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 ⌀ |
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trans ⦃f g h : C(X, Y)⦄ (h₀ : HomotopicWith f g P) (h₁ : HomotopicWith g h P) :
HomotopicWith f h P :=
⟨h₀.some.trans h₁.some⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | null |
HomotopyRel (f₀ f₁ : C(X, Y)) (S : Set X) :=
HomotopyWith f₀ f₁ fun f ↦ ∀ x ∈ S, f x = f₀ x | abbrev | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | HomotopyRel | A `HomotopyRel f₀ f₁ S` is a homotopy between `f₀` and `f₁` which is fixed on the points in `S`. |
eq_fst (F : HomotopyRel f₀ f₁ S) (t : I) {x : X} (hx : x ∈ S) : F (t, x) = f₀ x :=
F.prop t x hx | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | eq_fst | null |
eq_snd (F : HomotopyRel f₀ f₁ S) (t : I) {x : X} (hx : x ∈ S) : F (t, x) = f₁ x := by
rw [F.eq_fst t hx, ← F.eq_fst 1 hx, F.apply_one] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | eq_snd | null |
fst_eq_snd (F : HomotopyRel f₀ f₁ S) {x : X} (hx : x ∈ S) : f₀ x = f₁ x :=
F.eq_fst 0 hx ▸ F.eq_snd 0 hx | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | fst_eq_snd | null |
@[simps!]
refl (f : C(X, Y)) (S : Set X) : HomotopyRel f f S :=
HomotopyWith.refl f fun _ _ ↦ rfl | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | Given a map `f : C(X, Y)` and a set `S`, we can define a `HomotopyRel f f S` by setting
`F (t, x) = f x` for all `t`. This is defined using `HomotopyWith.refl`, but with the proof
filled in. |
@[simps!]
symm (F : HomotopyRel f₀ f₁ S) : HomotopyRel f₁ f₀ S where
toHomotopy := F.toHomotopy.symm
prop' := fun _ _ hx ↦ F.eq_snd _ hx
@[simp] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | Given a `HomotopyRel f₀ f₁ S`, we can define a `HomotopyRel f₁ f₀ S` by reversing the homotopy. |
symm_symm (F : HomotopyRel f₀ f₁ S) : F.symm.symm = F :=
HomotopyWith.symm_symm F | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_symm | null |
symm_bijective :
Function.Bijective (HomotopyRel.symm : HomotopyRel f₀ f₁ S → HomotopyRel f₁ f₀ S) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_bijective | null |
trans (F : HomotopyRel f₀ f₁ S) (G : HomotopyRel f₁ f₂ S) : HomotopyRel f₀ f₂ S where
toHomotopy := F.toHomotopy.trans G.toHomotopy
prop' t x hx := by
simp only [Homotopy.trans]
split_ifs
· simp [HomotopyWith.extendProp F (2 * t) x hx, F.fst_eq_snd hx, G.fst_eq_snd hx]
· simp [HomotopyWith.extendProp G (2 * t - 1) x hx, F.fst_eq_snd hx, G.fst_eq_snd hx] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | Given `HomotopyRel f₀ f₁ S` and `HomotopyRel f₁ f₂ S`, we can define a `HomotopyRel f₀ f₂ S`
by putting the first homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. |
trans_apply (F : HomotopyRel f₀ f₁ S) (G : HomotopyRel f₁ f₂ S) (x : I × X) :
(F.trans G) x =
if h : (x.1 : ℝ) ≤ 1 / 2 then
F (⟨2 * x.1, (unitInterval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2) :=
Homotopy.trans_apply _ _ _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans_apply | null |
symm_trans (F : HomotopyRel f₀ f₁ S) (G : HomotopyRel f₁ f₂ S) :
(F.trans G).symm = G.symm.trans F.symm :=
HomotopyWith.ext <| Homotopy.congr_fun <| Homotopy.symm_trans _ _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_trans | null |
@[simps!]
