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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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eq_id_iff_eq : A.eq_id ↔ A.1 = Δ | begin
split,
{ intro h,
dsimp at h,
rw h,
refl, },
{ intro h,
rcases A with ⟨Δ', ⟨f, hf⟩⟩,
simp only at h,
subst h,
refine ext _ _ rfl _,
{ haveI := hf,
simp only [eq_to_hom_refl, comp_id],
exact eq_id_of_epi f, }, },
end | lemma | simplicial_object.splitting.index_set.eq_id_iff_eq | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_iff_len_eq : A.eq_id ↔ A.1.unop.len = Δ.unop.len | begin
rw eq_id_iff_eq,
split,
{ intro h,
rw h, },
{ intro h,
rw ← unop_inj_iff,
ext,
exact h, },
end | lemma | simplicial_object.splitting.index_set.eq_id_iff_len_eq | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_iff_len_le : A.eq_id ↔ Δ.unop.len ≤ A.1.unop.len | begin
rw eq_id_iff_len_eq,
split,
{ intro h,
rw h, },
{ exact le_antisymm (len_le_of_epi (infer_instance : epi A.e)), },
end | lemma | simplicial_object.splitting.index_set.eq_id_iff_len_le | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_id_iff_mono : A.eq_id ↔ mono A.e | begin
split,
{ intro h,
dsimp at h,
subst h,
dsimp only [id, e],
apply_instance, },
{ intro h,
rw eq_id_iff_len_le,
exact len_le_of_mono h, }
end | lemma | simplicial_object.splitting.index_set.eq_id_iff_mono | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_comp {Δ₁ Δ₂ : simplex_categoryᵒᵖ} (A : index_set Δ₁) (p : Δ₁ ⟶ Δ₂) [epi p.unop] :
index_set Δ₂ | ⟨A.1, ⟨p.unop ≫ A.e, epi_comp _ _⟩⟩ | def | simplicial_object.splitting.index_set.epi_comp | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | Given `A : index_set Δ₁`, if `p.unop : unop Δ₂ ⟶ unop Δ₁` is an epi, this
is the obvious element in `A : index_set Δ₂` associated to the composition
of epimorphisms `p.unop ≫ A.e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pull : index_set Δ' | mk (factor_thru_image (θ.unop ≫ A.e)) | def | simplicial_object.splitting.index_set.pull | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | When `A : index_set Δ` and `θ : Δ → Δ'` is a morphism in `simplex_categoryᵒᵖ`,
an element in `index_set Δ'` can be defined by using the epi-mono factorisation
of `θ.unop ≫ A.e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fac_pull : (A.pull θ).e ≫ image.ι (θ.unop ≫ A.e) = θ.unop ≫ A.e | image.fac _ | lemma | simplicial_object.splitting.index_set.fac_pull | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summand (A : index_set Δ) : C | N A.1.unop.len | def | simplicial_object.splitting.summand | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | Given a sequences of objects `N : ℕ → C` in a category `C`, this is
a family of objects indexed by the elements `A : splitting.index_set Δ`.
The `Δ`-simplices of a split simplicial objects shall identify to the
coproduct of objects in such a family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod | ∐ summand N Δ | def | simplicial_object.splitting.coprod | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The coproduct of the family `summand N Δ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_coprod (A : index_set Δ) : N A.1.unop.len ⟶ coprod N Δ | sigma.ι _ A | def | simplicial_object.splitting.ι_coprod | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The inclusion of a summand in the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (Δ : simplex_categoryᵒᵖ) : coprod N Δ ⟶ X.obj Δ | sigma.desc (λ A, φ A.1.unop.len ≫ X.map A.e.op) | def | simplicial_object.splitting.map | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The canonical morphism `coprod N Δ ⟶ X.obj Δ` attached to a sequence
of objects `N` and a sequence of morphisms `N n ⟶ X _[n]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
splitting (X : simplicial_object C) | (N : ℕ → C)
(ι : Π n, N n ⟶ X _[n])
(map_is_iso' : ∀ (Δ : simplex_categoryᵒᵖ), is_iso (splitting.map X ι Δ)) | structure | simplicial_object.splitting | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | A splitting of a simplicial object `X` consists of the datum of a sequence
of objects `N`, a sequence of morphisms `ι : N n ⟶ X _[n]` such that
for all `Δ : simplex_categoryhᵒᵖ`, the canonical map `splitting.map X ι Δ`
is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_is_iso (Δ : simplex_categoryᵒᵖ) : is_iso (splitting.map X s.ι Δ) | s.map_is_iso' Δ | instance | simplicial_object.splitting.map_is_iso | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso (Δ : simplex_categoryᵒᵖ) : coprod s.N Δ ≅ X.obj Δ | as_iso (splitting.map X s.ι Δ) | def | simplicial_object.splitting.iso | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [
"iso"
] | The isomorphism on simplices given by the axiom `splitting.map_is_iso'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_summand {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :
s.N A.1.unop.len ⟶ X.obj Δ | splitting.ι_coprod s.N A ≫ (s.iso Δ).hom | def | simplicial_object.splitting.ι_summand | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | Via the isomorphism `s.iso Δ`, this is the inclusion of a summand
in the direct sum decomposition given by the splitting `s : splitting X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_summand_eq {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :
s.ι_summand A = s.ι A.1.unop.len ≫ X.map A.e.op | begin
dsimp only [ι_summand, iso.hom],
erw [colimit.ι_desc, cofan.mk_ι_app],
