fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[simps]
equivalenceLeftToRight (F : Arrow C) (X : CosimplicialObject.Augmented C)
(G : F.augmentedCechConerve ⟶ X) : F ⟶ Augmented.toArrow.obj X where
left := G.left
right := (WidePushout.ι _ 0 ≫ G.right.app ⦋0⦌ :)
w := by
dsimp
rw [@WidePushout.arrow_ι_assoc _ _ _ _ _ (fun (_ : Fin 1) => F.hom)
... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | equivalenceLeftToRight | A helper function used in defining the Čech conerve adjunction. |
@[simps!]
equivalenceRightToLeft (F : Arrow C) (X : CosimplicialObject.Augmented C)
(G : F ⟶ Augmented.toArrow.obj X) : F.augmentedCechConerve ⟶ X where
left := G.left
right :=
{ app := fun x =>
Limits.WidePushout.desc (G.left ≫ X.hom.app _)
(fun i => G.right ≫ X.right.map (SimplexCategory... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | equivalenceRightToLeft | A helper function used in defining the Čech conerve adjunction. |
@[simps]
cechConerveEquiv (F : Arrow C) (X : CosimplicialObject.Augmented C) :
(F.augmentedCechConerve ⟶ X) ≃ (F ⟶ Augmented.toArrow.obj X) where
toFun := equivalenceLeftToRight _ _
invFun := equivalenceRightToLeft _ _
left_inv := by
intro A
ext x : 2
· rfl
· refine WidePushout.hom_ext _ _ _ (... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechConerveEquiv | A helper function used in defining the Čech conerve adjunction. |
cechConerveAdjunction : augmentedCechConerve ⊣ (Augmented.toArrow : _ ⥤ Arrow C) :=
Adjunction.mkOfHomEquiv { homEquiv := cechConerveEquiv } | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechConerveAdjunction | The augmented Čech conerve construction is left adjoint to the `toArrow` functor. |
cechNerveTerminalFrom {C : Type u} [Category.{v} C] [HasFiniteProducts C] (X : C) :
SimplicialObject C where
obj n := ∏ᶜ fun _ : Fin (n.unop.len + 1) => X
map f := Limits.Pi.lift fun i => Limits.Pi.π _ (f.unop.toOrderHom i) | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechNerveTerminalFrom | Given an object `X : C`, the natural simplicial object sending `⦋n⦌` to `Xⁿ⁺¹`. |
wideCospan (X : C) : WidePullbackShape ι ⥤ C :=
WidePullbackShape.wideCospan (terminal C) (fun _ : ι => X) fun _ => terminal.from X | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | wideCospan | The diagram `Option ι ⥤ C` sending `none` to the terminal object and `some j` to `X`. |
uniqueToWideCospanNone (X Y : C) : Unique (Y ⟶ (wideCospan ι X).obj none) := by
dsimp [wideCospan]
infer_instance
variable [HasFiniteProducts C] | instance | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | uniqueToWideCospanNone | null |
wideCospan.limitCone [Finite ι] (X : C) : LimitCone (wideCospan ι X) where
cone :=
{ pt := ∏ᶜ fun _ : ι => X
π :=
{ app := fun X => Option.casesOn X (terminal.from _) fun i => limit.π _ ⟨i⟩
naturality := fun i j f => by
cases f
· cases i
all_goals simp... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | wideCospan.limitCone | The product `Xᶥ` is the vertex of a limit cone on `wideCospan ι X`. |
hasWidePullback [Finite ι] (X : C) :
HasWidePullback (Arrow.mk (terminal.from X)).right
(fun _ : ι => (Arrow.mk (terminal.from X)).left)
(fun _ => (Arrow.mk (terminal.from X)).hom) := by
cases nonempty_fintype ι
exact ⟨⟨wideCospan.limitCone ι X⟩⟩ | instance | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | hasWidePullback | null |
hasWidePullback' [Finite ι] (X : C) :
HasWidePullback (⊤_ C)
(fun _ : ι => X)
(fun _ => terminal.from X) :=
hasWidePullback _ _ | instance | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | hasWidePullback' | null |
hasLimit_wideCospan [Finite ι] (X : C) : HasLimit (wideCospan ι X) := hasWidePullback _ _ | instance | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | hasLimit_wideCospan | null |
wideCospan.limitIsoPi [Finite ι] (X : C) :
limit (wideCospan ι X) ≅ ∏ᶜ fun _ : ι => X :=
(IsLimit.conePointUniqueUpToIso (limit.isLimit _)
(wideCospan.limitCone ι X).2)
@[reassoc (attr := simp)] | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | wideCospan.limitIsoPi | the isomorphism to the product induced by the limit cone `wideCospan ι X` |
wideCospan.limitIsoPi_inv_comp_pi [Finite ι] (X : C) (j : ι) :
(wideCospan.limitIsoPi ι X).inv ≫ WidePullback.π _ j = Pi.π _ j :=
IsLimit.conePointUniqueUpToIso_inv_comp _ _ _
@[reassoc (attr := simp)] | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | wideCospan.limitIsoPi_inv_comp_pi | null |
wideCospan.limitIsoPi_hom_comp_pi [Finite ι] (X : C) (j : ι) :
(wideCospan.limitIsoPi ι X).hom ≫ Pi.π _ j = WidePullback.π _ j := by
rw [← wideCospan.limitIsoPi_inv_comp_pi, Iso.hom_inv_id_assoc] | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | wideCospan.limitIsoPi_hom_comp_pi | null |