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : HomotopyRel f₀ f₁ S) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
HomotopyRel g₀ g₁ S where
toHomotopy := Homotopy.cast F.toHomotopy h₀ h₁
prop' t x hx := by simpa only [← h₀, ← h₁] using F.prop t x hx | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | cast | Casting a `HomotopyRel f₀ f₁ S` to a `HomotopyRel g₀ g₁ S` where `f₀ = g₀` and `f₁ = g₁`. |
@[simps!] compContinuousMap {f₀ f₁ : C(X, Y)} (F : f₀.HomotopyRel f₁ S) (g : C(Y, Z)) :
(g.comp f₀).HomotopyRel (g.comp f₁) S where
toHomotopy := .comp (.refl g) F.toHomotopy
prop' t x hx := congr_arg g (F.prop t x hx) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | compContinuousMap | Post-compose a homotopy relative to a set by a continuous function. |
HomotopicRel (f₀ f₁ : C(X, Y)) (S : Set X) : Prop :=
Nonempty (HomotopyRel f₀ f₁ S) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | HomotopicRel | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic relative to a set `S` if
there exists a `HomotopyRel f₀ f₁ S`. |
protected homotopic {f₀ f₁ : C(X, Y)} (h : HomotopicRel f₀ f₁ S) : Homotopic f₀ f₁ :=
h.map fun F ↦ F.1 | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | homotopic | If two maps are homotopic relative to a set, then they are homotopic. |
refl (f : C(X, Y)) : HomotopicRel f f S :=
⟨HomotopyRel.refl f S⟩
@[symm] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | null |
symm ⦃f g : C(X, Y)⦄ (h : HomotopicRel f g S) : HomotopicRel g f S :=
h.map HomotopyRel.symm
@[trans] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | null |
trans ⦃f g h : C(X, Y)⦄ (h₀ : HomotopicRel f g S) (h₁ : HomotopicRel g h S) :
HomotopicRel f h S :=
h₀.map2 HomotopyRel.trans h₁ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | null |
equivalence : Equivalence fun f g : C(X, Y) => HomotopicRel f g S :=
⟨refl, by apply symm, by apply trans⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | equivalence | null |
comp_continuousMap ⦃f₀ f₁ : C(X, Y)⦄ (h : f₀.HomotopicRel f₁ S) (g : C(Y, Z)) :
(g.comp f₀).HomotopicRel (g.comp f₁) S := h.map (·.compContinuousMap g) | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | comp_continuousMap | null |
@[simp] homotopicRel_empty {f₀ f₁ : C(X, Y)} : HomotopicRel f₀ f₁ ∅ ↔ Homotopic f₀ f₁ :=
⟨fun h ↦ h.homotopic, fun ⟨F⟩ ↦ ⟨⟨F, fun _ _ ↦ False.elim⟩⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | homotopicRel_empty | null |
Nullhomotopic (f : C(X, Y)) : Prop :=
∃ y : Y, Homotopic f (ContinuousMap.const _ y) | def | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | Nullhomotopic | A map is nullhomotopic if it is homotopic to a constant map. |
nullhomotopic_of_constant (y : Y) : Nullhomotopic (ContinuousMap.const X y) :=
⟨y, by rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | nullhomotopic_of_constant | null |
Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
(g.comp f).Nullhomotopic := by
obtain ⟨y, hy⟩ := hf
use g y
exact .comp (.refl g) hy | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | Nullhomotopic.comp_right | null |
Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
(f.comp g).Nullhomotopic := by
obtain ⟨y, hy⟩ := hf
use y
exact .comp hy (.refl g) | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | Nullhomotopic.comp_left | null |
ContractibleSpace (X : Type*) [TopologicalSpace X] : Prop where
hequiv_unit' : Nonempty (X ≃ₕ Unit) | class | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | ContractibleSpace | A contractible space is one that is homotopy equivalent to `Unit`. |
ContractibleSpace.hequiv_unit (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
Nonempty (X ≃ₕ Unit) :=
ContractibleSpace.hequiv_unit' | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | ContractibleSpace.hequiv_unit | null |
id_nullhomotopic (X : Type*) [TopologicalSpace X] [ContractibleSpace X] :
(ContinuousMap.id X).Nullhomotopic := by
obtain ⟨hv⟩ := ContractibleSpace.hequiv_unit X
use hv.invFun ()
convert hv.left_inv.symm | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | id_nullhomotopic | null |
contractible_iff_id_nullhomotopic (Y : Type*) [TopologicalSpace Y] :
ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic := by
constructor