end | lemma | simplicial_object.splitting.ι_summand_eq | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_summand_id (n : ℕ) : s.ι_summand (index_set.id (op [n])) = s.ι n | by { erw [ι_summand_eq, X.map_id, comp_id], refl, } | lemma | simplicial_object.splitting.ι_summand_id | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
φ (f : X ⟶ Y) (n : ℕ) : s.N n ⟶ Y _[n] | s.ι n ≫ f.app (op [n]) | def | simplicial_object.splitting.φ | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | As it is stated in `splitting.hom_ext`, a morphism `f : X ⟶ Y` from a split
simplicial object to any simplicial object is determined by its restrictions
`s.φ f n : s.N n ⟶ Y _[n]` to the distinguished summands in each degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_summand_comp_app (f : X ⟶ Y) {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :
s.ι_summand A ≫ f.app Δ = s.φ f A.1.unop.len ≫ Y.map A.e.op | by simp only [ι_summand_eq_assoc, φ, nat_trans.naturality, assoc] | lemma | simplicial_object.splitting.ι_summand_comp_app | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext' {Z : C} {Δ : simplex_categoryᵒᵖ} (f g : X.obj Δ ⟶ Z)
(h : ∀ (A : index_set Δ), s.ι_summand A ≫ f = s.ι_summand A ≫ g) :
f = g | begin
rw ← cancel_epi (s.iso Δ).hom,
ext A,
discrete_cases,
simpa only [ι_summand_eq, iso_hom, colimit.ι_desc_assoc, cofan.mk_ι_app, assoc] using h A,
end | lemma | simplicial_object.splitting.hom_ext' | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext (f g : X ⟶ Y) (h : ∀ n : ℕ, s.φ f n = s.φ g n) : f = g | begin
ext Δ,
apply s.hom_ext',
intro A,
induction Δ using opposite.rec,
induction Δ using simplex_category.rec with n,
dsimp,
simp only [s.ι_summand_comp_app, h],
end | lemma | simplicial_object.splitting.hom_ext | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [
"hom_ext",
"opposite.rec",
"simplex_category.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc {Z : C} (Δ : simplex_categoryᵒᵖ)
(F : Π (A : index_set Δ), s.N A.1.unop.len ⟶ Z) : X.obj Δ ⟶ Z | (s.iso Δ).inv ≫ sigma.desc F | def | simplicial_object.splitting.desc | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The map `X.obj Δ ⟶ Z` obtained by providing a family of morphisms on all the
terms of decomposition given by a splitting `s : splitting X` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_desc {Z : C} (Δ : simplex_categoryᵒᵖ)
(F : Π (A : index_set Δ), s.N A.1.unop.len ⟶ Z) (A : index_set Δ) :
s.ι_summand A ≫ s.desc Δ F = F A | begin
dsimp only [ι_summand, desc],
simp only [assoc, iso.hom_inv_id_assoc, ι_coprod],
erw [colimit.ι_desc, cofan.mk_ι_app],
end | lemma | simplicial_object.splitting.ι_desc | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso (e : X ≅ Y) : splitting Y | { N := s.N,
ι := λ n, s.ι n ≫ e.hom.app (op [n]),
map_is_iso' := λ Δ, begin
convert (infer_instance : is_iso ((s.iso Δ).hom ≫ e.hom.app Δ)),
tidy,
end, } | def | simplicial_object.splitting.of_iso | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | A simplicial object that is isomorphic to a split simplicial object is split. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_summand_epi_naturality {Δ₁ Δ₂ : simplex_categoryᵒᵖ} (A : index_set Δ₁)
(p : Δ₁ ⟶ Δ₂) [epi p.unop] :
s.ι_summand A ≫ X.map p = s.ι_summand (A.epi_comp p) | begin
dsimp [ι_summand],
erw [colimit.ι_desc, colimit.ι_desc, cofan.mk_ι_app, cofan.mk_ι_app],
dsimp only [index_set.epi_comp, index_set.e],
rw [op_comp, X.map_comp, assoc, quiver.hom.op_unop],
end | lemma | simplicial_object.splitting.ι_summand_epi_naturality | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split | (X : simplicial_object C) (s : splitting X) | structure | simplicial_object.split | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The category `simplicial_object.split C` is the category of simplicial objects
in `C` equipped with a splitting, and morphisms are morphisms of simplicial objects
which are compatible with the splittings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' {X : simplicial_object C} (s : splitting X) : split C | ⟨X, s⟩ | def | simplicial_object.split.mk' | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [
"mk'"
] | The object in `simplicial_object.split C` attached to a splitting `s : splitting X`
of a simplicial object `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (S₁ S₂ : split C) | (F : S₁.X ⟶ S₂.X)
(f : Π (n : ℕ), S₁.s.N n ⟶ S₂.s.N n)
(comm' : ∀ (n : ℕ), S₁.s.ι n ≫ F.app (op [n]) = f n ≫ S₂.s.ι n) | structure | simplicial_object.split.hom | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | Morphisms in `simplicial_object.split C` are morphisms of simplicial objects that
are compatible with the splittings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.ext {S₁ S₂ : split C} (Φ₁ Φ₂ : hom S₁ S₂) (h : ∀ (n : ℕ), Φ₁.f n = Φ₂.f n) :
Φ₁ = Φ₂ | begin
rcases Φ₁ with ⟨F₁, f₁, c₁⟩,
rcases Φ₂ with ⟨F₂, f₂, c₂⟩,
have h' : f₁ = f₂ := by { ext, apply h, },
subst h',
simp only [eq_self_iff_true, and_true],
apply S₁.s.hom_ext,
intro n,
dsimp,
rw [c₁, c₂],
end | lemma | simplicial_object.split.hom.ext | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_F {S₁ S₂ : split C} {Φ₁ Φ₂ : S₁ ⟶ S₂} (h : Φ₁ = Φ₂) : Φ₁.F = Φ₂.F | by rw h | lemma | simplicial_object.split.congr_F | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_f {S₁ S₂ : split C} {Φ₁ Φ₂ : S₁ ⟶ S₂} (h : Φ₁ = Φ₂) (n : ℕ) :