iso (X : C) : (Arrow.mk (terminal.from X)).cechNerve ≅ cechNerveTerminalFrom X :=
NatIso.ofComponents (fun _ => wideCospan.limitIsoPi _ _) (fun {m n} f => by
dsimp only [cechNerveTerminalFrom, Arrow.cechNerve]
ext ⟨j⟩
simp) | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | iso | Given an object `X : C`, the Čech nerve of the hom to the terminal object `X ⟶ ⊤_ C` is
naturally isomorphic to a simplicial object sending `⦋n⦌` to `Xⁿ⁺¹` (when `C` is `G-Set`, this is
`EG`, the universal cover of the classifying space of `G`. |
@[ext]
ExtraDegeneracy (X : SimplicialObject.Augmented C) where
/-- a section of the augmentation in dimension `0` -/
s' : point.obj X ⟶ drop.obj X _⦋0⦌
/-- the extra degeneracy -/
s : ∀ n : ℕ, drop.obj X _⦋n⦌ ⟶ drop.obj X _⦋n + 1⦌
s'_comp_ε : s' ≫ X.hom.app (op ⦋0⦌) = 𝟙 _ := by cat_disch
s₀_comp_δ₁ : s 0 ... | structure | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ExtraDegeneracy | The datum of an extra degeneracy is a technical condition on
augmented simplicial objects. The morphisms `s'` and `s n` of the
structure formally behave like extra degeneracies `σ (-1)`. |
map {D : Type*} [Category D] {X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X)
(F : C ⥤ D) : ExtraDegeneracy (((whiskering _ _).obj F).obj X) where
s' := F.map ed.s'
s n := F.map (ed.s n)
s'_comp_ε := by
dsimp
rw [comp_id, ← F.map_comp, ed.s'_comp_ε]
dsimp only [point_obj]
rw [F.map_... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | map | If `ed` is an extra degeneracy for `X : SimplicialObject.Augmented C` and
`F : C ⥤ D` is a functor, then `ed.map F` is an extra degeneracy for the
augmented simplicial object in `D` obtained by applying `F` to `X`. |
ofIso {X Y : SimplicialObject.Augmented C} (e : X ≅ Y) (ed : ExtraDegeneracy X) :
ExtraDegeneracy Y where
s' := (point.mapIso e).inv ≫ ed.s' ≫ (drop.mapIso e).hom.app (op ⦋0⦌)
s n := (drop.mapIso e).inv.app (op ⦋n⦌) ≫ ed.s n ≫ (drop.mapIso e).hom.app (op ⦋n + 1⦌)
s'_comp_ε := by
simpa only [Functor.mapIso... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ofIso | If `X` and `Y` are isomorphic augmented simplicial objects, then an extra
degeneracy for `X` gives also an extra degeneracy for `Y` |
shiftFun {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) (i : Fin (n + 1)) : X :=
Matrix.vecCons 0 f i
@[simp] | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | shiftFun | When `[Zero X]`, the shift of a map `f : Fin n → X`
is a map `Fin (n + 1) → X` which sends `0` to `0` and `i.succ` to `f i`. |
shiftFun_zero {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) : shiftFun f 0 = 0 :=
rfl
@[deprecated (since := "2025-04-19")]
alias shiftFun_0 := shiftFun_zero
@[simp] | theorem | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | shiftFun_zero | null |
shiftFun_succ {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) (i : Fin n) :
shiftFun f i.succ = f i :=
rfl | theorem | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | shiftFun_succ | null |
@[simp]
shift {n : ℕ} {Δ : SimplexCategory} (f : ⦋n⦌ ⟶ Δ) : ⦋n + 1⦌ ⟶ Δ :=
SimplexCategory.Hom.mk
{ toFun := shiftFun f.toOrderHom
monotone' := fun i₁ i₂ hi => by
by_cases h₁ : i₁ = 0
· subst h₁
simp only [shiftFun_zero, Fin.zero_le]
· have h₂ : i₂ ≠ 0 := by
int... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | shift | The shift of a morphism `f : ⦋n⦌ → Δ` in `SimplexCategory` corresponds to
the monotone map which sends `0` to `0` and `i.succ` to `f.toOrderHom i`. |
protected noncomputable extraDegeneracy (Δ : SimplexCategory) :
SimplicialObject.Augmented.ExtraDegeneracy (stdSimplex.obj Δ) where
s' _ := objMk (OrderHom.const _ 0)
s _ f := objEquiv.symm (shift (objEquiv f))
s'_comp_ε := by
dsimp
subsingleton
s₀_comp_δ₁ := by
dsimp
ext1 x
apply objEqu... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | extraDegeneracy | The obvious extra degeneracy on the standard simplex. |
nonempty_extraDegeneracy_stdSimplex (Δ : SimplexCategory) :
Nonempty (SimplicialObject.Augmented.ExtraDegeneracy (stdSimplex.obj Δ)) :=
⟨StandardSimplex.extraDegeneracy Δ⟩ | instance | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | nonempty_extraDegeneracy_stdSimplex | null |
noncomputable ExtraDegeneracy.s (n : ℕ) :
f.cechNerve.obj (op ⦋n⦌) ⟶ f.cechNerve.obj (op ⦋n + 1⦌) :=
WidePullback.lift (WidePullback.base _)
(Fin.cases (WidePullback.base _ ≫ S.section_) (WidePullback.π _))
fun i => by
cases i using Fin.cases <;> simp | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ExtraDegeneracy.s | The extra degeneracy map on the Čech nerve of a split epi. It is
given on the `0`-projection by the given section of the split epi,
and by shifting the indices on the other projections. |