· intro
apply id_nullhomotopic
rintro ⟨p, h⟩
refine
{ hequiv_unit' :=
⟨{ toFun := ContinuousMap.const _ ()
invFun := ContinuousMap.const _ p
left_inv := ?_
right_inv := ?_ }⟩ }
· exact h.symm
· convert Homotopic.refl (ContinuousMap.id Unit)
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | contractible_iff_id_nullhomotopic | null |
protected ContinuousMap.HomotopyEquiv.contractibleSpace [ContractibleSpace Y] (e : X ≃ₕ Y) :
ContractibleSpace X :=
⟨(ContractibleSpace.hequiv_unit Y).map e.trans⟩ | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | ContinuousMap.HomotopyEquiv.contractibleSpace | null |
protected ContinuousMap.HomotopyEquiv.contractibleSpace_iff (e : X ≃ₕ Y) :
ContractibleSpace X ↔ ContractibleSpace Y :=
⟨fun _ => e.symm.contractibleSpace, fun _ => e.contractibleSpace⟩ | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | ContinuousMap.HomotopyEquiv.contractibleSpace_iff | null |
protected Homeomorph.contractibleSpace [ContractibleSpace Y] (e : X ≃ₜ Y) :
ContractibleSpace X :=
e.toHomotopyEquiv.contractibleSpace | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | Homeomorph.contractibleSpace | null |
protected Homeomorph.contractibleSpace_iff (e : X ≃ₜ Y) :
ContractibleSpace X ↔ ContractibleSpace Y :=
e.toHomotopyEquiv.contractibleSpace_iff | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | Homeomorph.contractibleSpace_iff | null |
hequiv [ContractibleSpace X] [ContractibleSpace Y] :
Nonempty (X ≃ₕ Y) := by
rcases ContractibleSpace.hequiv_unit' (X := X) with ⟨h⟩
rcases ContractibleSpace.hequiv_unit' (X := Y) with ⟨h'⟩
exact ⟨h.trans h'.symm⟩ | theorem | Topology | [
"Mathlib.Topology.Homotopy.Path",
"Mathlib.Topology.Homotopy.Equiv"
] | Mathlib/Topology/Homotopy/Contractible.lean | hequiv | null |
@[ext]
HomotopyEquiv (X : Type u) (Y : Type v) [TopologicalSpace X] [TopologicalSpace Y] where
/-- The forward map of an homotopy.
Do NOT use directly. Use the coercion instead. -/
toFun : C(X, Y)
/-- The backward map of an homotopy.
Do NOT use `e.invFun` directly. Use the coercion of `e.symm` instead. -/
invFun : C(Y, X)
left_inv : (invFun.comp toFun).Homotopic (ContinuousMap.id X)
right_inv : (toFun.comp invFun).Homotopic (ContinuousMap.id Y)
@[inherit_doc] scoped infixl:25 " ≃ₕ " => ContinuousMap.HomotopyEquiv | structure | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | HomotopyEquiv | A homotopy equivalence between topological spaces `X` and `Y` are a pair of functions
`toFun : C(X, Y)` and `invFun : C(Y, X)` such that `toFun.comp invFun` and `invFun.comp toFun`
are both homotopic to corresponding identity maps. |
@[continuity]
continuous (h : HomotopyEquiv X Y) : Continuous h :=
h.toFun.continuous | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | continuous | null |
toHomotopyEquiv (h : X ≃ₜ Y) : X ≃ₕ Y where
toFun := h
invFun := h.symm
left_inv := by rw [symm_comp_toContinuousMap]
right_inv := by rw [toContinuousMap_comp_symm]
@[simp] | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | toHomotopyEquiv | Any homeomorphism is a homotopy equivalence. |
coe_toHomotopyEquiv (h : X ≃ₜ Y) : (h.toHomotopyEquiv : X → Y) = h :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | coe_toHomotopyEquiv | null |
symm (h : X ≃ₕ Y) : Y ≃ₕ X where
toFun := h.invFun
invFun := h.toFun
left_inv := h.right_inv
right_inv := h.left_inv
@[simp] | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | symm | If `X` is homotopy equivalent to `Y`, then `Y` is homotopy equivalent to `X`. |
coe_invFun (h : HomotopyEquiv X Y) : (⇑h.invFun : Y → X) = ⇑h.symm :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | coe_invFun | null |
Simps.apply (h : X ≃ₕ Y) : X → Y :=
h | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.symm_apply (h : X ≃ₕ Y) : Y → X :=
h.symm
initialize_simps_projections HomotopyEquiv (toFun_toFun → apply, invFun_toFun → symm_apply,
-toFun, -invFun) | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | Simps.symm_apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
@[simps!]
refl (X : Type u) [TopologicalSpace X] : X ≃ₕ X :=
(Homeomorph.refl X).toHomotopyEquiv | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | refl | Any topological space is homotopy equivalent to itself. |
@[simps!]
trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : X ≃ₕ Z where
toFun := h₂.toFun.comp h₁.toFun
invFun := h₁.invFun.comp h₂.invFun
left_inv := by
refine Homotopic.trans ?_ h₁.left_inv
exact .comp (.refl _) (.comp h₂.left_inv (.refl _))
right_inv := by
refine Homotopic.trans ?_ h₂.right_inv
exact .comp (.refl _) <| .comp h₁.right_inv (.refl _) | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | trans | If `X` is homotopy equivalent to `Y`, and `Y` is homotopy equivalent to `Z`, then `X` is homotopy
equivalent to `Z`. |
symm_trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : (h₁.trans h₂).symm = h₂.symm.trans h₁.symm := rfl | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | symm_trans | null |
prodCongr (h₁ : X ≃ₕ Y) (h₂ : Z ≃ₕ Z') : (X × Z) ≃ₕ (Y × Z') where
toFun := h₁.toFun.prodMap h₂.toFun
invFun := h₁.invFun.prodMap h₂.invFun
left_inv := h₁.left_inv.prodMap h₂.left_inv
right_inv := h₁.right_inv.prodMap h₂.right_inv | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | prodCongr | If `X` is homotopy equivalent to `Y` and `Z` is homotopy equivalent to `Z'`, then `X × Z` is
homotopy equivalent to `Z × Z'`. |
piCongrRight {ι : Type*} {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, TopologicalSpace (Y i)] (h : ∀ i, X i ≃ₕ Y i) :
(∀ i, X i) ≃ₕ (∀ i, Y i) where
toFun := .piMap fun i ↦ (h i).toFun
invFun := .piMap fun i ↦ (h i).invFun
left_inv := .piMap fun i ↦ (h i).left_inv
right_inv := .piMap fun i ↦ (h i).right_inv | def | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | piCongrRight | If `X i` is homotopy equivalent to `Y i` for each `i`, then the space of functions (a.k.a. the
indexed product) `∀ i, X i` is homotopy equivalent to `∀ i, Y i`. |
@[simp]
refl_toHomotopyEquiv (X : Type u) [TopologicalSpace X] :
(Homeomorph.refl X).toHomotopyEquiv = HomotopyEquiv.refl X :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | refl_toHomotopyEquiv | null |
symm_toHomotopyEquiv (h : X ≃ₜ Y) : h.symm.toHomotopyEquiv = h.toHomotopyEquiv.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | symm_toHomotopyEquiv | null |
trans_toHomotopyEquiv (h₀ : X ≃ₜ Y) (h₁ : Y ≃ₜ Z) :
(h₀.trans h₁).toHomotopyEquiv = h₀.toHomotopyEquiv.trans h₁.toHomotopyEquiv :=
rfl | theorem | Topology | [
"Mathlib.Topology.Homotopy.Basic"
] | Mathlib/Topology/Homotopy/Equiv.lean | trans_toHomotopyEquiv | null |
boundary (N : Type*) : Set (I^N) :=
{y | ∃ i, y i = 0 ∨ y i = 1}
variable {N : Type*} [DecidableEq N] | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | boundary | `I^N` is notation (in the Topology namespace) for `N → I`,
i.e. the unit cube indexed by a type `N`. -/
scoped[Topology] notation "I^" N => N → I
namespace Cube
/-- The points in a cube with at least one projection equal to 0 or 1. |
splitAt (i : N) : (I^N) ≃ₜ I × I^{ j // j ≠ i } :=
funSplitAt I i | abbrev | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | splitAt | The forward direction of the homeomorphism
between the cube $I^N$ and $I × I^{N\setminus\{j\}}$. |
insertAt (i : N) : (I × I^{ j // j ≠ i }) ≃ₜ I^N :=
(funSplitAt I i).symm | abbrev | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | insertAt | The backward direction of the homeomorphism
between the cube $I^N$ and $I × I^{N\setminus\{j\}}$. |
insertAt_boundary (i : N) {t₀ : I} {t}
(H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i }) : insertAt i ⟨t₀, t⟩ ∈ boundary N := by
obtain H | ⟨j, H⟩ := H
· use i; rwa [funSplitAt_symm_apply, dif_pos rfl]
· use j; rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta] | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | insertAt_boundary | null |
LoopSpace :=
Path x x
@[inherit_doc] scoped[Topology.Homotopy] notation "Ω" => LoopSpace | abbrev | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | LoopSpace | The space of paths with both endpoints equal to a specified point `x : X`.
Denoted as `Ω`, within the `Topology.Homotopy` namespace. |
LoopSpace.inhabited : Inhabited (Path x x) :=
⟨Path.refl x⟩ | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | LoopSpace.inhabited | null |
GenLoop : Set C(I^N, X) :=
{p | ∀ y ∈ Cube.boundary N, p y = x}
@[inherit_doc] scoped[Topology.Homotopy] notation "Ω^" => GenLoop
open Topology.Homotopy
variable {N X x} | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | GenLoop | The `n`-dimensional generalized loops based at `x` in a space `X` are
continuous functions `I^n → X` that sends the boundary to `x`.
