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.

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

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[reducible, inline]

An abbreviation for X.fintypeDiagram ⋙ FintypeCat.toProfinite.

Equations
Instances For

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 := } }
Instances For
Equations
• =

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

Equations
Instances For
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
Instances For

X.asLimitCone is indeed a limit cone.

Equations
Instances For

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

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