ExtraDegeneracy.s_comp_π_0 (n : ℕ) :
ExtraDegeneracy.s f S n ≫ WidePullback.π _ 0 =
@WidePullback.base _ _ _ f.right (fun _ : Fin (n + 1) => f.left) (fun _ => f.hom) _ ≫
S.section_ := by
dsimp [ExtraDegeneracy.s]
simp | theorem | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ExtraDegeneracy.s_comp_π_0 | null |
ExtraDegeneracy.s_comp_π_succ (n : ℕ) (i : Fin (n + 1)) :
ExtraDegeneracy.s f S n ≫ WidePullback.π _ i.succ =
@WidePullback.π _ _ _ f.right (fun _ : Fin (n + 1) => f.left) (fun _ => f.hom) _ i := by
simp [ExtraDegeneracy.s] | theorem | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ExtraDegeneracy.s_comp_π_succ | null |
ExtraDegeneracy.s_comp_base (n : ℕ) :
ExtraDegeneracy.s f S n ≫ WidePullback.base _ = WidePullback.base _ := by
apply WidePullback.lift_base | theorem | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | ExtraDegeneracy.s_comp_base | null |
noncomputable extraDegeneracy :
SimplicialObject.Augmented.ExtraDegeneracy f.augmentedCechNerve where
s' := S.section_ ≫ WidePullback.lift f.hom (fun _ => 𝟙 _) fun i => by rw [id_comp]
s n := ExtraDegeneracy.s f S n
s'_comp_ε := by
dsimp
simp only [assoc, WidePullback.lift_base, SplitEpi.id]
s₀_com... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | extraDegeneracy | The augmented Čech nerve associated to a split epimorphism has an extra degeneracy. |
@[simps]
const (X : C) : ExtraDegeneracy (Augmented.const.obj X) where
s' := 𝟙 _
s _ := 𝟙 _ | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | const | The constant augmented simplicial object has an extra degeneracy. |
noncomputable homotopyEquiv [Preadditive C] [HasZeroObject C]
{X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X) :
HomotopyEquiv (AlgebraicTopology.AlternatingFaceMapComplex.obj (drop.obj X))
((ChainComplex.single₀ C).obj (point.obj X)) where
hom := AlternatingFaceMapComplex.ε.app X
inv := (C... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.AlternatingFaceMapComplex",
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.CechNerve",
"Mathlib.Algebra.Homology.Homotopy",
"Mathlib.Tactic.FinCases"
] | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | homotopyEquiv | If `C` is a preadditive category and `X` is an augmented simplicial object
in `C` that has an extra degeneracy, then the augmentation on the alternating
face map complex of `X` is a homotopy equivalence. |
objX : ∀ n : ℕ, Subobject (X.obj (op ⦋n⦌))
| 0 => ⊤
| n + 1 => Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ)
@[simp] theorem objX_zero : objX X 0 = ⊤ :=
rfl
@[simp] theorem objX_add_one (n) :
objX X (n + 1) = Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ) :=
rfl | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | objX | The normalized Moore complex in degree `n`, as a subobject of `X n`. |
@[simp]
objD : ∀ n : ℕ, (objX X (n + 1) : C) ⟶ (objX X n : C)
| 0 => Subobject.arrow _ ≫ X.δ (0 : Fin 2) ≫ inv (⊤ : Subobject _).arrow
| n + 1 => by
refine factorThru _ (arrow _ ≫ X.δ (0 : Fin (n + 3))) ?_
refine (finset_inf_factors _).mpr fun i _ => ?_
apply kernelSubobject_factors
dsimp [objX]
... | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | objD | The differentials in the normalized Moore complex. |
d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by
rcases n with _ | n <;> dsimp [objD]
· rw [Subobject.factorThru_arrow_assoc, Category.assoc, ← Fin.castSucc_zero,
← X.δ_comp_δ_assoc (Fin.zero_le (0 : Fin 2)),
← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin 2) (by simp)),
... | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | d_squared | null |
@[simps!]
obj (X : SimplicialObject C) : ChainComplex C ℕ :=
ChainComplex.of (fun n => (objX X n : C))
(-- the coercion here picks a representative of the subobject
objD X) (d_squared X)
variable {X} {Y : SimplicialObject C} (f : X ⟶ Y) | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | obj | The normalized Moore complex functor, on objects. |
@[simps!]
map (f : X ⟶ Y) : obj X ⟶ obj Y :=
ChainComplex.ofHom _ _ _ _ _ _
(fun n => factorThru _ (arrow _ ≫ f.app (op ⦋n⦌)) (by
cases n <;> dsimp
· apply top_factors
· refine (finset_inf_factors _).mpr fun i _ => kernelSubobject_factors _ _ ?_
rw [Category.assoc, SimplicialObject.δ, ← ... | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | map | The normalized Moore complex functor, on morphisms. |
@[simps]
normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
obj := obj
map f := map f | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | normalizedMooreComplex | The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.
The `n`-th object is intersection of
the kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.