We allow an arbitrary indexing type `N` in place of `Fin n` here. |
instFunLike : FunLike (Ω^ N X x) (I^N) X where
coe f := f.1
coe_injective' := fun ⟨⟨f, _⟩, _⟩ ⟨⟨g, _⟩, _⟩ _ => by congr
@[ext] | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | instFunLike | null |
ext (f g : Ω^ N X x) (H : ∀ y, f y = g y) : f = g :=
DFunLike.coe_injective' (funext H)
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | ext | null |
mk_apply (f : C(I^N, X)) (H y) : (⟨f, H⟩ : Ω^ N X x) y = f y :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | mk_apply | null |
instContinuousEval : ContinuousEval (Ω^ N X x) (I^N) X :=
.of_continuous_forget continuous_subtype_val | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | instContinuousEval | null |
instContinuousEvalConst : ContinuousEvalConst (Ω^ N X x) (I^N) X := inferInstance | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | instContinuousEvalConst | null |
copy (f : Ω^ N X x) (g : (I^N) → X) (h : g = f) : Ω^ N X x :=
⟨⟨g, h.symm ▸ f.1.2⟩, by convert f.2⟩ | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | copy | Copy of a `GenLoop` with a new map from the unit cube equal to the old one.
Useful to fix definitional equalities. |
coe_copy (f : Ω^ N X x) {g : (I^N) → X} (h : g = f) : ⇑(copy f g h) = g :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | coe_copy | null |
copy_eq (f : Ω^ N X x) {g : (I^N) → X} (h : g = f) : copy f g h = f := by
ext x
exact congr_fun h x | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | copy_eq | null |
boundary (f : Ω^ N X x) : ∀ y ∈ Cube.boundary N, f y = x :=
f.2 | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | boundary | null |
const : Ω^ N X x :=
⟨ContinuousMap.const _ x, fun _ _ => rfl⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | const | The constant `GenLoop` at `x`. |
const_apply {t} : (@const N X _ x) t = x :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | const_apply | null |
inhabited : Inhabited (Ω^ N X x) :=
⟨const⟩ | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | inhabited | null |
Homotopic (f g : Ω^ N X x) : Prop :=
f.1.HomotopicRel g.1 (Cube.boundary N) | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | Homotopic | The "homotopic relative to boundary" relation between `GenLoop`s. |
@[refl]
refl (f : Ω^ N X x) : Homotopic f f :=
ContinuousMap.HomotopicRel.refl _
@[symm]
nonrec theorem symm (H : Homotopic f g) : Homotopic g f :=
H.symm
@[trans]
nonrec theorem trans (H0 : Homotopic f g) (H1 : Homotopic g h) : Homotopic f h :=
H0.trans H1 | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | refl | null |
equiv : Equivalence (@Homotopic N X _ x) :=
⟨Homotopic.refl, Homotopic.symm, Homotopic.trans⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | equiv | null |
setoid (N) (x : X) : Setoid (Ω^ N X x) :=
⟨Homotopic, equiv⟩ | instance | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | setoid | null |
@[simps]
toLoop (i : N) (p : Ω^ N X x) : Ω (Ω^ { j // j ≠ i } X x) const where
toFun t :=
⟨(p.val.comp (Cube.insertAt i)).curry t, fun y yH =>
p.property (Cube.insertAt i (t, y)) (Cube.insertAt_boundary i <| Or.inr yH)⟩
source' := by ext t; refine p.property (Cube.insertAt i (0, t)) ⟨i, Or.inl ?_⟩; simp
target' := by ext t; refine p.property (Cube.insertAt i (1, t)) ⟨i, Or.inr ?_⟩; simp | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | toLoop | Loop from a generalized loop by currying $I^N → X$ into $I → (I^{N\setminus\{j\}} → X)$. |
continuous_toLoop (i : N) : Continuous (@toLoop N X _ x _ i) :=
Path.continuous_uncurry_iff.1 <|
Continuous.subtype_mk
(continuous_eval.comp <|
Continuous.prodMap
(ContinuousMap.continuous_curry.comp <|
(ContinuousMap.continuous_precomp _).comp continuous_subtype_val)
