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 ⌀ |
|---|---|---|---|---|---|---|
forget_reflectsIsomorphisms : (forget LightProfinite).ReflectsIsomorphisms := by
constructor
intro A B f hf
rw [isIso_iff_bijective] at hf
exact LightProfinite.isIso_of_bijective _ hf | instance | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | forget_reflectsIsomorphisms | null |
epi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f := by
constructor
· -- Note: in mathlib3 `contrapose` saw through `Function.Surjective`.
dsimp [Function.Surjective]
contrapose!
rintro ⟨y, hy⟩ hf
let C := Set.range f
have hC : IsClosed C := (isCompact... | theorem | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | epi_iff_surjective | null |
LightDiagram : Type (u+1) where
/-- The indexing diagram. -/
diagram : ℕᵒᵖ ⥤ FintypeCat
/-- The limit cone. -/
cone : Cone (diagram ⋙ FintypeCat.toProfinite.{u})
/-- The limit cone is limiting. -/
isLimit : IsLimit cone | structure | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightDiagram | A structure containing the data of sequential limit in `Profinite` of finite sets. |
toProfinite (S : LightDiagram) : Profinite := S.cone.pt
@[simps!] | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | toProfinite | The underlying `Profinite` of a `LightDiagram`. |
hasForget : ConcreteCategory LightDiagram (fun X Y => C(X.toProfinite, Y.toProfinite)) :=
InducedCategory.concreteCategory toProfinite | instance | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | hasForget | null |
@[simps!]
lightDiagramToProfinite : LightDiagram ⥤ Profinite := inducedFunctor _ | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | lightDiagramToProfinite | The fully faithful embedding `LightDiagram ⥤ Profinite` |
instCountableDiscreteQuotient (S : LightProfinite) :
Countable (DiscreteQuotient ((lightToProfinite.obj S))) :=
(DiscreteQuotient.finsetClopens_inj S).countable | instance | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | instCountableDiscreteQuotient | null |
noncomputable toLightDiagram (S : LightProfinite.{u}) : LightDiagram.{u} where
diagram := IsCofiltered.sequentialFunctor _ ⋙ (lightToProfinite.obj S).fintypeDiagram
cone := (Functor.Initial.limitConeComp (IsCofiltered.sequentialFunctor _)
(lightToProfinite.obj S).lim).cone
isLimit := (Functor.Initial.limitCon... | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | toLightDiagram | A profinite space which is light gives rise to a light profinite space. |
@[simps]
noncomputable lightProfiniteToLightDiagram : LightProfinite.{u} ⥤ LightDiagram.{u} where
obj X := X.toLightDiagram
map f := f
open scoped Classical in | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | lightProfiniteToLightDiagram | The functor part of the equivalence `LightProfinite ≌ LightDiagram` |
@[simps obj map]
lightDiagramToLightProfinite : LightDiagram.{u} ⥤ LightProfinite.{u} where
obj X := LightProfinite.of X.cone.pt
map f := f | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | lightDiagramToLightProfinite | The inverse part of the equivalence `LightProfinite ≌ LightDiagram` |
noncomputable LightProfinite.equivDiagram : LightProfinite.{u} ≌ LightDiagram.{u} where
functor := lightProfiniteToLightDiagram
inverse := lightDiagramToLightProfinite
unitIso := Iso.refl _
counitIso := NatIso.ofComponents
(fun _ ↦ lightDiagramToProfinite.preimageIso (Iso.refl _)) (by
intro _ _ f
... | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightProfinite.equivDiagram | The equivalence of categories `LightProfinite ≌ LightDiagram` |
LightDiagram' : Type u where
/-- The diagram takes values in a small category equivalent to `FintypeCat`. -/
diagram : ℕᵒᵖ ⥤ FintypeCat.Skeleton.{u} | structure | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightDiagram' | This is an auxiliary definition used to show that `LightDiagram` is essentially small.
