_complex, h] | lemma | homological_complex.single_map_homological_complex_hom_app_ne | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_map_homological_complex_inv_app_self (j : ι) (X : V) :
((single_map_homological_complex F c j).inv.app X).f j = eq_to_hom (by simp) | by simp [single_map_homological_complex] | lemma | homological_complex.single_map_homological_complex_inv_app_self | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_map_homological_complex_inv_app_ne
{i j : ι} (h : i ≠ j) (X : V):
((single_map_homological_complex F c j).inv.app X).f i = 0 | by simp [single_map_homological_complex, h] | lemma | homological_complex.single_map_homological_complex_inv_app_ne | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_homological_complex (F : V ⥤ W) [F.additive] :
single₀ V ⋙ F.map_homological_complex _ ≅ F ⋙ single₀ W | nat_iso.of_components (λ X,
{ hom := { f := λ i, match i with
| 0 := 𝟙 _
| (i+1) := F.map_zero_object.hom
end, },
inv := { f := λ i, match i with
| 0 := 𝟙 _
| (i+1) := F.map_zero_object.inv
end, },
hom_inv_id' := begin
ext (_|i),
{ unfold_aux, simp, },
{ unfold_aux,
dsimp... | def | chain_complex.single₀_map_homological_complex | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | Turning an object into a chain complex supported at zero then applying a functor is
the same as applying the functor then forming the complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single₀_map_homological_complex_hom_app_zero (F : V ⥤ W) [F.additive] (X : V) :
((single₀_map_homological_complex F).hom.app X).f 0 = 𝟙 _ | rfl | lemma | chain_complex.single₀_map_homological_complex_hom_app_zero | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_homological_complex_hom_app_succ
(F : V ⥤ W) [F.additive] (X : V) (n : ℕ) :
((single₀_map_homological_complex F).hom.app X).f (n+1) = 0 | rfl | lemma | chain_complex.single₀_map_homological_complex_hom_app_succ | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_homological_complex_inv_app_zero (F : V ⥤ W) [F.additive] (X : V) :
((single₀_map_homological_complex F).inv.app X).f 0 = 𝟙 _ | rfl | lemma | chain_complex.single₀_map_homological_complex_inv_app_zero | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_map_homological_complex_inv_app_succ
(F : V ⥤ W) [F.additive] (X : V) (n : ℕ) :
((single₀_map_homological_complex F).inv.app X).f (n+1) = 0 | rfl | lemma | chain_complex.single₀_map_homological_complex_inv_app_succ | algebra.homology | src/algebra/homology/additive.lean | [
"algebra.homology.homology",
"algebra.homology.single",
"category_theory.preadditive.additive_functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate [has_zero_morphisms V] : chain_complex V ℕ ⥤ chain_complex V ℕ | { obj := λ C,
{ X := λ i, C.X (i+1),
d := λ i j, C.d (i+1) (j+1),
shape' := λ i j w, by { apply C.shape, simpa }, },
map := λ C D f,
{ f := λ i, f.f (i+1), }, } | def | chain_complex.truncate | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | The truncation of a `ℕ`-indexed chain complex,
deleting the object at `0` and shifting everything else down. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_to [has_zero_object V] [has_zero_morphisms V] (C : chain_complex V ℕ) :
truncate.obj C ⟶ (single₀ V).obj (C.X 0) | (to_single₀_equiv (truncate.obj C) (C.X 0)).symm ⟨C.d 1 0, by tidy⟩ | def | chain_complex.truncate_to | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | There is a canonical chain map from the truncation of a chain map `C` to
the "single object" chain complex consisting of the truncated object `C.X 0` in degree 0.