continuous_id)
_ | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | continuous_toLoop | null |
@[simps]
fromLoop (i : N) (p : Ω (Ω^ { j // j ≠ i } X x) const) : Ω^ N X x :=
⟨(ContinuousMap.comp ⟨Subtype.val, by fun_prop⟩ p.toContinuousMap).uncurry.comp
(Cube.splitAt i),
by
rintro y ⟨j, Hj⟩
simp only [ContinuousMap.comp_apply, ContinuousMap.coe_coe,
funSplitAt_apply, ContinuousMap.uncurry_apply, ContinuousMap.coe_mk,
Function.uncurry_apply_pair]
obtain rfl | Hne := eq_or_ne j i
· rcases Hj with Hj | Hj <;> simp only [Hj, p.coe_toContinuousMap, p.source, p.target] <;> rfl
· exact GenLoop.boundary _ _ ⟨⟨j, Hne⟩, Hj⟩⟩ | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | fromLoop | Generalized loop from a loop by uncurrying $I → (I^{N\setminus\{j\}} → X)$ into $I^N → X$. |
continuous_fromLoop (i : N) : Continuous (@fromLoop N X _ x _ i) :=
((ContinuousMap.continuous_precomp _).comp <|
ContinuousMap.continuous_uncurry.comp <|
(ContinuousMap.continuous_postcomp _).comp continuous_induced_dom).subtype_mk
_ | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | continuous_fromLoop | null |
to_from (i : N) (p : Ω (Ω^ { j // j ≠ i } X x) const) : toLoop i (fromLoop i p) = p := by
simp_rw [toLoop, fromLoop, ContinuousMap.comp_assoc,
toContinuousMap_comp_symm, ContinuousMap.comp_id]
ext; rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | to_from | null |
@[simps]
loopHomeo (i : N) : Ω^ N X x ≃ₜ Ω (Ω^ { j // j ≠ i } X x) const where
toFun := toLoop i
invFun := fromLoop i
left_inv p := by ext; exact congr_arg p (by dsimp; exact Equiv.apply_symm_apply _ _)
right_inv := to_from i
continuous_toFun := continuous_toLoop i
continuous_invFun := continuous_fromLoop i | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | loopHomeo | The `n+1`-dimensional loops are in bijection with the loops in the space of
`n`-dimensional loops with base point `const`.
We allow an arbitrary indexing type `N` in place of `Fin n` here. |
toLoop_apply (i : N) {p : Ω^ N X x} {t} {tn} :
toLoop i p t tn = p (Cube.insertAt i ⟨t, tn⟩) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | toLoop_apply | null |
fromLoop_apply (i : N) {p : Ω (Ω^ { j // j ≠ i } X x) const} {t : I^N} :
fromLoop i p t = p (t i) (Cube.splitAt i t).snd :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | fromLoop_apply | null |
cCompInsert (i : N) : C(C(I^N, X), C(I × I^{ j // j ≠ i }, X)) :=
⟨fun f => f.comp (Cube.insertAt i),
(toContinuousMap <| Cube.insertAt i).continuous_precomp⟩ | abbrev | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | cCompInsert | Composition with `Cube.insertAt` as a continuous map. |
homotopyTo (i : N) {p q : Ω^ N X x} (H : p.1.HomotopyRel q.1 (Cube.boundary N)) :
C(I × I, C(I^{ j // j ≠ i }, X)) :=
((⟨_, ContinuousMap.continuous_curry⟩ : C(_, _)).comp <|
(cCompInsert i).comp H.toContinuousMap.curry).uncurry | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopyTo | A homotopy between `n+1`-dimensional loops `p` and `q` constant on the boundary
seen as a homotopy between two paths in the space of `n`-dimensional paths. |
homotopyTo_apply (i : N) {p q : Ω^ N X x} (H : p.1.HomotopyRel q.1 <| Cube.boundary N)
(t : I × I) (tₙ : I^{ j // j ≠ i }) :
homotopyTo i H t tₙ = H (t.fst, Cube.insertAt i (t.snd, tₙ)) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopyTo_apply | null |
homotopicTo (i : N) {p q : Ω^ N X x} :
Homotopic p q → (toLoop i p).Homotopic (toLoop i q) := by
refine Nonempty.map fun H => ⟨⟨⟨fun t => ⟨homotopyTo i H t, ?_⟩, ?_⟩, ?_, ?_⟩, ?_⟩
· rintro y ⟨i, iH⟩
rw [homotopyTo_apply, H.eq_fst, p.2]
all_goals apply Cube.insertAt_boundary; right; exact ⟨i, iH⟩
· fun_prop
iterate 2
intro
ext
dsimp