The differentials are induced from `X.δ 0`,
which maps each of these intersections of kernels to the next. |
normalizedMooreComplex_objD (X : SimplicialObject C) (n : ℕ) :
((normalizedMooreComplex C).obj X).d (n + 1) n = NormalizedMooreComplex.objD X n :=
ChainComplex.of_d _ _ (d_squared X) n | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.HomologicalComplex",
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Abelian.Basic"
] | Mathlib/AlgebraicTopology/MooreComplex.lean | normalizedMooreComplex_objD | null |
@[nolint unusedArguments]
SimplicialThickening (J : Type*) [LinearOrder J] : Type _ := J | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | SimplicialThickening | A type synonym for a linear order `J`, will be equipped with a simplicial category structure. |
@[ext]
Path {J : Type*} [LinearOrder J] (i j : J) where
/-- The underlying subset -/
I : Set J
left : i ∈ I := by simp
right : j ∈ I := by simp
left_le (k : J) (_ : k ∈ I) : i ≤ k := by simp
le_right (k : J) (_ : k ∈ I) : k ≤ j := by simp | structure | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | Path | A path from `i` to `j` in a linear order `J` is a subset of the interval `[i, j]` in `J` containing
the endpoints. |
Path.le {J : Type*} [LinearOrder J] {i j : J} (f : Path i j) : i ≤ j :=
f.left_le _ f.right | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | Path.le | null |
@[ext]
hom_ext {J : Type*} [LinearOrder J]
(i j : SimplicialThickening J) (x y : i ⟶ j) (h : ∀ t, t ∈ x.I ↔ t ∈ y.I) : x = y := by
apply Path.ext
ext
apply h | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | hom_ext | null |
@[simps]
compFunctor {J : Type*} [LinearOrder J]
(i j k : SimplicialThickening J) : (i ⟶ j) × (j ⟶ k) ⥤ (i ⟶ k) where
obj x := x.1 ≫ x.2
map f := ⟨⟨Set.union_subset_union f.1.1.1 f.2.1.1⟩⟩ | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | compFunctor | Composition of morphisms in `SimplicialThickening J`, as a functor `(i ⟶ j) × (j ⟶ k) ⥤ (i ⟶ k)` |
Hom (i j : SimplicialThickening J) : SSet := (nerve (i ⟶ j)) | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | Hom | The hom simplicial set of the simplicial category structure on `SimplicialThickening J` |
id (i : SimplicialThickening J) : 𝟙_ SSet ⟶ Hom i i :=
⟨fun _ _ ↦ (Functor.const _).obj (𝟙 _), fun _ _ _ ↦ by simp; rfl⟩ | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | id | The identity of the simplicial category structure on `SimplicialThickening J` |
comp (i j k : SimplicialThickening J) : Hom i j ⊗ Hom j k ⟶ Hom i k :=
⟨fun _ x ↦ x.1.prod' x.2 ⋙ compFunctor i j k, fun _ _ _ ↦ by simp; rfl⟩
@[simp] | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | comp | The composition of the simplicial category structure on `SimplicialThickening J` |
id_comp (i j : SimplicialThickening J) :
(λ_ (Hom i j)).inv ≫ id i ▷ Hom i j ≫ comp i i j = 𝟙 (Hom i j) := by
aesop
@[simp] | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | id_comp | null |
comp_id (i j : SimplicialThickening J) :
(ρ_ (Hom i j)).inv ≫ Hom i j ◁ id j ≫ comp i j j = 𝟙 (Hom i j) := by
aesop
@[simp] | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | comp_id | null |
assoc (i j k l : SimplicialThickening J) :
(α_ (Hom i j) (Hom j k) (Hom k l)).inv ≫ comp i j k ▷ Hom k l ≫ comp i k l =
Hom i j ◁ comp j k l ≫ comp i j l := by
aesop | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | assoc | null |
orderHom {J K : Type*} [LinearOrder J] [LinearOrder K] (f : J →o K) :
SimplicialThickening J →o SimplicialThickening K := f | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | orderHom | Auxiliary definition for `SimplicialThickening.functorMap` |
noncomputable functorMap {J K : Type u} [LinearOrder J] [LinearOrder K]
(f : J →o K) (i j : SimplicialThickening J) : (i ⟶ j) ⥤ ((orderHom f i) ⟶ (orderHom f j)) where
obj I := ⟨f '' I.I, Set.mem_image_of_mem f I.left, Set.mem_image_of_mem f I.right,
by rintro _ ⟨k, hk, rfl⟩; exact f.monotone (I.left_le k hk)... | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | functorMap | Auxiliary definition for `SimplicialThickening.functor` |
@[simps]
noncomputable functor {J K : Type u} [LinearOrder J] [LinearOrder K]
(f : J →o K) : EnrichedFunctor SSet (SimplicialThickening J) (SimplicialThickening K) where
obj := f
map i j := nerveMap ((functorMap f i j))
map_id i := by
ext
simp only [eId, EnrichedCategory.id]
exact Functor.ext (by ... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | functor | The simplicial thickening defines a functor from the category of linear orders to the category of
simplicial categories |
functor_id (J : Type u) [LinearOrder J] :
(functor (OrderHom.id (α := J))) = EnrichedFunctor.id _ _ := by
refine EnrichedFunctor.ext _ (fun _ ↦ rfl) fun i j ↦ ?_
ext
exact Functor.ext (by cat_disch) | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | functor_id | null |
functor_comp {J K L : Type u} [LinearOrder J] [LinearOrder K]
[LinearOrder L] (f : J →o K) (g : K →o L) :
functor (g.comp f) =
(functor f).comp _ (functor g) := by
refine EnrichedFunctor.ext _ (fun _ ↦ rfl) fun i j ↦ ?_
ext
exact Functor.ext (by cat_disch) | lemma | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | functor_comp | null |
noncomputable SimplicialNerve (C : Type u) [Category.{v} C] [SimplicialCategory C] :
SSet.{max u v} where
obj n := EnrichedFunctor SSet (SimplicialThickening (ULift (Fin (n.unop.len + 1)))) C
map f := (SimplicialThickening.functor f.unop.toOrderHom.uliftMap).comp (E := C) SSet
map_id i := by
change Enrich... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialCategory.Basic",
"Mathlib.AlgebraicTopology.SimplicialSet.Nerve"
] | Mathlib/AlgebraicTopology/SimplicialNerve.lean | SimplicialNerve | The simplicial nerve of a simplicial category `C` is defined as the simplicial set whose
`n`-simplices are given by the set of simplicial functors from the simplicial thickening of
the linear order `Fin (n + 1)` to `C` |
noncomputable TopCat.toSSet : TopCat.{u} ⥤ SSet.{u} :=
Presheaf.restrictedULiftYoneda.{0} SimplexCategory.toTop.{u} | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | TopCat.toSSet | The functor associating the *singular simplicial set* to a topological space.