Note that below we put a category instance on this structure which is completely different from the
category instance on `ℕᵒᵖ ⥤ FintypeCat.Skeleton.{u}`. Neither the morphisms nor the objects are the
same. |
LightDiagram'.toProfinite (S : LightDiagram') : Profinite :=
limit (S.diagram ⋙ FintypeCat.Skeleton.equivalence.functor ⋙ FintypeCat.toProfinite.{u}) | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightDiagram'.toProfinite | A `LightDiagram'` yields a `Profinite`. |
LightDiagram'.toLightFunctor : LightDiagram'.{u} ⥤ LightDiagram.{u} where
obj X := ⟨X.diagram ⋙ Skeleton.equivalence.functor, _, limit.isLimit _⟩
map f := f | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightDiagram'.toLightFunctor | The functor part of the equivalence of categories `LightDiagram' ≌ LightDiagram`. |
LightDiagram.equivSmall : LightDiagram.{u} ≌ LightDiagram'.{u} :=
LightDiagram'.toLightFunctor.asEquivalence.symm | def | Topology | [
"Mathlib.CategoryTheory.Limits.Shapes.Countable",
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.Topology.ClopenBox"
] | Mathlib/Topology/Category/LightProfinite/Basic.lean | LightDiagram.equivSmall | The equivalence between `LightDiagram` and a small category. |
effectiveEpi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
EffectiveEpi f ↔ Function.Surjective f := by
refine ⟨fun h ↦ ?_, fun h ↦ ⟨⟨effectiveEpiStruct f h⟩⟩⟩
rw [← epi_iff_surjective]
infer_instance | theorem | Topology | [
"Mathlib.Topology.Category.CompHausLike.EffectiveEpi",
"Mathlib.Topology.Category.LightProfinite.Limits"
] | Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean | effectiveEpi_iff_surjective | null |
@[simps]
functor : ℕᵒᵖ ⥤ StructuredArrow c.pt toLightProfinite where
obj i := StructuredArrow.mk (c.π.app i)
map f := StructuredArrow.homMk (F.map f) (c.w f) | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | functor | Given a sequential cone in `LightProfinite` consisting of finite sets,
we obtain a functor from the indexing category to `StructuredArrow c.pt toLightProfinite`. |
@[simps! obj map]
functorOp : ℕ ⥤ CostructuredArrow toLightProfinite.op ⟨c.pt⟩ :=
(functor c).rightOp ⋙ StructuredArrow.toCostructuredArrow _ _ | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | functorOp | Given a sequential cone in `LightProfinite` consisting of finite sets,
we obtain a functor from the opposite of the indexing category to
`CostructuredArrow toProfinite.op ⟨c.pt⟩`. |
functor_initial (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Initial (functor c) := by
rw [initial_iff_comp_equivalence _ (StructuredArrow.post _ _ lightToProfinite)]
have : ∀ i, Epi ((lightToProfinite.mapCone c).π.app i) :=
fun i ↦ inferInstanceAs (Epi (lightToProfinite.map (c.π.app i)))
exact Profinite.Extend.... | theorem | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | functor_initial | If the projection maps in the cone are epimorphic and the cone is limiting, then
`LightProfinite.Extend.functor` is initial. |
functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) := by
have := functor_initial c hc
have : ((StructuredArrow.toCostructuredArrow toLightProfinite c.pt)).IsEquivalence :=
(inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence )
have : (functor c).rightOp.Final :... | theorem | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | functorOp_final | If the projection maps in the cone are epimorphic and the cone is limiting, then
`LightProfinite.Extend.functorOp` is final. |
cone (S : LightProfinite) :
Cone (StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G) where
pt := G.obj S
π := {
app := fun i ↦ G.map i.hom
naturality := fun _ _ f ↦ (by
have := f.w
simp only [const_obj_obj, StructuredArrow.left_eq_id, const_obj_map, Category.id_comp,
Str... | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | cone | Given a functor `G` from `LightProfinite` and `S : LightProfinite`, we obtain a cone on
`(StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G)` with cone point `G.obj S`.
Whiskering this cone with `LightProfinite.Extend.functor c` gives `G.mapCone c` as we check in the
example below. |
noncomputable
isLimitCone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsLimit <| G.mapCone c) :
IsLimit (cone G c.pt) := (functor_initial c hc).isLimitWhiskerEquiv _ _ hc' | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | isLimitCone | If `c` and `G.mapCone c` are limit cones and the projection maps in `c` are epimorphic,
then `cone G c.pt` is a limit cone. |
@[simps]
cocone (S : LightProfinite) :
Cocone (CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G) where
pt := G.obj ⟨S⟩
ι := {
app := fun i ↦ G.map i.hom
naturality := fun _ _ f ↦ (by
have := f.w
simp only [op_obj, const_obj_obj, op_map, CostructuredArrow.right_eq_id, ... | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | cocone | Given a functor `G` from `LightProfiniteᵒᵖ` and `S : LightProfinite`, we obtain a cocone on
`(CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G)` with cocone point
`G.obj ⟨S⟩`.
