# mathlib3documentation

topology.category.Profinite.as_limit

# Profinite sets as limits of finite sets. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.fintype_diagram is the functor discrete_quotient X ⥤ Fintype whose limit is isomorphic to X (the limit taking place in Profinite via Fintype_to_Profinite, see 2).
2. X.diagram is an abbreviation for X.fintype_diagram ⋙ Fintype_to_Profinite.
3. X.as_limit_cone is the cone over X.diagram whose cone point is X.
4. X.iso_as_limit_cone_lift is the isomorphism X ≅ (Profinite.limit_cone X.diagram).X induced by lifting X.as_limit_cone.
5. X.as_limit_cone_iso is the isomorphism X.as_limit_cone ≅ (Profinite.limit_cone X.diagram) induced by X.iso_as_limit_cone_lift.
6. X.as_limit is a term of type is_limit X.as_limit_cone.
7. X.lim : category_theory.limits.limit_cone X.as_limit_cone is a bundled combination of 3 and 6.
noncomputable def Profinite.fintype_diagram (X : Profinite) :

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

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@[reducible]
noncomputable def Profinite.diagram (X : Profinite) :

An abbreviation for X.fintype_diagram ⋙ Fintype_to_Profinite.

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

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@[protected, instance]
noncomputable def Profinite.iso_as_limit_cone_lift (X : Profinite) :

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

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noncomputable def Profinite.as_limit_cone_iso (X : Profinite) :

The isomorphism of cones X.as_limit_cone and Profinite.limit_cone X.diagram. The underlying isomorphism is defeq to X.iso_as_limit_cone_lift.

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noncomputable def Profinite.as_limit (X : Profinite) :

X.as_limit_cone is indeed a limit cone.

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noncomputable def Profinite.lim (X : Profinite) :

A bundled version of X.as_limit_cone and X.as_limit.

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