rw [homotopyTo_apply, toLoop_apply]
swap
· apply H.apply_zero
· apply H.apply_one
intro t y yH
ext
dsimp
rw [homotopyTo_apply]
apply H.eq_fst; use i
rw [funSplitAt_symm_apply, dif_pos rfl]; exact yH | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopicTo | null |
@[simps!] homotopyFrom (i : N) {p q : Ω^ N X x} (H : (toLoop i p).Homotopy (toLoop i q)) :
C(I × I^N, X) :=
(ContinuousMap.comp ⟨_, ContinuousMap.continuous_uncurry⟩
(ContinuousMap.comp ⟨Subtype.val, by fun_prop⟩ H.toContinuousMap).curry).uncurry.comp <|
(ContinuousMap.id I).prodMap (Cube.splitAt i) | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopyFrom | The converse to `GenLoop.homotopyTo`: a homotopy between two loops in the space of
`n`-dimensional loops can be seen as a homotopy between two `n+1`-dimensional paths. |
homotopicFrom (i : N) {p q : Ω^ N X x} :
(toLoop i p).Homotopic (toLoop i q) → Homotopic p q := by
refine Nonempty.map fun H => ⟨⟨homotopyFrom i H, ?_, ?_⟩, ?_⟩
pick_goal 3
· rintro t y ⟨j, jH⟩
erw [homotopyFrom_apply]
obtain rfl | h := eq_or_ne j i
· simp only [Prod.map_apply, id_eq, funSplitAt_apply, Function.uncurry_apply_pair]
rw [H.eq_fst]
exacts [congr_arg p ((Cube.splitAt j).left_inv _), jH]
· rw [p.2 _ ⟨j, jH⟩]; apply boundary; exact ⟨⟨j, h⟩, jH⟩
all_goals
intro
apply (homotopyFrom_apply _ _ _).trans
simp only [Prod.map_apply, id_eq, funSplitAt_apply,
Function.uncurry_apply_pair, ContinuousMap.HomotopyWith.apply_zero,
ContinuousMap.HomotopyWith.apply_one, ne_eq, Path.coe_toContinuousMap, toLoop_apply_coe,
ContinuousMap.curry_apply, ContinuousMap.comp_apply]
first
| apply congr_arg p
| apply congr_arg q
apply (Cube.splitAt i).left_inv | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopicFrom | null |
transAt (i : N) (f g : Ω^ N X x) : Ω^ N X x :=
copy (fromLoop i <| (toLoop i f).trans <| toLoop i g)
(fun t => if (t i : ℝ) ≤ 1 / 2
then f (Function.update t i <| Set.projIcc 0 1 zero_le_one (2 * t i))
else g (Function.update t i <| Set.projIcc 0 1 zero_le_one (2 * t i - 1)))
(by
ext1; symm
dsimp only [Path.trans, fromLoop, Path.coe_mk_mk, Function.comp_apply, mk_apply,
ContinuousMap.comp_apply, ContinuousMap.coe_coe, funSplitAt_apply,
ContinuousMap.uncurry_apply, ContinuousMap.coe_mk, Function.uncurry_apply_pair]
split_ifs
· change f _ = _; congr 1
· change g _ = _; congr 1) | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | transAt | Concatenation of two `GenLoop`s along the `i`th coordinate. |
symmAt (i : N) (f : Ω^ N X x) : Ω^ N X x :=
(copy (fromLoop i (toLoop i f).symm) fun t => f fun j => if j = i then σ (t i) else t j) <| by
ext1; change _ = f _; congr; ext1; simp | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | symmAt | Reversal of a `GenLoop` along the `i`th coordinate. |
transAt_distrib {i j : N} (h : i ≠ j) (a b c d : Ω^ N X x) :
transAt i (transAt j a b) (transAt j c d) = transAt j (transAt i a c) (transAt i b d) := by
ext; simp_rw [transAt, coe_copy, Function.update_apply, if_neg h, if_neg h.symm]
split_ifs <;>
· congr 1; ext1; simp only [Function.update, eq_rec_constant, dite_eq_ite]
apply ite_ite_comm; rintro rfl; exact h.symm | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | transAt_distrib | null |
fromLoop_trans_toLoop {i : N} {p q : Ω^ N X x} :
fromLoop i ((toLoop i p).trans <| toLoop i q) = transAt i p q :=
(copy_eq _ _).symm | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | fromLoop_trans_toLoop | null |
fromLoop_symm_toLoop {i : N} {p : Ω^ N X x} : fromLoop i (toLoop i p).symm = symmAt i p :=
(copy_eq _ _).symm | theorem | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | fromLoop_symm_toLoop | null |
HomotopyGroup (N X : Type*) [TopologicalSpace X] (x : X) : Type _ :=