Whiskering this cocone with `LightProfinite.Extend.functorOp c` gives `G.mapCocone c.op` as we
check in the example bel... |
noncomputable
isColimitCocone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsColimit <| G.mapCocone c.op) :
IsColimit (cocone G c.pt) :=
haveI := functorOp_final c hc
(Functor.final_comp (opOpEquivalence ℕ).functor (functorOp c)).isColimitWhiskerEquiv _ _ hc' | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | isColimitCocone | If `c` is a limit cone, `G.mapCocone c.op` is a colimit cone and the projection maps in `c`
are epimorphic, then `cocone G c.pt` is a colimit cone. |
fintypeDiagram' : StructuredArrow S toLightProfinite ⥤ FintypeCat :=
StructuredArrow.proj S toLightProfinite | abbrev | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | fintypeDiagram' | A functor `StructuredArrow S toLightProfinite ⥤ FintypeCat` whose limit in `LightProfinite` is
isomorphic to `S`. |
diagram' : StructuredArrow S toLightProfinite ⥤ LightProfinite :=
S.fintypeDiagram' ⋙ toLightProfinite | abbrev | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | diagram' | An abbreviation for `S.fintypeDiagram' ⋙ toLightProfinite`. |
asLimitCone' : Cone (S.diagram') := cone (𝟭 _) S | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | asLimitCone' | A cone over `S.diagram'` whose cone point is `S`. |
noncomputable asLimit' : IsLimit S.asLimitCone' := isLimitCone _ (𝟭 _) S.asLimit S.asLimit | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | asLimit' | `S.asLimitCone'` is a limit cone. |
noncomputable lim' : LimitCone S.diagram' := ⟨S.asLimitCone', S.asLimit'⟩ | def | Topology | [
"Mathlib.Topology.Category.LightProfinite.AsLimit",
"Mathlib.Topology.Category.Profinite.Extend"
] | Mathlib/Topology/Category/LightProfinite/Extend.lean | lim' | A bundled version of `S.asLimitCone'` and `S.asLimit'`. |
isTerminalPUnit : IsTerminal (LightProfinite.of PUnit.{u + 1}) :=
CompHausLike.isTerminalPUnit | abbrev | Topology | [
"Mathlib.Topology.Category.CompHausLike.Limits",
"Mathlib.Topology.Category.LightProfinite.Basic"
] | Mathlib/Topology/Category/LightProfinite/Limits.lean | isTerminalPUnit | A one-element space is terminal in `Profinite` |
noncomputable natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where
toFun
| ∞ => 0
| OnePoint.some n => 1 / (n + 1 : ℝ)
continuous_toFun := OnePoint.continuous_iff_from_nat _ |>.mpr
tendsto_one_div_add_atTop_nhds_zero_nat | def | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Category.LightProfinite.Basic"
] | Mathlib/Topology/Category/LightProfinite/Sequence.lean | natUnionInftyEmbedding | The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. |
isClosedEmbedding_natUnionInftyEmbedding : IsClosedEmbedding natUnionInftyEmbedding := by
refine .of_continuous_injective_isClosedMap
natUnionInftyEmbedding.continuous ?_ ?_
· rintro (_ | n) (_ | m) h
· rfl
· simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, zero_eq_inv] at h
assum... | lemma | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Category.LightProfinite.Basic"
] | Mathlib/Topology/Category/LightProfinite/Sequence.lean | isClosedEmbedding_natUnionInftyEmbedding | The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed
embedding. |
NatUnionInfty : LightProfinite := of (OnePoint ℕ)
@[inherit_doc]
scoped notation "ℕ∪{∞}" => NatUnionInfty | abbrev | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Category.LightProfinite.Basic"
] | Mathlib/Topology/Category/LightProfinite/Sequence.lean | NatUnionInfty | The one point compactification of the natural numbers as a light profinite set. |
continuous_iff_convergent {Y : Type*} [TopologicalSpace Y] (f : ℕ∪{∞} → Y) :
Continuous f ↔ Tendsto (fun x : ℕ ↦ f x) atTop (𝓝 (f ∞)) :=
continuous_iff_from_nat f | lemma | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Topology.Category.LightProfinite.Basic"
] | Mathlib/Topology/Category/LightProfinite/Sequence.lean | continuous_iff_convergent | null |
fintypeDiagram : DiscreteQuotient X ⥤ FintypeCat where
obj S := @FintypeCat.of S (Fintype.ofFinite S)
map f := DiscreteQuotient.ofLE f.le
map_comp _ _ := by funext; cat_disch | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | fintypeDiagram | The functor `DiscreteQuotient X ⥤ Fintype` whose limit is isomorphic to `X`. |
diagram : DiscreteQuotient X ⥤ Profinite :=
X.fintypeDiagram ⋙ FintypeCat.toProfinite | abbrev | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | diagram | An abbreviation for `X.fintypeDiagram ⋙ FintypeCat.toProfinite`. |
asLimitCone : CategoryTheory.Limits.Cone X.diagram :=
{ pt := X
π := { app := fun S => CompHausLike.ofHom (Y := X.diagram.obj S) _
⟨S.proj, IsLocallyConstant.continuous (S.proj_isLocallyConstant)⟩ } } | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | asLimitCone | A cone over `X.diagram` whose cone point is `X`. |
isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
CompHausLike.isIso_of_bijective _
(by
refine ⟨fun a b h => ?_, fun a => ?_⟩
· refine DiscreteQuotient.eq_of_forall_proj_eq fun S => ?_
apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
... | instance | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | isIso_asLimitCone_lift | null |
isoAsLimitConeLift : X ≅ (limitCone.{u, u} X.diagram).pt :=
asIso <| (limitConeIsLimit.{u, u} _).lift X.asLimitCone | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | isoAsLimitConeLift | The isomorphism between `X` and the explicit limit of `X.diagram`,
induced by lifting `X.asLimitCone`. |
asLimitConeIso : X.asLimitCone ≅ limitCone.{u, u} _ :=
Limits.Cones.ext (isoAsLimitConeLift _) fun _ => rfl | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | asLimitConeIso | The isomorphism of cones `X.asLimitCone` and `Profinite.limitCone X.diagram`.