Φ₁.f n = Φ₂.f n | by rw h | lemma | simplicial_object.split.congr_f | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_F (S : split C) : (𝟙 S : S ⟶ S).F = 𝟙 (S.X) | rfl | lemma | simplicial_object.split.id_F | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_f (S : split C) (n : ℕ) : (𝟙 S : S ⟶ S).f n = 𝟙 (S.s.N n) | rfl | lemma | simplicial_object.split.id_f | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_F {S₁ S₂ S₃ : split C} (Φ₁₂ : S₁ ⟶ S₂) (Φ₂₃ : S₂ ⟶ S₃) :
(Φ₁₂ ≫ Φ₂₃).F = Φ₁₂.F ≫ Φ₂₃.F | rfl | lemma | simplicial_object.split.comp_F | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_f {S₁ S₂ S₃ : split C} (Φ₁₂ : S₁ ⟶ S₂) (Φ₂₃ : S₂ ⟶ S₃) (n : ℕ) :
(Φ₁₂ ≫ Φ₂₃).f n = Φ₁₂.f n ≫ Φ₂₃.f n | rfl | lemma | simplicial_object.split.comp_f | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_summand_naturality_symm {S₁ S₂ : split C} (Φ : S₁ ⟶ S₂)
{Δ : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) :
S₁.s.ι_summand A ≫ Φ.F.app Δ = Φ.f A.1.unop.len ≫ S₂.s.ι_summand A | by rw [S₁.s.ι_summand_eq, S₂.s.ι_summand_eq, assoc, Φ.F.naturality, ← Φ.comm_assoc] | lemma | simplicial_object.split.ι_summand_naturality_symm | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : split C ⥤ simplicial_object C | { obj := λ S, S.X,
map := λ S₁ S₂ Φ, Φ.F, } | def | simplicial_object.split.forget | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The functor `simplicial_object.split C ⥤ simplicial_object C` which forgets
the splitting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_N (n : ℕ) : split C ⥤ C | { obj := λ S, S.s.N n,
map := λ S₁ S₂ Φ, Φ.f n, } | def | simplicial_object.split.eval_N | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The functor `simplicial_object.split C ⥤ C` which sends a simplicial object equipped
with a splitting to its nondegenerate `n`-simplices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_ι_summand {Δ : simplex_categoryᵒᵖ} (A : splitting.index_set Δ) :
eval_N C A.1.unop.len ⟶ forget C ⋙ (evaluation simplex_categoryᵒᵖ C).obj Δ | { app := λ S, S.s.ι_summand A,
naturality' := λ S₁ S₂ Φ, (ι_summand_naturality_symm Φ A).symm, } | def | simplicial_object.split.nat_trans_ι_summand | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The inclusion of each summand in the coproduct decomposition of simplices
in split simplicial objects is a natural transformation of functors
`simplicial_object.split C ⥤ C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Top_obj (x : simplex_category) | { f : x → ℝ≥0 | ∑ i, f i = 1 } | def | simplex_category.to_Top_obj | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"simplex_category"
] | The topological simplex associated to `x : simplex_category`.
This is the object part of the functor `simplex_category.to_Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Top_obj.ext {x : simplex_category} (f g : x.to_Top_obj) :
(f : x → ℝ≥0) = g → f = g | subtype.ext | lemma | simplex_category.to_Top_obj.ext | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"simplex_category",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Top_map {x y : simplex_category} (f : x ⟶ y) : x.to_Top_obj → y.to_Top_obj | λ g, ⟨λ i, ∑ j in (finset.univ.filter (λ k, f k = i)), g j,
begin
simp only [finset.filter_congr_decidable, finset.sum_congr, to_Top_obj, set.mem_set_of],
rw ← finset.sum_bUnion,
convert g.2,
{ rw finset.eq_univ_iff_forall,
intros i,
rw finset.mem_bUnion,
exact ⟨f i, by simp, by simp⟩ },
{ intros ... | def | simplex_category.to_Top_map | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"disjoint_iff_inf_le",
"finset.eq_univ_iff_forall",
"finset.filter_congr_decidable",
"finset.inf_eq_inter",
"finset.mem_bUnion",
"finset.mem_filter",
"finset.mem_inter",
"finset.mem_univ",
"set.mem_set_of",
"simplex_category"
] | A morphism in `simplex_category` induces a map on the associated topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_Top_map {x y : simplex_category} (f : x ⟶ y) (g : x.to_Top_obj) (i : y) :
to_Top_map f g i = ∑ j in (finset.univ.filter (λ k, f k = i)), g j | rfl | lemma | simplex_category.coe_to_Top_map | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_Top_map {x y : simplex_category} (f : x ⟶ y) :
continuous (to_Top_map f) | continuous.subtype_mk (continuous_pi $ λ i, continuous_finset_sum _ $
λ j hj, (continuous_apply _).comp continuous_subtype_val) _ | lemma | simplex_category.continuous_to_Top_map | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"continuous",
"continuous.subtype_mk",
"continuous_apply",
"continuous_pi",
"continuous_subtype_val",
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Top : simplex_category ⥤ Top | { obj := λ x, Top.of x.to_Top_obj,
map := λ x y f, ⟨to_Top_map f⟩,
map_id' := begin
intros x,
ext f i : 3,
change (finset.univ.filter (λ k, k = i)).sum _ = _,
simp [finset.sum_filter]
end,
map_comp' := begin
intros x y z f g,
ext h i : 3,
dsimp,
erw ← finset.sum_bUnion,
apply... | def | simplex_category.to_Top | algebraic_topology | src/algebraic_topology/topological_simplex.lean | [
"algebraic_topology.simplex_category",
"topology.category.Top.basic",
"topology.instances.nnreal"
] | [
"Top",
"Top.of",
"disjoint_iff_inf_le",
"finset.ext",
"finset.inf_eq_inter",
"finset.mem_filter",
"finset.mem_inter",
"finset.mem_univ",
"simplex_category"
] | The functor associating the topological `n`-simplex to `[n] : simplex_category`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₀ : A ≌ B' | eA.trans e' | def | algebraic_topology.dold_kan.compatibility.equivalence₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | A basic equivalence `A ≅ B'` obtained by composing `eA : A ≅ A'` and `e' : A' ≅ B'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₁ : A ≌ B' | begin