Let `X : TopCat.{u}` be a topological space.
Then the singular simplicial set of `X`
has as `n`-simplices the continuous maps `ULift.{u} ⦋n⦌.toTop → X`.
Here, `⦋n⦌.toTop` is the standard topological `n`-simplex,
defined as `{ f : Fin (n+1) →... |
TopCat.toSSetObjEquiv (X : TopCat.{u}) (n : SimplexCategoryᵒᵖ) :
(toSSet.obj X).obj n ≃ C(n.unop.toTopObj, X) :=
Equiv.ulift.{0}.trans (ConcreteCategory.homEquiv.trans
(Homeomorph.ulift.continuousMapCongr (.refl _))) | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | TopCat.toSSetObjEquiv | If `X : TopCat.{u}` and `n : SimplexCategoryᵒᵖ`,
then `(toSSet.obj X).obj n` identifies to the type of continuous
maps from the standard simplex `n.unop.toTopObj` to `X`. |
noncomputable SSet.toTop : SSet.{u} ⥤ TopCat.{u} :=
stdSimplex.{u}.leftKanExtension SimplexCategory.toTop | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | SSet.toTop | The *geometric realization functor* is
the left Kan extension of `SimplexCategory.toTop` along the Yoneda embedding.
It is left adjoint to `TopCat.toSSet`, as witnessed by `sSetTopAdj`. |
noncomputable sSetTopAdj : SSet.toTop.{u} ⊣ TopCat.toSSet.{u} :=
Presheaf.uliftYonedaAdjunction
(SSet.stdSimplex.{u}.leftKanExtension SimplexCategory.toTop)
(SSet.stdSimplex.{u}.leftKanExtensionUnit SimplexCategory.toTop) | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | sSetTopAdj | Geometric realization is left adjoint to the singular simplicial set construction. |
noncomputable SSet.toTopSimplex :
SSet.stdSimplex.{u} ⋙ SSet.toTop ≅ SimplexCategory.toTop :=
Presheaf.isExtensionAlongULiftYoneda _ | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | SSet.toTopSimplex | The geometric realization of the representable simplicial sets agree
with the usual topological simplices. |
noncomputable TopCat.toSSetIsoConst (X : TopCat.{u}) [TotallyDisconnectedSpace X] :
TopCat.toSSet.obj X ≅ (Functor.const _).obj X :=
(NatIso.ofComponents (fun n ↦ Equiv.toIso
((TotallyDisconnectedSpace.continuousMapEquivOfConnectedSpace _ X).symm.trans
(X.toSSetObjEquiv n).symm))).symm | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex",
"Mathlib.AlgebraicTopology.TopologicalSimplex",
"Mathlib.CategoryTheory.Limits.Presheaf",
"Mathlib.Topology.Category.TopCat.Limits.Basic",
"Mathlib.Topology.Category.TopCat.ULift"
] | Mathlib/AlgebraicTopology/SingularSet.lean | TopCat.toSSetIsoConst | The singular simplicial set of a totally disconnected space is the constant simplicial set. |
toTopObj (x : SimplexCategory) := { f : ToType x → ℝ≥0 | ∑ i, f i = 1 } | def | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObj | The topological simplex associated to `x : SimplexCategory`.
