Profinite sets as limits of finite sets.

We show that any profinite set is isomorphic to the limit of its discrete (hence finite) quotients.

Definitions

There are a handful of definitions in this file, given X : Profinite:

1. X.fintypeDiagram is the functor DiscreteQuotient X ⥤ FintypeCat whose limit is isomorphic to X (the limit taking place in Profinite via FintypeCat.toProfinite, see 2).
2. X.diagram is an abbreviation for X.fintypeDiagram ⋙ FintypeCat.toProfinite.
3. X.asLimitCone is the cone over X.diagram whose cone point is X.
4. X.isoAsLimitConeLift is the isomorphism X ≅ (Profinite.limitCone X.diagram).X induced by lifting X.asLimitCone.
5. X.asLimitConeIso is the isomorphism X.asLimitCone ≅ (Profinite.limitCone X.diagram) induced by X.isoAsLimitConeLift.
6. X.asLimit is a term of type IsLimit X.asLimitCone.
7. X.lim : CategoryTheory.Limits.LimitCone X.asLimitCone is a bundled combination of 3 and 6.

def Profinite.fintypeDiagram (X : Profinite) : CategoryTheory.Functor (DiscreteQuotient ↑X.toCompHaus.toTop) FintypeCat

The functor DiscreteQuotient X ⥤ Fintype whose limit is isomorphic to X.

Equations

• One or more equations did not get rendered due to their size.

@[reducible, inline]

abbrev Profinite.diagram (X : Profinite) : CategoryTheory.Functor (DiscreteQuotient ↑X.toCompHaus.toTop) Profinite

An abbreviation for X.fintypeDiagram ⋙ FintypeCat.toProfinite.

Equations

• X.diagram = X.fintypeDiagram.comp FintypeCat.toProfinite

def Profinite.asLimitCone (X : Profinite) : CategoryTheory.Limits.Cone X.diagram

A cone over X.diagram whose cone point is X.

Equations

• X.asLimitCone = { pt := X, π := { app := fun (S : DiscreteQuotient ↑X.toCompHaus.toTop) => { toFun := S.proj, continuous_toFun := ⋯ }, naturality := ⋯ } }

instance Profinite.isIso_asLimitCone_lift (X : Profinite) : CategoryTheory.IsIso ((Profinite.limitConeIsLimit X.diagram).lift X.asLimitCone)

Equations

• ⋯ = ⋯

def Profinite.isoAsLimitConeLift (X : Profinite) : X ≅ (Profinite.limitCone X.diagram).pt

The isomorphism between X and the explicit limit of X.diagram, induced by lifting X.asLimitCone.

Equations

• X.isoAsLimitConeLift = CategoryTheory.asIso ((Profinite.limitConeIsLimit X.diagram).lift X.asLimitCone)

def Profinite.asLimitConeIso (X : Profinite) : X.asLimitCone ≅ Profinite.limitCone X.diagram

The isomorphism of cones X.asLimitCone and Profinite.limitCone X.diagram. The underlying isomorphism is defeq to X.isoAsLimitConeLift.

Equations

• X.asLimitConeIso = CategoryTheory.Limits.Cones.ext X.asLimitCone.pt.isoAsLimitConeLift ⋯

def Profinite.asLimit (X : Profinite) : CategoryTheory.Limits.IsLimit X.asLimitCone

X.asLimitCone is indeed a limit cone.

Equations

• X.asLimit = (Profinite.limitConeIsLimit X.diagram).ofIsoLimit X.asLimitConeIso.symm

def Profinite.lim (X : Profinite) : CategoryTheory.Limits.LimitCone X.diagram

A bundled version of X.asLimitCone and X.asLimit.

Equations

• X.lim = { cone := X.asLimitCone, isLimit := X.asLimit }