Quotient (GenLoop.Homotopic.setoid N x) | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | HomotopyGroup | The `n`th homotopy group at `x` defined as the quotient of `Ω^n x` by the
`GenLoop.Homotopic` relation. |
homotopyGroupEquivFundamentalGroup (i : N) :
HomotopyGroup N X x ≃ FundamentalGroup (Ω^ { j // j ≠ i } X x) const := by
refine Equiv.trans ?_ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
apply Quotient.congr (loopHomeo i).toEquiv
exact fun p q => ⟨homotopicTo i, homotopicFrom i⟩ | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopyGroupEquivFundamentalGroup | Equivalence between the homotopy group of X and the fundamental group of
`Ω^{j // j ≠ i} x`. |
HomotopyGroup.Pi (n) (X : Type*) [TopologicalSpace X] (x : X) :=
HomotopyGroup (Fin n) _ x
@[inherit_doc] scoped[Topology] notation "π_" => HomotopyGroup.Pi | abbrev | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | HomotopyGroup.Pi | Homotopy group of finite index, denoted as `π_n` within the Topology namespace. |
genLoopHomeoOfIsEmpty (N x) [IsEmpty N] : Ω^ N X x ≃ₜ X where
toFun f := f 0
invFun y := ⟨ContinuousMap.const _ y, fun _ ⟨i, _⟩ => isEmptyElim i⟩
left_inv f := by ext; exact congr_arg f (Subsingleton.elim _ _)
continuous_invFun := ContinuousMap.const'.2.subtype_mk _ | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | genLoopHomeoOfIsEmpty | The 0-dimensional generalized loops based at `x` are in bijection with `X`. |
homotopyGroupEquivZerothHomotopyOfIsEmpty (N x) [IsEmpty N] :
HomotopyGroup N X x ≃ ZerothHomotopy X :=
Quotient.congr (genLoopHomeoOfIsEmpty N x).toEquiv
(by
intro a₁ a₂
constructor <;> rintro ⟨H⟩
exacts
[⟨{ toFun := fun t => H ⟨t, isEmptyElim⟩
source' := (H.apply_zero _).trans (congr_arg a₁ <| Subsingleton.elim _ _)
target' := (H.apply_one _).trans (congr_arg a₂ <| Subsingleton.elim _ _) }⟩,
⟨{ toFun := fun t0 => H t0.fst
map_zero_left := fun _ => H.source.trans (congr_arg a₁ <| Subsingleton.elim _ _)
map_one_left := fun _ => H.target.trans (congr_arg a₂ <| Subsingleton.elim _ _)
prop' := fun _ _ ⟨i, _⟩ => isEmptyElim i }⟩]) | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | homotopyGroupEquivZerothHomotopyOfIsEmpty | The homotopy "group" indexed by an empty type is in bijection with
the path components of `X`, aka the `ZerothHomotopy`. |
HomotopyGroup.pi0EquivZerothHomotopy : π_ 0 X x ≃ ZerothHomotopy X :=
homotopyGroupEquivZerothHomotopyOfIsEmpty (Fin 0) x | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | HomotopyGroup.pi0EquivZerothHomotopy | The 0th homotopy "group" is in bijection with `ZerothHomotopy`. |
genLoopEquivOfUnique (N) [Unique N] : Ω^ N X x ≃ Ω X x where
toFun p :=
Path.mk ⟨fun t => p fun _ => t, by continuity⟩
(GenLoop.boundary _ (fun _ => 0) ⟨default, Or.inl rfl⟩)
(GenLoop.boundary _ (fun _ => 1) ⟨default, Or.inr rfl⟩)
invFun p :=
⟨⟨fun c => p (c default), by continuity⟩,
by
rintro y ⟨i, iH | iH⟩ <;> cases Unique.eq_default i <;> apply (congr_arg p iH).trans
exacts [p.source, p.target]⟩
left_inv p := by ext y; exact congr_arg p (eq_const_of_unique y).symm
/- TODO (?): deducing this from `homotopyGroupEquivFundamentalGroup` would require
combination of `CategoryTheory.Functor.mapAut` and
`FundamentalGroupoid.fundamentalGroupoidFunctor` applied to `genLoopHomeoOfIsEmpty`,
with possibly worse defeq. -/ | def | Topology | [
"Mathlib.Algebra.Group.Ext",
"Mathlib.Algebra.Group.TransferInstance",
"Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup",
"Mathlib.GroupTheory.EckmannHilton"
] | Mathlib/Topology/Homotopy/HomotopyGroup.lean | genLoopEquivOfUnique | The 1-dimensional generalized loops based at `x` are in bijection with loops at `x`. |
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