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.

• X : Type u

  The underlying type of a two-pointed type.

• toTwoPointing : TwoPointing self.X

  The two points of a bipointed type, bundled together as a pair of distinct elements.

instance TwoP.instCoeSortType : CoeSort TwoP (Type u_3)

@[reducible, inline]

abbrev TwoP.of {X : Type u_3} (toTwoPointing : TwoPointing X) : TwoP

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

theorem TwoP.coe_of {X : Type u_3} (toTwoPointing : TwoPointing X) : (of toTwoPointing).X = X

def TwoPointing.TwoP {X : Type u_3} (toTwoPointing : TwoPointing X) : _root_.TwoP

Alias of TwoP.of.

---

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

instance TwoP.instInhabited : Inhabited TwoP

noncomputable def TwoP.toBipointed (X : TwoP) : Bipointed

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

@[simp]

theorem TwoP.coe_toBipointed (X : TwoP) : X.toBipointed.X = X.X

noncomputable instance TwoP.largeCategory : CategoryTheory.LargeCategory TwoP

noncomputable instance TwoP.concreteCategory : CategoryTheory.ConcreteCategory TwoP fun (X Y : TwoP) => X.toBipointed.HomSubtype Y.toBipointed

noncomputable instance TwoP.hasForgetToBipointed : CategoryTheory.HasForget₂ TwoP Bipointed

theorem TwoP.hom_ext {X Y : TwoP} {f g : X ⟶ Y} (h : f.hom = g.hom) : f = g

theorem TwoP.hom_ext_iff {X Y : TwoP} {f g : X ⟶ Y} : f = g ↔ f.hom = g.hom

noncomputable def TwoP.swap : CategoryTheory.Functor TwoP TwoP

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

@[simp]

theorem TwoP.swap_map {X✝ Y✝ : TwoP} (f : X✝ ⟶ Y✝) : swap.map f = CategoryTheory.InducedCategory.homMk { toFun := ⇑(CategoryTheory.ConcreteCategory.hom f.hom), map_fst := ⋯, map_snd := ⋯ }

@[simp]

theorem TwoP.swap_obj_toTwoPointing (X : TwoP) : (swap.obj X).toTwoPointing = X.toTwoPointing.swap

@[simp]

theorem TwoP.swap_obj_X (X : TwoP) : (swap.obj X).X = X.X

noncomputable def TwoP.swapEquiv : TwoP ≌ TwoP

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

@[simp]

theorem TwoP.swapEquiv_counitIso_hom_app_hom_toFun (X : TwoP) (a : ((swap.comp swap).obj X).toBipointed.X) : (swapEquiv.counitIso.hom.app X).hom.toFun a = a

@[simp]

theorem TwoP.swapEquiv_counitIso_inv_app_hom_toFun (X : TwoP) (a : ((swap.comp swap).obj X).toBipointed.X) : (swapEquiv.counitIso.inv.app X).hom.toFun a = a

@[simp]

theorem TwoP.swapEquiv_inverse_map_hom_toFun {X✝ Y✝ : TwoP} (f : X✝ ⟶ Y✝) (a : X✝.toBipointed.X) : (swapEquiv.inverse.map f).hom.toFun a = (CategoryTheory.ConcreteCategory.hom f.hom) a

@[simp]

theorem TwoP.swapEquiv_unitIso_inv_app_hom_toFun (X : TwoP) (a : ((CategoryTheory.Functor.id TwoP).obj X).toBipointed.X) : (swapEquiv.unitIso.inv.app X).hom.toFun a = a

@[simp]

theorem TwoP.swapEquiv_functor_obj_toTwoPointing_toProd (X : TwoP) : (swapEquiv.functor.obj X).toTwoPointing.toProd = (X.toTwoPointing.toProd.2, X.toTwoPointing.toProd.1)