letI : is_equivalence F :=
is_equivalence.of_iso hF (is_equivalence.of_equivalence (equivalence₀ eA e')),
exact F.as_equivalence,
end | def | algebraic_topology.dold_kan.compatibility.equivalence₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | An intermediate equivalence `A ≅ B'` whose functor is `F` and whose inverse is
`e'.inverse ⋙ eA.inverse`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse | rfl | lemma | algebraic_topology.dold_kan.compatibility.equivalence₁_inverse | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence₁_counit_iso :
(e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' | calc (e'.inverse ⋙ eA.inverse) ⋙ F
≅ (e'.inverse ⋙ eA.inverse) ⋙ (eA.functor ⋙ e'.functor) : iso_whisker_left _ hF.symm
... ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor) ⋙ e'.functor : iso.refl _
... ≅ e'.inverse ⋙ 𝟭 _ ⋙ e'.functor : iso_whisker_left _ (iso_whisker_right eA.counit_iso _)
... ≅ e'.inverse ⋙ e'.functor :... | def | algebraic_topology.dold_kan.compatibility.equivalence₁_counit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The counit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₁_counit_iso_eq : (equivalence₁ hF).counit_iso = equivalence₁_counit_iso hF | begin
ext Y,
dsimp [equivalence₀, equivalence₁, is_equivalence.inverse, is_equivalence.of_equivalence],
simp only [equivalence₁_counit_iso_hom_app, category_theory.functor.map_id, comp_id],
end | lemma | algebraic_topology.dold_kan.compatibility.equivalence₁_counit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence₁_unit_iso :
𝟭 A ≅ F ⋙ (e'.inverse ⋙ eA.inverse) | calc 𝟭 A ≅ eA.functor ⋙ eA.inverse : eA.unit_iso
... ≅ eA.functor ⋙ 𝟭 A' ⋙ eA.inverse : iso.refl _
... ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse) ⋙ eA.inverse :
iso_whisker_left _ (iso_whisker_right e'.unit_iso _)
... ≅ (eA.functor ⋙ e'.functor) ⋙ (e'.inverse ⋙ eA.inverse) : iso.refl _
... ≅ F ⋙ (e'.inverse ⋙ e... | def | algebraic_topology.dold_kan.compatibility.equivalence₁_unit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The unit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₁_unit_iso_eq : (equivalence₁ hF).unit_iso = equivalence₁_unit_iso hF | begin
ext X,
dsimp [equivalence₀, equivalence₁, nat_iso.hcomp,
is_equivalence.of_equivalence],
simp only [id_comp, assoc, equivalence₁_unit_iso_hom_app],
end | lemma | algebraic_topology.dold_kan.compatibility.equivalence₁_unit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence₂ : A ≌ B | (equivalence₁ hF).trans eB.symm | def | algebraic_topology.dold_kan.compatibility.equivalence₂ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | An intermediate equivalence `A ≅ B` obtained as the composition of `equivalence₁` and
the inverse of `eB : B ≌ B'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₂_inverse : (equivalence₂ eB hF).inverse =
eB.functor ⋙ e'.inverse ⋙ eA.inverse | rfl | lemma | algebraic_topology.dold_kan.compatibility.equivalence₂_inverse | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence₂_counit_iso :
(eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ (F ⋙ eB.inverse) ≅ 𝟭 B | calc (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ (F ⋙ eB.inverse)
≅ eB.functor ⋙ (e'.inverse ⋙ eA.inverse ⋙ F) ⋙ eB.inverse : iso.refl _
... ≅ eB.functor ⋙ 𝟭 _ ⋙ eB.inverse :
iso_whisker_left _ (iso_whisker_right (equivalence₁_counit_iso hF) _)
... ≅ eB.functor ⋙ eB.inverse : iso.refl _
... ≅ 𝟭 B : eB.unit_i... | def | algebraic_topology.dold_kan.compatibility.equivalence₂_counit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The counit isomorphism of the equivalence `equivalence₂` between `A` and `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₂_counit_iso_eq :
(equivalence₂ eB hF).counit_iso = equivalence₂_counit_iso eB hF | begin
ext Y',
dsimp [equivalence₂, iso.refl],
simp only [equivalence₁_counit_iso_eq, equivalence₂_counit_iso_hom_app,
equivalence₁_counit_iso_hom_app, functor.map_comp, assoc],
end | lemma | algebraic_topology.dold_kan.compatibility.equivalence₂_counit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence₂_unit_iso :
𝟭 A ≅ (F ⋙ eB.inverse) ⋙ (eB.functor ⋙ e'.inverse ⋙ eA.inverse) | calc 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse : equivalence₁_unit_iso hF
... ≅ F ⋙ 𝟭 B' ⋙ (e'.inverse ⋙ eA.inverse) : iso.refl _
... ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :
iso_whisker_left _ (iso_whisker_right eB.counit_iso.symm _)
... ≅ (F ⋙ eB.inverse) ⋙ (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ... | def | algebraic_topology.dold_kan.compatibility.equivalence₂_unit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The unit isomorphism of the equivalence `equivalence₂` between `A` and `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence₂_unit_iso_eq :
(equivalence₂ eB hF).unit_iso = equivalence₂_unit_iso eB hF | begin
ext X,
dsimp [equivalence₂],
simpa only [equivalence₂_unit_iso_hom_app, equivalence₁_unit_iso_eq,
equivalence₁_unit_iso_hom_app, assoc, nat_iso.cancel_nat_iso_hom_left],
end | lemma | algebraic_topology.dold_kan.compatibility.equivalence₂_unit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : A ≌ B | begin
letI : is_equivalence G := begin
refine is_equivalence.of_iso _ (is_equivalence.of_equivalence (equivalence₂ eB hF).symm),