The components of this chain map are `C.d 1 0` in degree 0, and zero otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment (C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
chain_complex V ℕ | { X := λ i, match i with
| 0 := X
| (i+1) := C.X i
end,
d := λ i j, match i, j with
| 1, 0 := f
| (i+1), (j+1) := C.d i j
| _, _ := 0
end,
shape' := λ i j s, begin
simp at s,
rcases i with _|_|i; cases j; unfold_aux; try { simp },
{ simpa using s, },
{ rw [C.shape], simpa [← ne.def, na... | def | chain_complex.augment | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex",
"nat.succ_ne_succ"
] | We can "augment" a chain complex by inserting an arbitrary object in degree zero
(shifting everything else up), along with a suitable differential. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_X_zero (C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
(augment C f w).X 0 = X | rfl | lemma | chain_complex.augment_X_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_X_succ (C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
(i : ℕ) :
(augment C f w).X (i+1) = C.X i | rfl | lemma | chain_complex.augment_X_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_d_one_zero
(C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
(augment C f w).d 1 0 = f | rfl | lemma | chain_complex.augment_d_one_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_d_succ_succ
(C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i j : ℕ) :
(augment C f w).d (i+1) (j+1) = C.d i j | by { dsimp [augment], rcases i with _|i, refl, refl, } | lemma | chain_complex.augment_d_succ_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_augment (C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
truncate.obj (augment C f w) ≅ C | { hom :=
{ f := λ i, 𝟙 _, },
inv :=
{ f := λ i, by { exact 𝟙 _, },
comm' := λ i j, by { cases j; { dsimp, simp, }, }, },
hom_inv_id' := by { ext i, cases i; { dsimp, simp, }, },
inv_hom_id' := by { ext i, cases i; { dsimp, simp, }, }, }. | def | chain_complex.truncate_augment | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | Truncating an augmented chain complex is isomorphic (with components the identity)
to the original complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_augment_hom_f
(C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i : ℕ) :
(truncate_augment C f w).hom.f i = 𝟙 (C.X i) | rfl | lemma | chain_complex.truncate_augment_hom_f | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_augment_inv_f
(C : chain_complex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i : ℕ) :
(truncate_augment C f w).inv.f i = 𝟙 ((truncate.obj (augment C f w)).X i) | rfl | lemma | chain_complex.truncate_augment_inv_f | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
chain_complex_d_succ_succ_zero (C : chain_complex V ℕ) (i : ℕ) :
C.d (i+2) 0 = 0 | by { rw C.shape, simpa using i.succ_succ_ne_one.symm } | lemma | chain_complex.chain_complex_d_succ_succ_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate (C : chain_complex V ℕ) :
augment (truncate.obj C) (C.d 1 0) (C.d_comp_d _ _ _) ≅ C | { hom :=
{ f := λ i, by { cases i; exact 𝟙 _, },
comm' := λ i j, by { rcases i with _|_|i; cases j; { dsimp, simp, }, }, },
inv :=
{ f := λ i, by { cases i; exact 𝟙 _, },
comm' := λ i j, by { rcases i with _|_|i; cases j; { dsimp, simp, }, }, },
hom_inv_id' := by { ext i, cases i; { dsimp, simp, }, },... | def | chain_complex.augment_truncate | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_truncate_hom_f_zero (C : chain_complex V ℕ) :
(augment_truncate C).hom.f 0 = 𝟙 (C.X 0) | rfl | lemma | chain_complex.augment_truncate_hom_f_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_hom_f_succ (C : chain_complex V ℕ) (i : ℕ) :
(augment_truncate C).hom.f (i+1) = 𝟙 (C.X (i+1)) | rfl | lemma | chain_complex.augment_truncate_hom_f_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_inv_f_zero (C : chain_complex V ℕ) :
(augment_truncate C).inv.f 0 = 𝟙 (C.X 0) | rfl | lemma | chain_complex.augment_truncate_inv_f_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_inv_f_succ (C : chain_complex V ℕ) (i : ℕ) :
(augment_truncate C).inv.f (i+1) = 𝟙 (C.X (i+1)) | rfl | lemma | chain_complex.augment_truncate_inv_f_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_single₀_as_complex
[has_zero_object V] (C : chain_complex V ℕ) (X : V) (f : C ⟶ (single₀ V).obj X) :
chain_complex V ℕ | let ⟨f, w⟩ := to_single₀_equiv C X f in augment C f w | def | chain_complex.to_single₀_as_complex | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"chain_complex"
] | A chain map from a chain complex to a single object chain complex in degree zero
can be reinterpreted as a chain complex.
Ths is the inverse construction of `truncate_to`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate [has_zero_morphisms V] : cochain_complex V ℕ ⥤ cochain_complex V ℕ | { obj := λ C,
{ X := λ i, C.X (i+1),
d := λ i j, C.d (i+1) (j+1),
shape' := λ i j w, by { apply C.shape, simpa }, },
map := λ C D f,
{ f := λ i, f.f (i+1), }, } | def | cochain_complex.truncate | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | The truncation of a `ℕ`-indexed cochain complex,
deleting the object at `0` and shifting everything else down. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_truncate [has_zero_object V] [has_zero_morphisms V] (C : cochain_complex V ℕ) :
(single₀ V).obj (C.X 0) ⟶ truncate.obj C | (from_single₀_equiv (truncate.obj C) (C.X 0)).symm ⟨C.d 0 1, by tidy⟩ | def | cochain_complex.to_truncate | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | There is a canonical chain map from the truncation of a cochain complex `C` to
the "single object" cochain complex consisting of the truncated object `C.X 0` in degree 0.