The underlying isomorphism is defeq to `X.isoAsLimitConeLift`. |
asLimit : CategoryTheory.Limits.IsLimit X.asLimitCone :=
Limits.IsLimit.ofIsoLimit (limitConeIsLimit _) X.asLimitConeIso.symm | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | asLimit | `X.asLimitCone` is indeed a limit cone. |
lim : Limits.LimitCone X.diagram :=
⟨X.asLimitCone, X.asLimit⟩ | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.DiscreteQuotient"
] | Mathlib/Topology/Category/Profinite/AsLimit.lean | lim | A bundled version of `X.asLimitCone` and `X.asLimit`. |
and is a fully faithful subcategory of `TopCat`. The fully faithful functor
is called `Profinite.toTop`. | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | and | null |
Profinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X) | abbrev | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | Profinite | The type of profinite topological spaces. |
of (X : Type*) [TopologicalSpace X] [CompactSpace X] [T2Space X]
[TotallyDisconnectedSpace X] : Profinite :=
CompHausLike.of _ X | abbrev | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | of | Construct a term of `Profinite` from a type endowed with the structure of a
compact, Hausdorff and totally disconnected topological space. |
profiniteToCompHaus : Profinite ⥤ CompHaus :=
compHausLikeToCompHaus _ | abbrev | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | profiniteToCompHaus | The fully faithful embedding of `Profinite` in `CompHaus`. |
Profinite.toTopCat : Profinite ⥤ TopCat :=
CompHausLike.compHausLikeToTop _ | abbrev | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | Profinite.toTopCat | The fully faithful embedding of `Profinite` in `TopCat`.
This is definitionally the same as the obvious composite. |
@[stacks 0900]
CompHaus.toProfiniteObj (X : CompHaus.{u}) : Profinite.{u} where
toTop := TopCat.of (ConnectedComponents X)
is_compact := Quotient.compactSpace
is_hausdorff := ConnectedComponents.t2
prop := ConnectedComponents.totallyDisconnectedSpace | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | CompHaus.toProfiniteObj | (Implementation) The object part of the connected_components functor from compact Hausdorff spaces
to Profinite spaces, given by quotienting a space by its connected components. |
Profinite.toCompHausEquivalence (X : CompHaus.{u}) (Y : Profinite.{u}) :
(CompHaus.toProfiniteObj X ⟶ Y) ≃ (X ⟶ profiniteToCompHaus.obj Y) where
toFun f := ofHom _ (f.hom.comp ⟨Quotient.mk'', continuous_quotient_mk'⟩)
invFun g := TopCat.ofHom
{ toFun := Continuous.connectedComponentsLift g.hom.2
conti... | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | Profinite.toCompHausEquivalence | (Implementation) The bijection of homsets to establish the reflective adjunction of Profinite
spaces in compact Hausdorff spaces. |
CompHaus.toProfinite : CompHaus ⥤ Profinite :=
Adjunction.leftAdjointOfEquiv Profinite.toCompHausEquivalence fun _ _ _ _ _ => rfl | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | CompHaus.toProfinite | The connected_components functor from compact Hausdorff spaces to profinite spaces,
left adjoint to the inclusion functor. |
CompHaus.toProfinite_obj' (X : CompHaus) :
↥(CompHaus.toProfinite.obj X) = ConnectedComponents X :=
rfl | theorem | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | CompHaus.toProfinite_obj' | null |
FintypeCat.botTopology (A : FintypeCat) : TopologicalSpace A := ⊥ | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | FintypeCat.botTopology | Finite types are given the discrete topology. |
FintypeCat.discreteTopology (A : FintypeCat) : DiscreteTopology A :=
⟨rfl⟩