calc eB.functor ⋙ e'.inverse ⋙ eA.inverse
≅ (eB.functor ⋙ e'.inverse) ⋙ eA.inverse : iso.refl _
... ≅ (G ⋙ eA.functor) ⋙ eA.inverse : iso_whisker_right hG _
... ≅ G ... | def | algebraic_topology.dold_kan.compatibility.equivalence | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The equivalence `A ≅ B` whose functor is `F ⋙ eB.inverse` and
whose inverse is `G : B ≅ A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_functor : (equivalence hF hG).functor = F ⋙ eB.inverse | rfl | lemma | algebraic_topology.dold_kan.compatibility.equivalence_functor | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
τ₀ : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor | calc eB.functor ⋙ e'.inverse ⋙ e'.functor
≅ eB.functor ⋙ 𝟭 _ : iso_whisker_left _ e'.counit_iso
... ≅ eB.functor : functor.right_unitor _ | def | algebraic_topology.dold_kan.compatibility.τ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the counit isomorphism of `e'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
τ₁ (η : G ⋙ F ≅ eB.functor) :
eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor | calc eB.functor ⋙ e'.inverse ⋙ e'.functor
≅ (eB.functor ⋙ e'.inverse) ⋙ e'.functor : iso.refl _
... ≅ (G ⋙ eA.functor) ⋙ e'.functor : iso_whisker_right hG _
... ≅ G ⋙ (eA.functor ⋙ e'.functor) : by refl
... ≅ G ⋙ F : iso_whisker_left _ hF
... ≅ eB.functor : η | def | algebraic_topology.dold_kan.compatibility.τ₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the isomorphisms `hF : eA.functor ⋙ e'.functor ≅ F`,
`hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor` and the datum of
an isomorphism `η : G ⋙ F ≅ eB.functor`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_counit_iso : G ⋙ (F ⋙ eB.inverse) ≅ 𝟭 B | calc G ⋙ (F ⋙ eB.inverse) ≅ (G ⋙ F) ⋙ eB.inverse : iso.refl _
... ≅ eB.functor ⋙ eB.inverse : iso_whisker_right η _
... ≅ 𝟭 B : eB.unit_iso.symm | def | algebraic_topology.dold_kan.compatibility.equivalence_counit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The counit isomorphism of `equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_counit_iso_eq :
(equivalence hF hG).counit_iso = equivalence_counit_iso η | begin
ext1, apply nat_trans.ext, ext Y,
dsimp [equivalence, equivalence_counit_iso, is_equivalence.of_equivalence],
simp only [equivalence₂_counit_iso_eq eB hF],
erw [nat_trans.id_app, nat_trans.id_app],
dsimp [equivalence₂, equivalence₁],
simp only [assoc, comp_id, F.map_id, id_comp,
equivalence₂_couni... | lemma | algebraic_topology.dold_kan.compatibility.equivalence_counit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
υ : eA.functor ≅ F ⋙ e'.inverse | calc eA.functor ≅ eA.functor ⋙ 𝟭 A' : (functor.left_unitor _).symm
... ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse) : iso_whisker_left _ e'.unit_iso
... ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse : iso.refl _
... ≅ F ⋙ e'.inverse : iso_whisker_right hF _ | def | algebraic_topology.dold_kan.compatibility.υ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The isomorphism `eA.functor ≅ F ⋙ e'.inverse` deduced from the
unit isomorphism of `e'` and the isomorphism `hF : eA.functor ⋙ e'.functor ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_unit_iso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ G | calc 𝟭 A ≅ eA.functor ⋙ eA.inverse : eA.unit_iso
... ≅ (F ⋙ e'.inverse) ⋙ eA.inverse : iso_whisker_right ε _
... ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse : iso.refl _
... ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ (e'.inverse ⋙ eA.inverse) :
iso_whisker_left _ (iso_whisker_right eB.counit_iso.symm _)
... ≅ (F ⋙ eB.inverse) ... | def | algebraic_topology.dold_kan.compatibility.equivalence_unit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | The unit isomorphism of `equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_unit_iso_eq :
(equivalence hF hG).unit_iso = equivalence_unit_iso hG ε | begin
ext1, apply nat_trans.ext, ext X,
dsimp [equivalence, iso.refl, nat_iso.hcomp, is_equivalence.inverse,
is_equivalence.of_equivalence],
erw [nat_trans.id_app, id_comp, G.map_id, comp_id, comp_id],
simp only [equivalence₂_unit_iso_eq eB hF, equivalence₂_unit_iso_hom_app],
dsimp [equivalence₂, equivale... | lemma | algebraic_topology.dold_kan.compatibility.equivalence_unit_iso_eq | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/compatibility.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposition_Q (n q : ℕ) :
((Q q).f (n+1) : X _[n+1] ⟶ X _[n+1]) =
∑ (i : fin (n+1)) in finset.filter (λ i : fin(n+1), (i:ℕ)<q) finset.univ,
(P i).f (n+1) ≫ X.δ (i.rev).succ ≫ X.σ i.rev | begin
induction q with q hq,
{ simp only [Q_eq_zero, homological_complex.zero_f_apply, nat.not_lt_zero,
finset.filter_false, finset.sum_empty], },
{ by_cases hqn : q+1 ≤ n+1, swap,
{ rw [Q_is_eventually_constant (show n+1≤q, by linarith), hq],
congr,
ext,
have hx := x.is_lt,
simp... | lemma | algebraic_topology.dold_kan.decomposition_Q | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [
"fin.coe_mk",
"fin.ext_iff",
"finset.filter",
"finset.filter_false",
"finset.mem_filter",
"finset.mem_insert",
"finset.mem_univ",
"finset.univ",
"homological_complex.comp_f",
"homological_complex.sub_f_apply",
"homological_complex.zero_f_apply",
"lt_self_iff_false",
"nat.lt_succ_iff_lt_or_eq... | In each positive degree, this lemma decomposes the idempotent endomorphism
`Q q` as a sum of morphisms which are postcompositions with suitable degeneracies.