@[simp]

theorem TwoP.swapEquiv_inverse_obj_toTwoPointing_toProd (X : TwoP) : (swapEquiv.inverse.obj X).toTwoPointing.toProd = (X.toTwoPointing.toProd.2, X.toTwoPointing.toProd.1)

@[simp]

theorem TwoP.swapEquiv_inverse_obj_X (X : TwoP) : (swapEquiv.inverse.obj X).X = X.X

@[simp]

theorem TwoP.swapEquiv_functor_map_hom_toFun {X✝ Y✝ : TwoP} (f : X✝ ⟶ Y✝) (a : X✝.toBipointed.X) : (swapEquiv.functor.map f).hom.toFun a = (CategoryTheory.ConcreteCategory.hom f.hom) a

@[simp]

theorem TwoP.swapEquiv_unitIso_hom_app_hom_toFun (X : TwoP) (a : ((CategoryTheory.Functor.id TwoP).obj X).toBipointed.X) : (swapEquiv.unitIso.hom.app X).hom.toFun a = a

@[simp]

theorem TwoP.swapEquiv_functor_obj_X (X : TwoP) : (swapEquiv.functor.obj X).X = X.X

@[simp]

theorem TwoP.swapEquiv_symm : swapEquiv.symm = swapEquiv

@[simp]

theorem TwoP_swap_comp_forget_to_Bipointed : TwoP.swap.comp (CategoryTheory.forget₂ TwoP Bipointed) = (CategoryTheory.forget₂ TwoP Bipointed).comp Bipointed.swap

noncomputable def pointedToTwoPFst : CategoryTheory.Functor Pointed TwoP

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

@[simp]

theorem pointedToTwoPFst_obj_X (X : Pointed) : (pointedToTwoPFst.obj X).X = Option X.X

@[simp]

theorem pointedToTwoPFst_map_hom_toFun {X✝ Y✝ : Pointed} (f : X✝ ⟶ Y✝) (a✝ : Option X✝.X) : (pointedToTwoPFst.map f).hom.toFun a✝ = Option.map f.toFun a✝

@[simp]

theorem pointedToTwoPFst_obj_toTwoPointing_toProd (X : Pointed) : (pointedToTwoPFst.obj X).toTwoPointing.toProd = (some X.point, none)

noncomputable def pointedToTwoPSnd : CategoryTheory.Functor Pointed TwoP

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

@[simp]

theorem pointedToTwoPSnd_obj_toTwoPointing_toProd (X : Pointed) : (pointedToTwoPSnd.obj X).toTwoPointing.toProd = (none, some X.point)

@[simp]

theorem pointedToTwoPSnd_map_hom_toFun {X✝ Y✝ : Pointed} (f : X✝ ⟶ Y✝) (a✝ : Option X✝.X) : (pointedToTwoPSnd.map f).hom.toFun a✝ = Option.map f.toFun a✝

@[simp]

theorem pointedToTwoPSnd_obj_X (X : Pointed) : (pointedToTwoPSnd.obj X).X = Option X.X

@[simp]

theorem pointedToTwoPFst_comp_swap : pointedToTwoPFst.comp TwoP.swap = pointedToTwoPSnd

@[simp]

theorem pointedToTwoPSnd_comp_swap : pointedToTwoPSnd.comp TwoP.swap = pointedToTwoPFst

@[simp]

theorem pointedToTwoPFst_comp_forget_to_bipointed : pointedToTwoPFst.comp (CategoryTheory.forget₂ TwoP Bipointed) = pointedToBipointedFst

@[simp]

theorem pointedToTwoPSnd_comp_forget_to_bipointed : pointedToTwoPSnd.comp (CategoryTheory.forget₂ TwoP Bipointed) = pointedToBipointedSnd

noncomputable def pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction : pointedToTwoPFst ⊣ (CategoryTheory.forget₂ TwoP Bipointed).comp bipointedToPointedFst

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

noncomputable def pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction : pointedToTwoPSnd ⊣ (CategoryTheory.forget₂ TwoP Bipointed).comp bipointedToPointedSnd

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