attribute [local instance] FintypeCat.discreteTopology | theorem | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | FintypeCat.discreteTopology | null |
@[simps! -isSimp map_hom_apply]
FintypeCat.toProfinite : FintypeCat ⥤ Profinite where
obj A := Profinite.of A
map f := ofHom _ ⟨f, by continuity⟩ | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | FintypeCat.toProfinite | The natural functor from `Fintype` to `Profinite`, endowing a finite type with the
discrete topology. |
FintypeCat.toProfiniteFullyFaithful : toProfinite.FullyFaithful where
preimage f := (f : _ → _)
map_preimage _ := rfl
preimage_map _ := rfl | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | FintypeCat.toProfiniteFullyFaithful | `FintypeCat.toLightProfinite` is fully faithful. |
limitCone {J : Type v} [SmallCategory J] (F : J ⥤ Profinite.{max u v}) : Limits.Cone F where
pt :=
{ toTop := (CompHaus.limitCone.{v, u} (F ⋙ profiniteToCompHaus)).pt.toTop
prop := by
change TotallyDisconnectedSpace ({ u : ∀ j : J, F.obj j | _ } : Type _)
exact Subtype.totallyDisconnectedSpa... | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | limitCone | An explicit limit cone for a functor `F : J ⥤ Profinite`, defined in terms of
`CompHaus.limitCone`, which is defined in terms of `TopCat.limitCone`. |
limitConeIsLimit {J : Type v} [SmallCategory J] (F : J ⥤ Profinite.{max u v}) :
Limits.IsLimit (limitCone F) where
lift S :=
(CompHaus.limitConeIsLimit.{v, u} (F ⋙ profiniteToCompHaus)).lift
(profiniteToCompHaus.mapCone S)
uniq S _ h := (CompHaus.limitConeIsLimit.{v, u} _).uniq (profiniteToCompHaus.ma... | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | limitConeIsLimit | The limit cone `Profinite.limitCone F` is indeed a limit cone. |
toProfiniteAdjToCompHaus : CompHaus.toProfinite ⊣ profiniteToCompHaus :=
Adjunction.adjunctionOfEquivLeft _ _ | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | toProfiniteAdjToCompHaus | The adjunction between CompHaus.to_Profinite and Profinite.to_CompHaus |
toCompHaus.reflective : Reflective profiniteToCompHaus where
L := CompHaus.toProfinite
adj := Profinite.toProfiniteAdjToCompHaus | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | toCompHaus.reflective | The category of profinite sets is reflective in the category of compact Hausdorff spaces |
noncomputable toCompHaus.createsLimits : CreatesLimits profiniteToCompHaus :=
monadicCreatesLimits _ | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | toCompHaus.createsLimits | null |
noncomputable toTopCat.reflective : Reflective Profinite.toTopCat :=
Reflective.comp profiniteToCompHaus compHausToTop | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | toTopCat.reflective | null |
noncomputable toTopCat.createsLimits : CreatesLimits Profinite.toTopCat :=
monadicCreatesLimits _ | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | toTopCat.createsLimits | null |
hasLimits : Limits.HasLimits Profinite :=
hasLimits_of_hasLimits_createsLimits Profinite.toTopCat | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | hasLimits | null |
hasColimits : Limits.HasColimits Profinite :=
hasColimits_of_reflective profiniteToCompHaus | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | hasColimits | null |
forget_preservesLimits : Limits.PreservesLimits (forget Profinite) := by
apply Limits.comp_preservesLimits Profinite.toTopCat (forget TopCat) | instance | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | forget_preservesLimits | null |
epi_iff_surjective {X Y : Profinite.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by
constructor
· dsimp [Function.Surjective]
contrapose!