As `Q q` is the complement projection to `P q`, this implies that in the case of
simplicial abelian groups, any $(n+1)$-simplex $x$ can be decomposed as
$x = x' +... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morph_components (n : ℕ) (Z : C) | (a : X _[n+1] ⟶ Z)
(b : fin (n+1) → (X _[n] ⟶ Z)) | structure | algebraic_topology.dold_kan.morph_components | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | The structure `morph_components` is an ad hoc structure that is used in
the proof that `N₁ : simplicial_object C ⥤ karoubi (chain_complex C ℕ))`
reflects isomorphisms. The fields are the data that are needed in order to
construct a morphism `X _[n+1] ⟶ Z` (see `φ`) using the decomposition of the
identity given by `deco... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
φ {Z : C} (f : morph_components X n Z) : X _[n+1] ⟶ Z | P_infty.f (n+1) ≫ f.a + ∑ (i : fin (n+1)), (P i).f (n+1) ≫ X.δ i.rev.succ ≫ f.b i.rev | def | algebraic_topology.dold_kan.morph_components.φ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | The morphism `X _[n+1] ⟶ Z ` associated to `f : morph_components X n Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : morph_components X n (X _[n+1]) | { a := P_infty.f (n+1),
b := λ i, X.σ i, } | def | algebraic_topology.dold_kan.morph_components.id | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | the canonical `morph_components` whose associated morphism is the identity
(see `F_id`) thanks to `decomposition_Q n (n+1)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_φ : (id X n).φ = 𝟙 _ | begin
simp only [← P_add_Q_f (n+1) (n+1), φ],
congr' 1,
{ simp only [id, P_infty_f, P_f_idem], },
{ convert (decomposition_Q n (n+1)).symm,
ext i,
simpa only [finset.mem_univ, finset.mem_filter, true_and, true_iff] using fin.is_lt i, },
end | lemma | algebraic_topology.dold_kan.morph_components.id_φ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [
"fin.is_lt",
"finset.mem_filter",
"finset.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
post_comp : morph_components X n Z' | { a := f.a ≫ h,
b := λ i, f.b i ≫ h } | def | algebraic_topology.dold_kan.morph_components.post_comp | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | A `morph_components` can be postcomposed with a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
post_comp_φ : (f.post_comp h).φ = f.φ ≫ h | begin
unfold φ post_comp,
simp only [add_comp, sum_comp, assoc],
end | lemma | algebraic_topology.dold_kan.morph_components.post_comp_φ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_comp : morph_components X' n Z | { a := g.app (op [n+1]) ≫ f.a,
b := λ i, g.app (op [n]) ≫ f.b i } | def | algebraic_topology.dold_kan.morph_components.pre_comp | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | A `morph_components` can be precomposed with a morphism of simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_comp_φ : (f.pre_comp g).φ = g.app (op [n+1]) ≫ f.φ | begin
unfold φ pre_comp,
simp only [P_infty_f, comp_add],
congr' 1,
{ simp only [P_f_naturality_assoc], },
{ simp only [comp_sum, P_f_naturality_assoc, simplicial_object.δ_naturality_assoc], }
end | lemma | algebraic_topology.dold_kan.morph_components.pre_comp_φ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/decomposition.lean | [
"algebraic_topology.dold_kan.p_infty"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
higher_faces_vanish.comp_σ {Y : C} {X : simplicial_object C} {n b q : ℕ} {φ : Y ⟶ X _[n+1]}
(v : higher_faces_vanish q φ) (hnbq : n + 1 = b + q) :
higher_faces_vanish q (φ ≫ X.σ ⟨b,
by simpa only [hnbq, nat.lt_succ_iff, le_add_iff_nonneg_right] using zero_le q⟩) | λ j hj, begin
rw [assoc, simplicial_object.δ_comp_σ_of_gt', fin.pred_succ,
v.comp_δ_eq_zero_assoc _ _ hj, zero_comp],
{ intro hj',
simpa only [hj', hnbq, fin.coe_zero, zero_add, add_comm b, add_assoc, false_and,
add_le_iff_nonpos_right, le_zero_iff, add_eq_zero_iff, nat.one_ne_zero] using hj, },
{ s... | lemma | algebraic_topology.dold_kan.higher_faces_vanish.comp_σ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/degeneracies.lean | [
"algebraic_topology.dold_kan.decomposition",
"tactic.fin_cases"
] | [
"fin.coe_mk",
"fin.coe_succ",
"fin.coe_zero",
"fin.lt_iff_coe_lt_coe",
"fin.pred_succ",
"fin.succ_mk",
"le_zero_iff",
"nat.lt_iff_add_one_le",
"nat.lt_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_P_eq_zero (X : simplicial_object C)
{n q : ℕ} (i : fin (n + 1)) (hi : n + 1 ≤ i + q) : (X.σ i) ≫ (P q).f (n + 1) = 0 | begin
induction q with q hq generalizing i hi,
{ exfalso,
have h := fin.is_lt i,
linarith, },
{ by_cases n+1 ≤ (i : ℕ) + q,
{ unfold P,
simp only [homological_complex.comp_f, ← assoc],
rw [hq i h, zero_comp], },
{ have hi' : n = (i : ℕ) + q,
{ cases le_iff_exists_add.mp hi with j... | lemma | algebraic_topology.dold_kan.σ_comp_P_eq_zero | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/degeneracies.lean | [
"algebraic_topology.dold_kan.decomposition",
"tactic.fin_cases"
] | [
"fin.cast_succ",
"fin.cast_succ_mk",
"fin.cast_succ_zero",
"fin.coe_mk",
"fin.coe_one",
"fin.coe_succ",
"fin.coe_two",
"fin.coe_zero",
"fin.eta",
"fin.is_lt",
"fin.le_iff_coe_le_coe",
"fin.mk_one",
"fin.mk_zero",
"fin.rev_eq",