This is the object part of the functor `SimplexCategory.toTop`. |
@[ext]
toTopObj.ext {x : SimplexCategory} (f g : x.toTopObj) : (f : ToType x → ℝ≥0) = g → f = g :=
Subtype.ext
@[simp] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObj.ext | null |
toTopObj_zero_apply_zero (f : ⦋0⦌.toTopObj) : f 0 = 1 := by
simpa [toType_apply] using show ∑ _, _ = _ from f.2 | lemma | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObj_zero_apply_zero | null |
toTopObj_one_add_eq_one (f : ⦋1⦌.toTopObj) : f 0 + f 1 = 1 := by
simpa [toType_apply, Finset.sum] using show ∑ _, _ = _ from f.2 | lemma | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObj_one_add_eq_one | null |
toTopObj_one_coe_add_coe_eq_one (f : ⦋1⦌.toTopObj) : (f 0 : ℝ) + f 1 = 1 := by
norm_cast
rw [toTopObj_one_add_eq_one] | lemma | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObj_one_coe_add_coe_eq_one | null |
toTopObjOneHomeo : ⦋1⦌.toTopObj ≃ₜ I where
toFun f := ⟨f 0, (f 0).2, toTopObj_one_coe_add_coe_eq_one f ▸ le_add_of_nonneg_right (f 1).2⟩
invFun x := ⟨![toNNReal x, toNNReal (σ x)],
show ∑ _, _ = _ by ext; simp [toType_apply, Finset.sum]⟩
left_inv f := by ext i; fin_cases i <;> simp [← toTopObj_one_coe_add_coe... | def | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopObjOneHomeo | The one-dimensional topological simplex is homeomorphic to the unit interval. |
toTopMap {x y : SimplexCategory} (f : x ⟶ y) (g : x.toTopObj) : y.toTopObj :=
⟨fun i => ∑ j ∈ Finset.univ.filter (f · = i), g j, by
simp only [toTopObj, Set.mem_setOf]
rw [← Finset.sum_biUnion]
· have hg : ∑ i : ToType x, g i = 1 := g.2
convert hg
simp [Finset.eq_univ_iff_forall]
· convert... | def | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTopMap | A morphism in `SimplexCategory` induces a map on the associated topological spaces. |
coe_toTopMap {x y : SimplexCategory} (f : x ⟶ y) (g : x.toTopObj) (i : ToType y) :
toTopMap f g i = ∑ j ∈ Finset.univ.filter (f · = i), g j :=
rfl
@[continuity, fun_prop] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | coe_toTopMap | null |
continuous_toTopMap {x y : SimplexCategory} (f : x ⟶ y) : Continuous (toTopMap f) := by
refine Continuous.subtype_mk (continuous_pi fun i => ?_) _
dsimp only [coe_toTopMap]
exact continuous_finset_sum _ (fun j _ => (continuous_apply _).comp continuous_subtype_val) | theorem | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | continuous_toTopMap | null |
@[simps obj map]
toTop₀ : SimplexCategory ⥤ TopCat.{0} where
obj x := TopCat.of x.toTopObj
map f := TopCat.ofHom ⟨toTopMap f, by fun_prop⟩
map_id := by
classical
intro Δ
ext f
simp [Finset.sum_filter]
map_comp := fun f g => by
classical
ext h : 3
dsimp
rw [← Finset.sum_biUnion]
... | def | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTop₀ | The functor `SimplexCategory ⥤ TopCat.{0}`
associating the topological `n`-simplex to `⦋n⦌ : SimplexCategory`. |
@[simps! obj map, pp_with_univ]
toTop : SimplexCategory ⥤ TopCat.{u} :=
toTop₀ ⋙ TopCat.uliftFunctor | def | AlgebraicTopology | [
"Mathlib.Algebra.BigOperators.Ring.Finset",
"Mathlib.AlgebraicTopology.SimplexCategory.Basic",
"Mathlib.Topology.Category.TopCat.ULift",
"Mathlib.Topology.Connected.PathConnected"
] | Mathlib/AlgebraicTopology/TopologicalSimplex.lean | toTop | The functor `SimplexCategory ⥤ TopCat.{u}`
associating the topological `n`-simplex to `⦋n⦌ : SimplexCategory`. |
ae_differentiableWithinAt_of_mem_real {f : ℝ → ℝ} {s : Set ℝ}
(h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x := by
obtain ⟨p, q, hp, hq, rfl⟩ : ∃ p q, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q :=
h.exists_monotoneOn_sub_monotoneOn
filter_upwards [hp.ae_differentiableWit... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | ae_differentiableWithinAt_of_mem_real | A bounded variation function into `ℝ` is differentiable almost everywhere. Superseded by
`ae_differentiableWithinAt_of_mem`. |
ae_differentiableWithinAt_of_mem_pi {ι : Type*} [Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ}
(h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x := by
have A : ∀ i : ι, LipschitzWith 1 fun x : ι → ℝ => x i := fun i => LipschitzWith.eval i
have : ∀ i : ι, ∀ᵐ x, x ∈ s → DifferentiableWit... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | ae_differentiableWithinAt_of_mem_pi | A bounded variation function into a finite-dimensional product vector space is differentiable
almost everywhere. Superseded by `ae_differentiableWithinAt_of_mem`. |
ae_differentiableWithinAt_of_mem {f : ℝ → V} {s : Set ℝ}
(h : LocallyBoundedVariationOn f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x := by
let A := (Module.Basis.ofVectorSpace ℝ V).equivFun.toContinuousLinearEquiv
suffices H : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ (A ∘ f) s x by
filter_upwards [H] w... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | ae_differentiableWithinAt_of_mem | A real function into a finite-dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. |
ae_differentiableWithinAt {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s)
(hs : MeasurableSet s) : ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x := by
rw [ae_restrict_iff' hs]
