Documentation

Mathlib.CategoryTheory.Category.TwoP

The category of two-pointed types #

This defines TwoP, the category of two-pointed types.

References #

• [nLab, coalgebra of the real interval] (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)
structure TwoP :
Type (u + 1)

The category of two-pointed types.

Instances For
def TwoP.of {X : Type u_3} (toTwoPointing : ) :

Turns a two-pointing into a two-pointed type.

Instances For
@[simp]
theorem TwoP.coe_of {X : Type u_3} (toTwoPointing : ) :
(TwoP.of toTwoPointing).X = X
def TwoPointing.TwoP {X : Type u_3} (toTwoPointing : ) :

Alias of TwoP.of.

Turns a two-pointing into a two-pointed type.

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noncomputable def TwoP.toBipointed (X : TwoP) :

Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are distinct.

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@[simp]
theorem TwoP.coe_toBipointed (X : TwoP) :
().X = X.X
noncomputable instance TwoP.largeCategory :
noncomputable instance TwoP.concreteCategory :
noncomputable instance TwoP.hasForgetToBipointed :
@[simp]
theorem TwoP.swap_obj_X (X : TwoP) :
(TwoP.swap.obj X).X = X.X
@[simp]
theorem TwoP.swap_map_toFun :
∀ {X Y : TwoP} (f : X Y) (a : ().X), Bipointed.Hom.toFun (TwoP.swap.map f) a =
@[simp]
theorem TwoP.swap_obj_toTwoPointing (X : TwoP) :
(TwoP.swap.obj X).toTwoPointing = TwoPointing.swap X.toTwoPointing
noncomputable def TwoP.swap :

Swaps the pointed elements of a two-pointed type. TwoPointing.swap as a functor.

Instances For
@[simp]
theorem TwoP.swapEquiv_functor_obj_toTwoPointing_toProd (X : TwoP) :
(TwoP.swapEquiv.functor.obj X).toTwoPointing.toProd = (X.toTwoPointing.snd, X.toTwoPointing.fst)
@[simp]
theorem TwoP.swapEquiv_inverse_obj_toTwoPointing_toProd (X : TwoP) :
(TwoP.swapEquiv.inverse.obj X).toTwoPointing.toProd = (X.toTwoPointing.snd, X.toTwoPointing.fst)
@[simp]
theorem TwoP.swapEquiv_functor_obj_X (X : TwoP) :
(TwoP.swapEquiv.functor.obj X).X = X.X
@[simp]
theorem TwoP.swapEquiv_counitIso_hom_app_toFun (X : TwoP) (a : ().X) :
Bipointed.Hom.toFun (TwoP.swapEquiv.counitIso.hom.app X) a = a
@[simp]
theorem TwoP.swapEquiv_inverse_obj_X (X : TwoP) :
(TwoP.swapEquiv.inverse.obj X).X = X.X
@[simp]
theorem TwoP.swapEquiv_inverse_map_toFun :
∀ {X Y : TwoP} (f : X Y) (a : ().X), Bipointed.Hom.toFun (TwoP.swapEquiv.inverse.map f) a =
@[simp]
theorem TwoP.swapEquiv_unitIso_inv_app_toFun (X : TwoP) :
∀ (a : ().X), Bipointed.Hom.toFun (TwoP.swapEquiv.unitIso.inv.app X) a = Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.comp (TwoP.swap.map (TwoP.swap.map { toFun := id, map_fst := (_ : id ().toProd.fst = id ().toProd.fst), map_snd := (_ : id ().toProd.snd = id ().toProd.snd) })) (CategoryTheory.CategoryStruct.comp (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.fst = id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.fst), map_snd := (_ : id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.snd = id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.snd) }) { toFun := id, map_fst := (_ : id ().toProd.fst = id ().toProd.fst), map_snd := (_ : id ().toProd.snd = id ().toProd.snd) })) (Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.id (TwoP.swap.obj (TwoP.swap.obj X))) a)
@[simp]
theorem TwoP.swapEquiv_counitIso_inv_app_toFun (X : TwoP) (a : ().X) :
Bipointed.Hom.toFun (TwoP.swapEquiv.counitIso.inv.app X) a = a
@[simp]
theorem TwoP.swapEquiv_unitIso_hom_app_toFun (X : TwoP) :
∀ (a : ().X), Bipointed.Hom.toFun (TwoP.swapEquiv.unitIso.hom.app X) a = Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.id (TwoP.swap.obj (TwoP.swap.obj X))) (Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.comp { toFun := id, map_fst := (_ : id ().toProd.fst = id ().toProd.fst), map_snd := (_ : id ().toProd.snd = id ().toProd.snd) } (CategoryTheory.CategoryStruct.comp (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.fst = id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.fst), map_snd := (_ : id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.snd = id (TwoP.toBipointed (().obj (TwoP.swap.obj X))).toProd.snd) }) (TwoP.swap.map (TwoP.swap.map { toFun := id, map_fst := (_ : id ().toProd.fst = id ().toProd.fst), map_snd := (_ : id ().toProd.snd = id ().toProd.snd) })))) a)
@[simp]
theorem TwoP.swapEquiv_functor_map_toFun :
∀ {X Y : TwoP} (f : X Y) (a : ().X), Bipointed.Hom.toFun (TwoP.swapEquiv.functor.map f) a =
noncomputable def TwoP.swapEquiv :

The equivalence between TwoP and itself induced by Prod.swap both ways.

Instances For
@[simp]
@[simp]
@[simp]
theorem pointedToTwoPFst_obj_toTwoPointing_toProd (X : Pointed) :
(pointedToTwoPFst.obj X).toTwoPointing.toProd = (some X.point, none)
@[simp]
theorem pointedToTwoPFst_map_toFun :
∀ {X Y : Pointed} (f : X Y) (a : Option X.X), Bipointed.Hom.toFun (pointedToTwoPFst.map f) a = Option.map f.toFun a
noncomputable def pointedToTwoPFst :

The functor from Pointed to TwoP which adds a second point.

Instances For
@[simp]
@[simp]
theorem pointedToTwoPSnd_map_toFun :
∀ {X Y : Pointed} (f : X Y) (a : Option X.X), Bipointed.Hom.toFun (pointedToTwoPSnd.map f) a = Option.map f.toFun a
@[simp]
theorem pointedToTwoPSnd_obj_toTwoPointing_toProd (X : Pointed) :
(pointedToTwoPSnd.obj X).toTwoPointing.toProd = (none, some X.point)
noncomputable def pointedToTwoPSnd :

The functor from Pointed to TwoP which adds a first point.

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Adding a second point is left adjoint to forgetting the second point.

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Adding a first point is left adjoint to forgetting the first point.

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