"fin.succ_mk",
"fin.succ_one_eq_two",
"fin.succ_zero_eq_o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_P_infty (X : simplicial_object C) {n : ℕ} (i : fin (n+1)) :
(X.σ i) ≫ P_infty.f (n+1) = 0 | begin
rw [P_infty_f, σ_comp_P_eq_zero X i],
simp only [le_add_iff_nonneg_left, zero_le],
end | lemma | algebraic_topology.dold_kan.σ_comp_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/degeneracies.lean | [
"algebraic_topology.dold_kan.decomposition",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degeneracy_comp_P_infty (X : simplicial_object C)
(n : ℕ) {Δ' : simplex_category} (θ : [n] ⟶ Δ') (hθ : ¬mono θ) :
X.map θ.op ≫ P_infty.f n = 0 | begin
rw simplex_category.mono_iff_injective at hθ,
cases n,
{ exfalso,
apply hθ,
intros x y h,
fin_cases x,
fin_cases y, },
{ obtain ⟨i, α, h⟩ := simplex_category.eq_σ_comp_of_not_injective θ hθ,
rw [h, op_comp, X.map_comp, assoc, (show X.map (simplex_category.σ i).op = X.σ i, by refl),
... | lemma | algebraic_topology.dold_kan.degeneracy_comp_P_infty | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/degeneracies.lean | [
"algebraic_topology.dold_kan.decomposition",
"tactic.fin_cases"
] | [
"simplex_category",
"simplex_category.eq_σ_comp_of_not_injective",
"simplex_category.mono_iff_injective",
"simplex_category.σ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N : simplicial_object A ⥤ chain_complex A ℕ | algebraic_topology.normalized_Moore_complex A | def | category_theory.abelian.dold_kan.N | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence.lean | [
"algebraic_topology.dold_kan.equivalence_pseudoabelian",
"algebraic_topology.dold_kan.normalized"
] | [
"algebraic_topology.normalized_Moore_complex",
"chain_complex"
] | The functor `N` for the equivalence is `normalized_Moore_complex A` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : chain_complex A ℕ ⥤ simplicial_object A | idempotents.dold_kan.Γ | def | category_theory.abelian.dold_kan.Γ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence.lean | [
"algebraic_topology.dold_kan.equivalence_pseudoabelian",
"algebraic_topology.dold_kan.normalized"
] | [
"chain_complex"
] | The functor `Γ` for the equivalence is the same as in the pseudoabelian case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comparison_N : (N : simplicial_object A ⥤ _) ≅ idempotents.dold_kan.N | calc N ≅ N ⋙ 𝟭 _ : functor.left_unitor N
... ≅ N ⋙ ((to_karoubi_equivalence _).functor ⋙ (to_karoubi_equivalence _).inverse) :
iso_whisker_left _ (to_karoubi_equivalence _).unit_iso
... ≅ (N ⋙ (to_karoubi_equivalence _).functor) ⋙ (to_karoubi_equivalence _).inverse :
iso.refl _
... ≅ N₁ ⋙ (to_karoubi_equivalence _... | def | category_theory.abelian.dold_kan.comparison_N | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence.lean | [
"algebraic_topology.dold_kan.equivalence_pseudoabelian",
"algebraic_topology.dold_kan.normalized"
] | [] | The comparison isomorphism between `normalized_Moore_complex A` and
the functor `idempotents.dold_kan.N` from the pseudoabelian case | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence : simplicial_object A ≌ chain_complex A ℕ | begin
let F : simplicial_object A ⥤ _ := idempotents.dold_kan.N,
let hF : is_equivalence F := is_equivalence.of_equivalence idempotents.dold_kan.equivalence,
letI : is_equivalence (N : simplicial_object A ⥤ _ ) :=
is_equivalence.of_iso comparison_N.symm hF,
exact N.as_equivalence,
end | def | category_theory.abelian.dold_kan.equivalence | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence.lean | [
"algebraic_topology.dold_kan.equivalence_pseudoabelian",
"algebraic_topology.dold_kan.normalized"
] | [
"chain_complex"
] | The Dold-Kan equivalence for abelian categories | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_inverse : (equivalence : simplicial_object A ≌ _).inverse = Γ | rfl | lemma | category_theory.abelian.dold_kan.equivalence_inverse | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence.lean | [
"algebraic_topology.dold_kan.equivalence_pseudoabelian",
"algebraic_topology.dold_kan.normalized"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
N : karoubi (simplicial_object C) ⥤ karoubi (chain_complex C ℕ) | N₂ | def | category_theory.preadditive.dold_kan.N | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_additive.lean | [
"algebraic_topology.dold_kan.n_comp_gamma"
] | [
"chain_complex"
] | The functor `karoubi (simplicial_object C) ⥤ karoubi (chain_complex C ℕ)` of
the Dold-Kan equivalence for additive categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : karoubi (chain_complex C ℕ) ⥤ karoubi (simplicial_object C) | Γ₂ | def | category_theory.preadditive.dold_kan.Γ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_additive.lean | [
"algebraic_topology.dold_kan.n_comp_gamma"
] | [
"chain_complex"
] | The inverse functor `karoubi (chain_complex C ℕ) ⥤ karoubi (simplicial_object C)` of
the Dold-Kan equivalence for additive categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence : karoubi (simplicial_object C) ≌ karoubi (chain_complex C ℕ) | { functor := N,
inverse := Γ,
unit_iso := Γ₂N₂,
counit_iso := N₂Γ₂,
functor_unit_iso_comp' := λ P, begin
let α := N.map_iso (Γ₂N₂.app P),
