File: Mathlib/CategoryTheory/Galois/Examples.lean

full_name: CategoryTheory.FintypeCat.imageComplement
code: /-- Complement of the image of a morphism `f : X ⟶ Y` in `FintypeCat`. -/
noncomputable def imageComplement {X Y : FintypeCat.{u}} (f : X ⟶ Y) :
    FintypeCat.{u} 
goal: FintypeCat

full_name: CategoryTheory.FintypeCat.imageComplementIncl
code: /-- The inclusion from the complement of the image of `f : X ⟶ Y` into `Y`. -/
def imageComplementIncl {X Y : FintypeCat.{u}}
    (f : X ⟶ Y) : imageComplement f ⟶ Y 
goal: CategoryTheory.FintypeCat.imageComplement f ⟶ Y

full_name: CategoryTheory.FintypeCat.Action.imageComplement
code: /-- Given `f : X ⟶ Y` for `X Y : Action FintypeCat (MonCat.of G)`, the complement of the image
of `f` has a natural `G`-action. -/
noncomputable def Action.imageComplement {X Y : Action FintypeCat (MonCat.of G)}
    (f : X ⟶ Y) : Action FintypeCat (MonCat.of G) 
goal: Action FintypeCat (MonCat.of G)

full_name: CategoryTheory.FintypeCat.Action.imageComplementIncl
code: /-- The inclusion from the complement of the image of `f : X ⟶ Y` into `Y`. -/
def Action.imageComplementIncl {X Y : Action FintypeCat (MonCat.of G)} (f : X ⟶ Y) :
    Action.imageComplement G f ⟶ Y 
goal: CategoryTheory.FintypeCat.Action.imageComplement G f ⟶ Y

full_name: CategoryTheory.FintypeCat.instMonoActionFintypeCatOfImageComplementIncl
code: instance {X Y : Action FintypeCat (MonCat.of G)} (f : X ⟶ Y) :
    Mono (Action.imageComplementIncl G f) 
goal: CategoryTheory.Mono (CategoryTheory.FintypeCat.Action.imageComplementIncl G f)

full_name: CategoryTheory.FintypeCat.instHasColimitsOfShapeSingleObjFintypeCatOfFinite
code: /-- The category of finite sets has quotients by finite groups in arbitrary universes. -/
instance [Finite G] : HasColimitsOfShape (SingleObj G) FintypeCat.{w} 
goal: CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.SingleObj G) FintypeCat

full_name: CategoryTheory.FintypeCat.instPreservesFiniteLimitsActionFintypeCatOfForget
code: noncomputable instance : PreservesFiniteLimits (forget (Action FintypeCat (MonCat.of G))) 
goal: CategoryTheory.Limits.PreservesFiniteLimits (CategoryTheory.forget (Action FintypeCat (MonCat.of G)))

full_name: CategoryTheory.FintypeCat.instPreGaloisCategoryActionFintypeCatOf
code: /-- The category of finite `G`-sets is a `PreGaloisCategory`. -/
instance : PreGaloisCategory (Action FintypeCat (MonCat.of G)) 
goal: CategoryTheory.PreGaloisCategory (Action FintypeCat (MonCat.of G))

full_name: CategoryTheory.FintypeCat.instFiberFunctorActionFintypeCatOfForget
code: /-- The forgetful functor from finite `G`-sets to sets is a `FiberFunctor`. -/
noncomputable instance : FiberFunctor (Action.forget FintypeCat (MonCat.of G)) 
goal: CategoryTheory.PreGaloisCategory.FiberFunctor (Action.forget FintypeCat (MonCat.of G))

full_name: CategoryTheory.FintypeCat.instGaloisCategoryActionFintypeCatOf
code: /-- The category of finite `G`-sets is a `GaloisCategory`. -/
instance : GaloisCategory (Action FintypeCat (MonCat.of G)) 
goal: CategoryTheory.GaloisCategory (Action FintypeCat (MonCat.of G))

full_name: CategoryTheory.FintypeCat.Action.pretransitive_of_isConnected
code: /-- The `G`-action on a connected finite `G`-set is transitive. -/
theorem Action.pretransitive_of_isConnected (X : Action FintypeCat (MonCat.of G))
    [IsConnected X] : MulAction.IsPretransitive G X.V 
goal: MulAction.IsPretransitive G ↑X.V

full_name: CategoryTheory.FintypeCat.Action.isConnected_of_transitive
code: /-- A nonempty `G`-set with transitive `G`-action is connected. -/
theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
    [MulAction.IsPretransitive G X] [h : Nonempty X] :
    IsConnected (Action.FintypeCat.ofMulAction G X) 
goal: CategoryTheory.PreGaloisCategory.IsConnected (Action.FintypeCat.ofMulAction G X)

full_name: CategoryTheory.FintypeCat.Action.isConnected_iff_transitive
code: /-- A nonempty finite `G`-set is connected if and only if the `G`-action is transitive. -/
theorem Action.isConnected_iff_transitive (X : Action FintypeCat (MonCat.of G)) [Nonempty X.V] :
    IsConnected X ↔ MulAction.IsPretransitive G X.V 
goal: CategoryTheory.PreGaloisCategory.IsConnected X ↔ MulAction.IsPretransitive G ↑X.V