rintro ⟨y, hy⟩ hf
let C := Set.range f
have hC : IsClosed C := (isCompact_range f.hom.continuous).isClosed
let U := Cᶜ
have hyU : y ∈ U := by
... | theorem | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | epi_iff_surjective | null |
pi {α : Type u} (β : α → Profinite) : Profinite := .of (Π (a : α), β a) | def | Topology | [
"Mathlib.CategoryTheory.FintypeCat",
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.Separation.Profinite"
] | Mathlib/Topology/Category/Profinite/Basic.lean | pi | The pi-type of profinite spaces is profinite. |
exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V := by
have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, v} (F ⋙ Profinite.toTopCat)
(Profinite.toTopCat.mapCone C) (isLimitOfPreserves _ hC) (fun j => {W ... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.DiscreteQuotient",
"Mathlib.Topology.Category.TopCat.Limits.Cofiltered",
"Mathlib.Topology.Category.TopCat.Limits.Konig"
] | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | exists_isClopen_of_cofiltered | If `X` is a cofiltered limit of profinite sets, then any clopen subset of `X` arises from
a clopen set in one of the terms in the limit. |
exists_locallyConstant_fin_two (hC : IsLimit C) (f : LocallyConstant C.pt (Fin 2)) :
∃ (j : J) (g : LocallyConstant (F.obj j) (Fin 2)), f = g.comap (C.π.app _).hom := by
let U := f ⁻¹' {0}
have hU : IsClopen U := f.isLocallyConstant.isClopen_fiber _
obtain ⟨j, V, hV, h⟩ := exists_isClopen_of_cofiltered C hC h... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.DiscreteQuotient",
"Mathlib.Topology.Category.TopCat.Limits.Cofiltered",
"Mathlib.Topology.Category.TopCat.Limits.Konig"
] | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | exists_locallyConstant_fin_two | null |
exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom := by
cases nonempty_fintype α
let ι : α → α → Fin 2 := fun x y => i... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.DiscreteQuotient",
"Mathlib.Topology.Category.TopCat.Limits.Cofiltered",
"Mathlib.Topology.Category.TopCat.Limits.Konig"
] | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | exists_locallyConstant_finite_aux | null |
exists_locallyConstant_finite_nonempty {α : Type*} [Finite α] [Nonempty α]
(hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom := by
inhabit α
obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f
classical
let ι : α → α → Fin ... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.DiscreteQuotient",
"Mathlib.Topology.Category.TopCat.Limits.Cofiltered",
"Mathlib.Topology.Category.TopCat.Limits.Konig"
] | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | exists_locallyConstant_finite_nonempty | null |
exists_locallyConstant {α : Type*} (hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom := by
let S := f.discreteQuotient
let ff : S → α := f.lift
cases isEmpty_or_nonempty S
· suffices ∃ j, IsEmpty (F.obj j) by
refine this.imp fun j ... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.LocallyConstant.Basic",
"Mathlib.Topology.DiscreteQuotient",
"Mathlib.Topology.Category.TopCat.Limits.Cofiltered",
"Mathlib.Topology.Category.TopCat.Limits.Konig"
] | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | exists_locallyConstant | Any locally constant function from a cofiltered limit of profinite sets factors through
one of the components. |
effectiveEpi_tfae
{B X : Profinite.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 2 ↔ 3 := epi_iff_surjective π
tfae_have 3 → 1 := fun hπ ↦ ⟨⟨CompHausLike.effectiveEpiStruct π hπ⟩⟩
tfae_finish | theorem | Topology | [
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Profinite.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | effectiveEpi_tfae | null |
noncomputable profiniteToCompHausEffectivePresentation (X : CompHaus) :
profiniteToCompHaus.EffectivePresentation X where
p := Stonean.toProfinite.obj X.presentation
f := CompHaus.presentation.π X
effectiveEpi := ((CompHaus.effectiveEpi_tfae _).out 0 1).mpr (inferInstance : Epi _) | def | Topology | [
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Profinite.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | profiniteToCompHausEffectivePresentation | An effective presentation of an `X : Profinite` with respect to the inclusion functor from `Stonean` |
effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : Profinite.{u}}
(X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by
tfae_have 2 → 1
| _ => by
simpa [← effectiveEpi_desc_iff_effectiv... | theorem | Topology | [
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Profinite.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | effectiveEpiFamily_tfae | null |
effectiveEpiFamily_of_jointly_surjective
{α : Type} [Finite α] {B : Profinite.{u}}
(X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B))
(surj : ∀ b : B, ∃ (a : α) (x : X a), π a x = b) :
EffectiveEpiFamily X π :=
((effectiveEpiFamily_tfae X π).out 2 0).mp surj | theorem | Topology | [
"Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular",
"Mathlib.Topology.Category.CompHaus.EffectiveEpi",
"Mathlib.Topology.Category.Profinite.Limits",
"Mathlib.Topology.Category.Stonean.Basic"
] | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | effectiveEpiFamily_of_jointly_surjective | null |
exists_hom (hc : IsLimit c) {X : FintypeCat} (f : c.pt ⟶ toProfinite.obj X) :
∃ (i : I) (g : F.obj i ⟶ X), f = c.π.app i ≫ toProfinite.map g := by
let _ : TopologicalSpace X := ⊥
have : DiscreteTopology (toProfinite.obj X) := ⟨rfl⟩
let f' : LocallyConstant c.pt (toProfinite.obj X) :=
⟨f, (IsLocallyConstan... | lemma | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | exists_hom | A continuous map from a profinite set to a finite set factors through one of the components of
the profinite set when written as a cofiltered limit of finite sets. |
@[simps]
functor : I ⥤ StructuredArrow c.pt toProfinite where
obj i := StructuredArrow.mk (c.π.app i)
map f := StructuredArrow.homMk (F.map f) (c.w f) | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | functor | Given a cone in `Profinite`, consisting of finite sets and indexed by a cofiltered category,
we obtain a functor from the indexing category to `StructuredArrow c.pt toProfinite`. |
@[simps! obj map]
functorOp : Iᵒᵖ ⥤ CostructuredArrow toProfinite.op ⟨c.pt⟩ :=
(functor c).op ⋙ StructuredArrow.toCostructuredArrow _ _ | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | functorOp | Given a cone in `Profinite`, consisting of finite sets and indexed by a cofiltered category,
we obtain a functor from the opposite of the indexing category to
`CostructuredArrow toProfinite.op ⟨c.pt⟩`. |
functor_initial (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Initial (functor c) := by
let e : I ≌ ULiftHom.{w} (ULift.{w} I) := ULiftHomULiftCategory.equiv _
suffices (e.inverse ⋙ functor c).Initial from initial_of_equivalence_comp e.inverse (functor c)
rw [initial_iff_of_isCofiltered (F := e.inverse ⋙ functor c)]
... | lemma | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | functor_initial | If the projection maps in the cone are epimorphic and the cone is limiting, then
`Profinite.Extend.functor` is initial.