let β := N₂Γ₂.app (N.obj P),
symmetry,
change 𝟙 _ = α.hom ≫ β.hom,
rw [← iso.inv_comp_eq, comp_id, ← comp_id β.hom, ← iso.inv_comp_eq],
exact algebra... | def | category_theory.preadditive.dold_kan.equivalence | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_additive.lean | [
"algebraic_topology.dold_kan.n_comp_gamma"
] | [
"algebraic_topology.dold_kan.identity_N₂_objectwise",
"chain_complex"
] | The Dold-Kan equivalence `karoubi (simplicial_object C) ≌ karoubi (chain_complex C ℕ)`
for additive categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
N : simplicial_object C ⥤ chain_complex C ℕ | N₁ ⋙ (to_karoubi_equivalence _).inverse | def | category_theory.idempotents.dold_kan.N | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | The functor `N` for the equivalence is obtained by composing
`N' : simplicial_object C ⥤ karoubi (chain_complex C ℕ)` and the inverse
of the equivalence `chain_complex C ℕ ≌ karoubi (chain_complex C ℕ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : chain_complex C ℕ ⥤ simplicial_object C | Γ₀ | def | category_theory.idempotents.dold_kan.Γ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | The functor `Γ` for the equivalence is `Γ'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hN₁ : (to_karoubi_equivalence (simplicial_object C)).functor ⋙
preadditive.dold_kan.equivalence.functor = N₁ | functor.congr_obj (functor_extension₁_comp_whiskering_left_to_karoubi _ _) N₁ | lemma | category_theory.idempotents.dold_kan.hN₁ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hΓ₀ : (to_karoubi_equivalence (chain_complex C ℕ)).functor ⋙
preadditive.dold_kan.equivalence.inverse = Γ ⋙ (to_karoubi_equivalence _).functor | functor.congr_obj (functor_extension₂_comp_whiskering_left_to_karoubi _ _) Γ₀ | lemma | category_theory.idempotents.dold_kan.hΓ₀ | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : simplicial_object C ≌ chain_complex C ℕ | compatibility.equivalence (eq_to_iso hN₁) (eq_to_iso hΓ₀) | def | category_theory.idempotents.dold_kan.equivalence | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | The Dold-Kan equivalence for pseudoabelian categories given
by the functors `N` and `Γ`. It is obtained by applying the results in
`compatibility.lean` to the equivalence `preadditive.dold_kan.equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_functor : (equivalence : simplicial_object C ≌ _).functor = N | rfl | lemma | category_theory.idempotents.dold_kan.equivalence_functor | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence_inverse : (equivalence : simplicial_object C ≌ _).inverse = Γ | rfl | lemma | category_theory.idempotents.dold_kan.equivalence_inverse | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hη : compatibility.τ₀ =
compatibility.τ₁ (eq_to_iso hN₁) (eq_to_iso hΓ₀)
(N₁Γ₀ : Γ ⋙ N₁ ≅ (to_karoubi_equivalence (chain_complex C ℕ)).functor) | begin
ext K : 3,
simpa only [compatibility.τ₀_hom_app, compatibility.τ₁_hom_app, eq_to_iso.hom,
preadditive.dold_kan.equivalence_counit_iso, N₂Γ₂_to_karoubi_iso_hom, eq_to_hom_map,
eq_to_hom_trans_assoc, eq_to_hom_app] using N₂Γ₂_compatible_with_N₁Γ₀ K,
end | lemma | category_theory.idempotents.dold_kan.hη | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | The natural isomorphism `NΓ' satisfies the compatibility that is needed
for the construction of our counit isomorphism `η` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
η : Γ ⋙ N ≅ 𝟭 (chain_complex C ℕ) | compatibility.equivalence_counit_iso
(N₁Γ₀ : (Γ : chain_complex C ℕ ⥤ _ ) ⋙ N₁ ≅ (to_karoubi_equivalence _).functor) | def | category_theory.idempotents.dold_kan.η | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | The counit isomorphism induced by `N₁Γ₀` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_counit_iso :
dold_kan.equivalence.counit_iso = (η : Γ ⋙ N ≅ 𝟭 (chain_complex C ℕ)) | compatibility.equivalence_counit_iso_eq hη | lemma | category_theory.idempotents.dold_kan.equivalence_counit_iso | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [
"chain_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hε : compatibility.υ (eq_to_iso hN₁) =
(Γ₂N₁ : (to_karoubi_equivalence _).functor ≅ (N₁ : simplicial_object C ⥤ _) ⋙
preadditive.dold_kan.equivalence.inverse) | begin
ext X : 4,
erw [nat_trans.comp_app, compatibility_Γ₂N₁_Γ₂N₂_nat_trans],
simp only [compatibility.υ_hom_app, compatibility_Γ₂N₁_Γ₂N₂,
preadditive.dold_kan.equivalence_unit_iso, Γ₂N₂, iso.symm_hom, as_iso_inv, assoc],
erw [← nat_trans.comp_app_assoc, is_iso.hom_inv_id],
dsimp,
simpa only [id_comp, e... | lemma | category_theory.idempotents.dold_kan.hε | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ε : 𝟭 (simplicial_object C) ≅ N ⋙ Γ | compatibility.equivalence_unit_iso (eq_to_iso hΓ₀) Γ₂N₁ | def | category_theory.idempotents.dold_kan.ε | algebraic_topology.dold_kan | src/algebraic_topology/dold_kan/equivalence_pseudoabelian.lean | [
"algebraic_topology.dold_kan.equivalence_additive",
"algebraic_topology.dold_kan.compatibility",
"category_theory.idempotents.simplicial_object"
] | [] | The unit isomorphism induced by `Γ₂N₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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