exact h.ae_differentiableWithinAt_of_mem | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | ae_differentiableWithinAt | A real function into a finite-dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. |
ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) :
∀ᵐ x, DifferentiableAt ℝ f x := by
filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx
rw [differentiableWithinAt_univ] at hx
exact hx (mem_univ _) | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | ae_differentiableAt | A real function into a finite-dimensional real vector space with bounded variation
is differentiable almost everywhere. |
LipschitzOnWith.ae_differentiableWithinAt_of_mem_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
(h : LipschitzOnWith C f s) : ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x :=
h.locallyBoundedVariationOn.ae_differentiableWithinAt_of_mem | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | LipschitzOnWith.ae_differentiableWithinAt_of_mem_real | A real function into a finite-dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzOnWith.ae_differentiableWithinAt_of_mem`. |
LipschitzOnWith.ae_differentiableWithinAt_real {C : ℝ≥0} {f : ℝ → V} {s : Set ℝ}
(h : LipschitzOnWith C f s) (hs : MeasurableSet s) :
∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x :=
h.locallyBoundedVariationOn.ae_differentiableWithinAt hs | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | LipschitzOnWith.ae_differentiableWithinAt_real | A real function into a finite-dimensional real vector space which is Lipschitz on a set
is differentiable almost everywhere in this set. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzOnWith.ae_differentiableWithinAt`. |
LipschitzWith.ae_differentiableAt_real {C : ℝ≥0} {f : ℝ → V} (h : LipschitzWith C f) :
∀ᵐ x, DifferentiableAt ℝ f x :=
(h.locallyBoundedVariationOn univ).ae_differentiableAt | theorem | Analysis | [
"Mathlib.Analysis.Calculus.FDeriv.Add",
"Mathlib.Analysis.Calculus.FDeriv.Equiv",
"Mathlib.Analysis.Calculus.FDeriv.Prod",
"Mathlib.Analysis.Calculus.Monotone",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/BoundedVariation.lean | LipschitzWith.ae_differentiableAt_real | A real Lipschitz function into a finite-dimensional real vector space is differentiable
almost everywhere. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzWith.ae_differentiableAt`. |
HasConstantSpeedOnWith :=
∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (l * (y - x))
variable {f s l} | def | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasConstantSpeedOnWith | `f` has constant speed `l` on `s` if the variation of `f` on `s ∩ Icc x y` is equal to
`l * (y - x)` for any `x y` in `s`. |
HasConstantSpeedOnWith.hasLocallyBoundedVariationOn (h : HasConstantSpeedOnWith f s l) :
LocallyBoundedVariationOn f s := fun x y hx hy => by
simp only [BoundedVariationOn, h hx hy, Ne, ENNReal.ofReal_ne_top, not_false_iff] | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasConstantSpeedOnWith.hasLocallyBoundedVariationOn | null |
hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
(l : ℝ≥0) : HasConstantSpeedOnWith f s l := by
rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero] | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | hasConstantSpeedOnWith_of_subsingleton | null |
hasConstantSpeedOnWith_iff_ordered :
HasConstantSpeedOnWith f s l ↔ ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s),
x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (l * (y - x)) := by
refine ⟨fun h x xs y ys _ => h xs ys, fun h x xs y ys => ?_⟩
rcases le_total x y with (xy | yx)
· exact h xs ys xy
· rw [eVar... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | hasConstantSpeedOnWith_iff_ordered | null |
hasConstantSpeedOnWith_iff_variationOnFromTo_eq :
HasConstantSpeedOnWith f s l ↔ LocallyBoundedVariationOn f s ∧
∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), variationOnFromTo f s x y = l * (y - x) := by
constructor
· rintro h; refine ⟨h.hasLocallyBoundedVariationOn, fun x xs y ys => ?_⟩
rw [hasConstantSpeedOnW... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | hasConstantSpeedOnWith_iff_variationOnFromTo_eq | null |
HasConstantSpeedOnWith.union {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l)
(hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x) :
HasConstantSpeedOnWith f (s ∪ t) l := by
rw [hasConstantSpeedOnWith_iff_ordered] at hfs hft ⊢
rintro z (zs | zt) y (ys | yt) zy
· have : (s ... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasConstantSpeedOnWith.union | null |
HasConstantSpeedOnWith.Icc_Icc {x y z : ℝ} (hfs : HasConstantSpeedOnWith f (Icc x y) l)
(hft : HasConstantSpeedOnWith f (Icc y z) l) : HasConstantSpeedOnWith f (Icc x z) l := by
rcases le_total x y with (xy | yx)
· rcases le_total y z with (yz | zy)
· rw [← Set.Icc_union_Icc_eq_Icc xy yz]
exact hfs.un... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasConstantSpeedOnWith.Icc_Icc | null |
hasConstantSpeedOnWith_zero_iff :
HasConstantSpeedOnWith f s 0 ↔ ∀ᵉ (x ∈ s) (y ∈ s), edist (f x) (f y) = 0 := by
dsimp [HasConstantSpeedOnWith]
simp only [zero_mul, ENNReal.ofReal_zero, ← eVariationOn.eq_zero_iff]
constructor
· by_contra!
