The category of bipointed types #

This defines Bipointed, the category of bipointed types.

TODO #

Monoidal structure

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structure Bipointed :

Type (u + 1)

The category of bipointed types.

X : Type u
The underlying type of a bipointed type.
toProd : self.X × self.X
The two points of a bipointed type, bundled together as a pair.

Instances For

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instance Bipointed.instCoeSortType :

CoeSort Bipointed (Type u_1)

Equations

Bipointed.instCoeSortType = { coe := Bipointed.X }

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def Bipointed.of {X : Type u_1} (to_prod : X × X) :

Bipointed

Turns a bipointing into a bipointed type.

Equations

Bipointed.of to_prod = { X := X, toProd := to_prod }

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theorem Bipointed.coe_of {X : Type u_1} (to_prod : X × X) :

(of to_prod).X = X

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def Prod.Bipointed {X : Type u_1} (to_prod : X × X) :

_root_.Bipointed

Alias of Bipointed.of.

Turns a bipointing into a bipointed type.

Equations

@Prod.Bipointed = @Bipointed.of

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instance Bipointed.instInhabited :

Inhabited Bipointed

Equations

Bipointed.instInhabited = { default := Bipointed.of ((), ()) }

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structure Bipointed.Hom (X Y : Bipointed) :

Type u

Morphisms in Bipointed.

toFun : X.X → Y.X
The underlying function of a morphism of bipointed types.
map_fst : self.toFun X.toProd.1 = Y.toProd.1
map_snd : self.toFun X.toProd.2 = Y.toProd.2

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theorem Bipointed.Hom.ext {X Y : Bipointed} {x y : X.Hom Y} (toFun : x.toFun = y.toFun) :

x = y

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def Bipointed.Hom.id (X : Bipointed) :

X.Hom X

The identity morphism of X : Bipointed.

Equations

Bipointed.Hom.id X = { toFun := id, map_fst := ⋯, map_snd := ⋯ }

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@[simp]

theorem Bipointed.Hom.id_toFun (X : Bipointed) (a : X.X) :

(id X).toFun a = _root_.id a

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instance Bipointed.Hom.instInhabited (X : Bipointed) :

Inhabited (X.Hom X)

Equations

Bipointed.Hom.instInhabited X = { default := Bipointed.Hom.id X }

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def Bipointed.Hom.comp {X Y Z : Bipointed} (f : X.Hom Y) (g : Y.Hom Z) :

X.Hom Z

Composition of morphisms of Bipointed.

Equations

f.comp g = { toFun := g.toFun ∘ f.toFun, map_fst := ⋯, map_snd := ⋯ }

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@[simp]

theorem Bipointed.Hom.comp_toFun {X Y Z : Bipointed} (f : X.Hom Y) (g : Y.Hom Z) (a✝ : X.X) :

(f.comp g).toFun a✝ = (g.toFun ∘ f.toFun) a✝

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instance Bipointed.largeCategory :

CategoryTheory.LargeCategory Bipointed

Equations

Bipointed.largeCategory = CategoryTheory.Category.mk @Bipointed.largeCategory.proof_1 @Bipointed.largeCategory.proof_2 @Bipointed.largeCategory.proof_3

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instance Bipointed.hasForget :

CategoryTheory.HasForget Bipointed

Equations

Bipointed.hasForget = CategoryTheory.HasForget.mk { obj := Bipointed.X, map := @Bipointed.Hom.toFun, map_id := Bipointed.hasForget.proof_1, map_comp := @Bipointed.hasForget.proof_2 }

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def Bipointed.swap :

CategoryTheory.Functor Bipointed Bipointed

Swaps the pointed elements of a bipointed type. Prod.swap as a functor.

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@[simp]

theorem Bipointed.swap_map_toFun {X✝ Y✝ : Bipointed} (f : X✝ ⟶ Y✝) (a✝ : X✝.X) :

(swap.map f).toFun a✝ = f.toFun a✝

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@[simp]

theorem Bipointed.swap_obj_toProd (X : Bipointed) :

(swap.obj X).toProd = X.toProd.swap

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@[simp]

theorem Bipointed.swap_obj_X (X : Bipointed) :

(swap.obj X).X = X.X

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def Bipointed.swapEquiv :

Bipointed ≌ Bipointed

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

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theorem Bipointed.swapEquiv_functor_obj_toProd (X : Bipointed) :

(swapEquiv.functor.obj X).toProd = X.toProd.swap

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theorem Bipointed.swapEquiv_inverse_map_toFun {X✝ Y✝ : Bipointed} (f : X✝ ⟶ Y✝) (a✝ : X✝.X) :

(swapEquiv.inverse.map f).toFun a✝ = f.toFun a✝

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theorem Bipointed.swapEquiv_counitIso_hom_app_toFun (X : Bipointed) (a : ((swap.comp swap).obj X).X) :

(swapEquiv.counitIso.hom.app X).toFun a = a

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theorem Bipointed.swapEquiv_functor_obj_X (X : Bipointed) :

(swapEquiv.functor.obj X).X = X.X

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@[simp]

theorem Bipointed.swapEquiv_inverse_obj_X (X : Bipointed) :