TODO: investigate how to weaken the assumption `∀ i, Epi (c.π.app i)` to
`∀ i, ∃ j (_ : j ⟶ i), Epi (c.π.app j)`. |
functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) := by
have := functor_initial c hc
have : ((StructuredArrow.toCostructuredArrow toProfinite c.pt)).IsEquivalence :=
(inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence )
exact Functor.final_comp (functor c).op... | lemma | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | functorOp_final | If the projection maps in the cone are epimorphic and the cone is limiting, then
`Profinite.Extend.functorOp` is final. |
@[simps]
cone (S : Profinite) :
Cone (StructuredArrow.proj S toProfinite ⋙ toProfinite ⋙ G) where
pt := G.obj S
π := {
app := fun i ↦ G.map i.hom
naturality := fun _ _ f ↦ (by simp [← map_comp]) } | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | cone | Given a functor `G` from `Profinite` and `S : Profinite`, we obtain a cone on
`(StructuredArrow.proj S toProfinite ⋙ toProfinite ⋙ G)` with cone point `G.obj S`.
Whiskering this cone with `Profinite.Extend.functor c` gives `G.mapCone c` as we check in the
example below. |
noncomputable
isLimitCone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsLimit <| G.mapCone c) :
IsLimit (cone G c.pt) := (functor_initial c hc).isLimitWhiskerEquiv _ _ hc' | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | isLimitCone | If `c` and `G.mapCone c` are limit cones and the projection maps in `c` are epimorphic,
then `cone G c.pt` is a limit cone. |
@[simps]
cocone (S : Profinite) :
Cocone (CostructuredArrow.proj toProfinite.op ⟨S⟩ ⋙ toProfinite.op ⋙ G) where
pt := G.obj ⟨S⟩
ι := {
app := fun i ↦ G.map i.hom
naturality := fun _ _ f ↦ (by
have := f.w
simp only [op_obj, const_obj_obj, op_map, CostructuredArrow.right_eq_id, const_obj_map,
... | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | cocone | Given a functor `G` from `Profiniteᵒᵖ` and `S : Profinite`, we obtain a cocone on
`(CostructuredArrow.proj toProfinite.op ⟨S⟩ ⋙ toProfinite.op ⋙ G)` with cocone point `G.obj ⟨S⟩`.