The components of this chain map are `C.d 0 1` in degree 0, and zero otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment (C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
cochain_complex V ℕ | { X := λ i, match i with
| 0 := X
| (i+1) := C.X i
end,
d := λ i j, match i, j with
| 0, 1 := f
| (i+1), (j+1) := C.d i j
| _, _ := 0
end,
shape' := λ i j s, begin
simp at s,
rcases j with _|_|j; cases i; unfold_aux; try { simp },
{ simpa using s, },
{ rw [C.shape], simp only [complex_... | def | cochain_complex.augment | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex",
"nat.one_lt_succ_succ"
] | We can "augment" a cochain complex by inserting an arbitrary object in degree zero
(shifting everything else up), along with a suitable differential. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_X_zero
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
(augment C f w).X 0 = X | rfl | lemma | cochain_complex.augment_X_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_X_succ
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ) :
(augment C f w).X (i+1) = C.X i | rfl | lemma | cochain_complex.augment_X_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_d_zero_one
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
(augment C f w).d 0 1 = f | rfl | lemma | cochain_complex.augment_d_zero_one | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_d_succ_succ
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i j : ℕ) :
(augment C f w).d (i+1) (j+1) = C.d i j | rfl | lemma | cochain_complex.augment_d_succ_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_augment (C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
truncate.obj (augment C f w) ≅ C | { hom :=
{ f := λ i, 𝟙 _, },
inv :=
{ f := λ i, by { exact 𝟙 _, },
comm' := λ i j, by { cases j; { dsimp, simp, }, }, },
hom_inv_id' := by { ext i, cases i; { dsimp, simp, }, },
inv_hom_id' := by { ext i, cases i; { dsimp, simp, }, }, }. | def | cochain_complex.truncate_augment | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | Truncating an augmented cochain complex is isomorphic (with components the identity)
to the original complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_augment_hom_f
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ) :
(truncate_augment C f w).hom.f i = 𝟙 (C.X i) | rfl | lemma | cochain_complex.truncate_augment_hom_f | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_augment_inv_f
(C : cochain_complex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ) :
(truncate_augment C f w).inv.f i = 𝟙 ((truncate.obj (augment C f w)).X i) | rfl | lemma | cochain_complex.truncate_augment_inv_f | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cochain_complex_d_succ_succ_zero (C : cochain_complex V ℕ) (i : ℕ) :
C.d 0 (i+2) = 0 | by { rw C.shape, simp only [complex_shape.up_rel, zero_add], exact (nat.one_lt_succ_succ _).ne } | lemma | cochain_complex.cochain_complex_d_succ_succ_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex",
"nat.one_lt_succ_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate (C : cochain_complex V ℕ) :
augment (truncate.obj C) (C.d 0 1) (C.d_comp_d _ _ _) ≅ C | { hom :=
{ f := λ i, by { cases i; exact 𝟙 _, },
comm' := λ i j, by { rcases j with _|_|j; cases i; { dsimp, simp, }, }, },
inv :=
{ f := λ i, by { cases i; exact 𝟙 _, },
comm' := λ i j, by { rcases j with _|_|j; cases i; { dsimp, simp, }, }, },
hom_inv_id' := by { ext i, cases i; { dsimp, simp, }, },... | def | cochain_complex.augment_truncate | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | Augmenting a truncated complex with the original object and morphism is isomorphic
(with components the identity) to the original complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_truncate_hom_f_zero (C : cochain_complex V ℕ) :
(augment_truncate C).hom.f 0 = 𝟙 (C.X 0) | rfl | lemma | cochain_complex.augment_truncate_hom_f_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_hom_f_succ (C : cochain_complex V ℕ) (i : ℕ) :
(augment_truncate C).hom.f (i+1) = 𝟙 (C.X (i+1)) | rfl | lemma | cochain_complex.augment_truncate_hom_f_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_inv_f_zero (C : cochain_complex V ℕ) :
(augment_truncate C).inv.f 0 = 𝟙 (C.X 0) | rfl | lemma | cochain_complex.augment_truncate_inv_f_zero | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
augment_truncate_inv_f_succ (C : cochain_complex V ℕ) (i : ℕ) :
(augment_truncate C).inv.f (i+1) = 𝟙 (C.X (i+1)) | rfl | lemma | cochain_complex.augment_truncate_inv_f_succ | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_as_complex
[has_zero_object V] (C : cochain_complex V ℕ) (X : V) (f : (single₀ V).obj X ⟶ C) :
cochain_complex V ℕ | let ⟨f, w⟩ := from_single₀_equiv C X f in augment C f w | def | cochain_complex.from_single₀_as_complex | algebra.homology | src/algebra/homology/augment.lean | [
"algebra.homology.single"
] | [
"cochain_complex"
] | A chain map from a single object cochain complex in degree zero to a cochain complex
can be reinterpreted as a cochain complex.
Ths is the inverse construction of `to_truncate`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.