(swapEquiv.inverse.obj X).X = X.X

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theorem Bipointed.swapEquiv_unitIso_inv_app_toFun (X : Bipointed) (a : ((CategoryTheory.Functor.id Bipointed).obj X).X) :

(swapEquiv.unitIso.inv.app X).toFun a = a

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theorem Bipointed.swapEquiv_functor_map_toFun {X✝ Y✝ : Bipointed} (f : X✝ ⟶ Y✝) (a✝ : X✝.X) :

(swapEquiv.functor.map f).toFun a✝ = f.toFun a✝

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theorem Bipointed.swapEquiv_counitIso_inv_app_toFun (X : Bipointed) (a : ((swap.comp swap).obj X).X) :

(swapEquiv.counitIso.inv.app X).toFun a = a

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theorem Bipointed.swapEquiv_unitIso_hom_app_toFun (X : Bipointed) (a : ((CategoryTheory.Functor.id Bipointed).obj X).X) :

(swapEquiv.unitIso.hom.app X).toFun a = a

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@[simp]

theorem Bipointed.swapEquiv_inverse_obj_toProd (X : Bipointed) :

(swapEquiv.inverse.obj X).toProd = X.toProd.swap

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@[simp]

theorem Bipointed.swapEquiv_symm :

swapEquiv.symm = swapEquiv

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def bipointedToPointedFst :

CategoryTheory.Functor Bipointed Pointed

The forgetful functor from Bipointed to Pointed which forgets about the second point.

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def bipointedToPointedSnd :

CategoryTheory.Functor Bipointed Pointed

The forgetful functor from Bipointed to Pointed which forgets about the first point.

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@[simp]

theorem bipointedToPointedFst_comp_forget :

bipointedToPointedFst.comp (CategoryTheory.forget Pointed) = CategoryTheory.forget Bipointed

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@[simp]

theorem bipointedToPointedSnd_comp_forget :

bipointedToPointedSnd.comp (CategoryTheory.forget Pointed) = CategoryTheory.forget Bipointed

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@[simp]

theorem swap_comp_bipointedToPointedFst :

Bipointed.swap.comp bipointedToPointedFst = bipointedToPointedSnd

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@[simp]

theorem swap_comp_bipointedToPointedSnd :

Bipointed.swap.comp bipointedToPointedSnd = bipointedToPointedFst

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def pointedToBipointed :

CategoryTheory.Functor Pointed Bipointed

The functor from Pointed to Bipointed which bipoints the point.

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def pointedToBipointedFst :

CategoryTheory.Functor Pointed Bipointed

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

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def pointedToBipointedSnd :

CategoryTheory.Functor Pointed Bipointed

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

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theorem pointedToBipointedFst_comp_swap :

pointedToBipointedFst.comp Bipointed.swap = pointedToBipointedSnd

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@[simp]

theorem pointedToBipointedSnd_comp_swap :

pointedToBipointedSnd.comp Bipointed.swap = pointedToBipointedFst

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def pointedToBipointedCompBipointedToPointedFst :

pointedToBipointed.comp bipointedToPointedFst ≅ CategoryTheory.Functor.id Pointed

BipointedToPointed_fst is inverse to PointedToBipointed.

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theorem pointedToBipointedCompBipointedToPointedFst_hom_app_toFun (X : Pointed) (a : ((pointedToBipointed.comp bipointedToPointedFst).obj X).X) :

(pointedToBipointedCompBipointedToPointedFst.hom.app X).toFun a = a

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theorem pointedToBipointedCompBipointedToPointedFst_inv_app_toFun (X : Pointed) (a : ((CategoryTheory.Functor.id Pointed).obj X).X) :

(pointedToBipointedCompBipointedToPointedFst.inv.app X).toFun a = a

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def pointedToBipointedCompBipointedToPointedSnd :

pointedToBipointed.comp bipointedToPointedSnd ≅ CategoryTheory.Functor.id Pointed

BipointedToPointed_snd is inverse to PointedToBipointed.

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theorem pointedToBipointedCompBipointedToPointedSnd_inv_app_toFun (X : Pointed) (a : ((CategoryTheory.Functor.id Pointed).obj X).X) :

(pointedToBipointedCompBipointedToPointedSnd.inv.app X).toFun a = a

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theorem pointedToBipointedCompBipointedToPointedSnd_hom_app_toFun (X : Pointed) (a : ((pointedToBipointed.comp bipointedToPointedSnd).obj X).X) :

(pointedToBipointedCompBipointedToPointedSnd.hom.app X).toFun a = a

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def pointedToBipointedFstBipointedToPointedFstAdjunction :

pointedToBipointedFst ⊣ bipointedToPointedFst

The free/forgetful adjunction between PointedToBipointed_fst and BipointedToPointed_fst.

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def pointedToBipointedSndBipointedToPointedSndAdjunction :

pointedToBipointedSnd ⊣ bipointedToPointedSnd

The free/forgetful adjunction between PointedToBipointed_snd and BipointedToPointed_snd.

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Documentation

Mathlib.CategoryTheory.Category.Bipointed

The category of bipointed types #

TODO #