Whiskering this cocone with `Profinite.Extend.functorOp c` gives `G.mapCocone c.op` as we check in
the example below. |
noncomputable
isColimitCocone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsColimit <| G.mapCocone c.op) :
IsColimit (cocone G c.pt) := (functorOp_final c hc).isColimitWhiskerEquiv _ _ hc' | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | isColimitCocone | If `c` is a limit cone, `G.mapCocone c.op` is a colimit cone and the projection maps in `c`
are epimorphic, then `cocone G c.pt` is a colimit cone. |
fintypeDiagram' : StructuredArrow S toProfinite ⥤ FintypeCat :=
StructuredArrow.proj S toProfinite | abbrev | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | fintypeDiagram' | A functor `StructuredArrow S toProfinite ⥤ FintypeCat` whose limit in `Profinite` is isomorphic
to `S`. |
diagram' : StructuredArrow S toProfinite ⥤ Profinite :=
S.fintypeDiagram' ⋙ toProfinite | abbrev | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | diagram' | An abbreviation for `S.fintypeDiagram' ⋙ toProfinite`. |
asLimitCone' : Cone (S.diagram') := cone (𝟭 _) S | abbrev | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | asLimitCone' | A cone over `S.diagram'` whose cone point is `S`. |
noncomputable asLimit' : IsLimit S.asLimitCone' := isLimitCone _ (𝟭 _) S.asLimit S.asLimit | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | asLimit' | `S.asLimitCone'` is a limit cone. |
noncomputable lim' : LimitCone S.diagram' := ⟨S.asLimitCone', S.asLimit'⟩ | def | Topology | [
"Mathlib.Topology.Category.Profinite.AsLimit",
"Mathlib.Topology.Category.Profinite.CofilteredLimit",
"Mathlib.CategoryTheory.Filtered.Final"
] | Mathlib/Topology/Category/Profinite/Extend.lean | lim' | A bundled version of `S.asLimitCone'` and `S.asLimit'`. |
isTerminalPUnit : IsTerminal (Profinite.of PUnit.{u + 1}) := CompHausLike.isTerminalPUnit | abbrev | Topology | [
"Mathlib.Topology.Category.Profinite.Basic",
"Mathlib.Topology.Category.CompHausLike.Limits"
] | Mathlib/Topology/Category/Profinite/Limits.lean | isTerminalPUnit | A one-element space is terminal in `Profinite` |
obj : Set ((i : {i : ι // J i}) → X i) := ContinuousMap.precomp (Subtype.val (p := J)) '' C | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | obj | The object part of the functor `indexFunctor : (Finset ι)ᵒᵖ ⥤ Profinite`. |
π_app : C(C, obj C J) :=
⟨Set.MapsTo.restrict (precomp (Subtype.val (p := J))) _ _ (Set.mapsTo_image _ _),
Continuous.restrict _ (Pi.continuous_precomp' _)⟩
variable {J K} | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | π_app | The projection maps in the limit cone `indexCone`. |
map (h : ∀ i, J i → K i) : C(obj C K, obj C J) :=
⟨Set.MapsTo.restrict (precomp (Set.inclusion h)) _ _ (fun _ hx ↦ by
obtain ⟨y, hy⟩ := hx
rw [← hy.2]
exact ⟨y, hy.1, rfl⟩), Continuous.restrict _ (Pi.continuous_precomp' _)⟩ | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | map | The morphism part of the functor `indexFunctor : (Finset ι)ᵒᵖ ⥤ Profinite`. |
surjective_π_app :
Function.Surjective (π_app C J) := by
intro x
obtain ⟨y, hy⟩ := x.prop
exact ⟨⟨y, hy.1⟩, Subtype.ext hy.2⟩ | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | surjective_π_app | null |
map_comp_π_app (h : ∀ i, J i → K i) : map C h ∘ π_app C K = π_app C J := rfl
variable {C} | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | map_comp_π_app | null |
eq_of_forall_π_app_eq (a b : C)
(h : ∀ (J : Finset ι), π_app C (· ∈ J) a = π_app C (· ∈ J) b) : a = b := by
ext i
specialize h ({i} : Finset ι)
rw [Subtype.ext_iff] at h
simp only [π_app, ContinuousMap.precomp, ContinuousMap.coe_mk,
Set.MapsTo.val_restrict_apply] at h
exact congr_fun h ⟨i, Finset.mem_... | theorem | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | eq_of_forall_π_app_eq | null |
noncomputable
indexFunctor (hC : IsCompact C) : (Finset ι)ᵒᵖ ⥤ Profinite.{u} where
obj J := @Profinite.of (obj C (· ∈ (unop J))) _
(by rw [← isCompact_iff_compactSpace]; exact hC.image (Pi.continuous_precomp' _)) _ _
map h := TopCat.ofHom (map C (leOfHom h.unop)) | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | indexFunctor | The functor from the poset of finsets of `ι` to `Profinite`, indexing the limit. |
noncomputable
indexCone (hC : IsCompact C) : Cone (indexFunctor hC) where
pt := @Profinite.of C _ (by rwa [← isCompact_iff_compactSpace]) _ _
π := { app := fun J ↦ TopCat.ofHom (π_app C (· ∈ unop J)) }
variable (hC : IsCompact C) | def | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | indexCone | The limit cone on `indexFunctor` |
isIso_indexCone_lift :
IsIso ((limitConeIsLimit.{u, u} (indexFunctor hC)).lift (indexCone hC)) :=
haveI : CompactSpace C := by rwa [← isCompact_iff_compactSpace]
CompHausLike.isIso_of_bijective _
(by
refine ⟨fun a b h ↦ ?_, fun a ↦ ?_⟩
· refine eq_of_forall_π_app_eq a b (fun J ↦ ?_)
appl... | instance | Topology | [
"Mathlib.Topology.Category.Profinite.Basic"
] | Mathlib/Topology/Category/Profinite/Product.lean | isIso_indexCone_lift | null |
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