obtain ⟨h, hfs⟩ := this
simp_rw [ne_eq, eVariationOn.eq_zero_... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | hasConstantSpeedOnWith_zero_iff | null |
HasConstantSpeedOnWith.ratio {l' : ℝ≥0} (hl' : l' ≠ 0) {φ : ℝ → ℝ} (φm : MonotoneOn φ s)
(hfφ : HasConstantSpeedOnWith (f ∘ φ) s l) (hf : HasConstantSpeedOnWith f (φ '' s) l') ⦃x : ℝ⦄
(xs : x ∈ s) : EqOn φ (fun y => l / l' * (y - x) + φ x) s := by
rintro y ys
rw [← sub_eq_iff_eq_add, mul_comm, ← mul_div_ass... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasConstantSpeedOnWith.ratio | null |
HasUnitSpeedOn (f : ℝ → E) (s : Set ℝ) :=
HasConstantSpeedOnWith f s 1 | def | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasUnitSpeedOn | `f` has unit speed on `s` if it is linearly parameterized by `l = 1` on `s`. |
HasUnitSpeedOn.union {t : Set ℝ} {x : ℝ} (hfs : HasUnitSpeedOn f s)
(hft : HasUnitSpeedOn f t) (hs : IsGreatest s x) (ht : IsLeast t x) :
HasUnitSpeedOn f (s ∪ t) :=
HasConstantSpeedOnWith.union hfs hft hs ht | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasUnitSpeedOn.union | null |
HasUnitSpeedOn.Icc_Icc {x y z : ℝ} (hfs : HasUnitSpeedOn f (Icc x y))
(hft : HasUnitSpeedOn f (Icc y z)) : HasUnitSpeedOn f (Icc x z) :=
HasConstantSpeedOnWith.Icc_Icc hfs hft | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | HasUnitSpeedOn.Icc_Icc | null |
unique_unit_speed {φ : ℝ → ℝ} (φm : MonotoneOn φ s) (hfφ : HasUnitSpeedOn (f ∘ φ) s)
(hf : HasUnitSpeedOn f (φ '' s)) ⦃x : ℝ⦄ (xs : x ∈ s) : EqOn φ (fun y => y - x + φ x) s := by
dsimp only [HasUnitSpeedOn] at hf hfφ
convert HasConstantSpeedOnWith.ratio one_ne_zero φm hfφ hf xs using 3
simp | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | unique_unit_speed | If both `f` and `f ∘ φ` have unit speed (on `t` and `s` respectively) and `φ`
monotonically maps `s` onto `t`, then `φ` is just a translation (on `s`). |
unique_unit_speed_on_Icc_zero {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) {φ : ℝ → ℝ}
(φm : MonotoneOn φ <| Icc 0 s) (φst : φ '' Icc 0 s = Icc 0 t)
(hfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)) (hf : HasUnitSpeedOn f (Icc 0 t)) :
EqOn φ id (Icc 0 s) := by
rw [← φst] at hf
convert unique_unit_speed φm hfφ hf ⟨le_rfl,... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | unique_unit_speed_on_Icc_zero | If both `f` and `f ∘ φ` have unit speed (on `Icc 0 t` and `Icc 0 s` respectively)
and `φ` monotonically maps `Icc 0 s` onto `Icc 0 t`, then `φ` is the identity on `Icc 0 s` |
noncomputable naturalParameterization (f : α → E) (s : Set α) (a : α) : ℝ → E :=
f ∘ @Function.invFunOn _ _ ⟨a⟩ (variationOnFromTo f s a) s | def | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | naturalParameterization | The natural parameterization of `f` on `s`, which, if `f` has locally bounded variation on `s`,
* has unit speed on `s` (by `has_unit_speed_naturalParameterization`).
* composed with `variationOnFromTo f s a`, is at distance zero from `f`
(by `edist_naturalParameterization_eq_zero`). |
edist_naturalParameterization_eq_zero {f : α → E} {s : Set α}
(hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) {b : α} (bs : b ∈ s) :
edist (naturalParameterization f s a (variationOnFromTo f s a b)) (f b) = 0 := by
dsimp only [naturalParameterization]
haveI : Nonempty α := ⟨a⟩
obtain ⟨cs, hc⟩ :=... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | edist_naturalParameterization_eq_zero | null |
has_unit_speed_naturalParameterization (f : α → E) {s : Set α}
(hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
HasUnitSpeedOn (naturalParameterization f s a) (variationOnFromTo f s a '' s) := by
dsimp only [HasUnitSpeedOn]
rw [hasConstantSpeedOnWith_iff_ordered]
rintro _ ⟨b, bs, rfl⟩ _ ⟨c, cs,... | theorem | Analysis | [
"Mathlib.Data.Set.Function",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.EMetricSpace.BoundedVariation"
] | Mathlib/Analysis/ConstantSpeed.lean | has_unit_speed_naturalParameterization | null |
convolution_integrand_bound_right_of_le_of_subset {C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G}
{s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t := by
refine le_indicator (f := fun t ↦ ‖L (f t) (g (x - t))‖) (fun t _ => ?_) (fun t ht => ?_) t
... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_integrand_bound_right_of_le_of_subset | null |
_root_.HasCompactSupport.convolution_integrand_bound_right_of_subset
(hcg : HasCompactSupport g) (hg : Continuous g)
{x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t := by
refine convolution_integrand_bound_right_of... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolution_integrand_bound_right_of_subset | null |
_root_.HasCompactSupport.convolution_integrand_bound_right (hcg : HasCompactSupport g)
(hg : Continuous g) {x t : G} {s : Set G} (hx : x ∈ s) :
‖L (f t) (g (x - t))‖ ≤ (-tsupport g + s).indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t :=
hcg.convolution_integrand_bound_right_of_subset L hg hx Subset.rfl | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolution_integrand_bound_right | null |
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