mathlib documentation

category_theory.category.Bipointed

The category of bipointed types #

This defines Bipointed, the category of bipointed types.

TODO #

Monoidal structure

structure Bipointed  :
Type (u+1)
  • X : Type ?
  • to_prod : self.X × self.X

The category of bipointed types.

Instances for Bipointed
@[protected, instance]
Equations
def Bipointed.of {X : Type u_1} (to_prod : X × X) :

Turns a bipointing into a bipointed type.

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@[simp]
theorem Bipointed.coe_of {X : Type u_1} (to_prod : X × X) :
(Bipointed.of to_prod) = X
def prod.Bipointed {X : Type u_1} (to_prod : X × X) :

Alias of Bipointed.of`.

@[protected, instance]
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theorem Bipointed.hom.ext {X Y : Bipointed} (x y : X.hom Y) (h : x.to_fun = y.to_fun) :
x = y
theorem Bipointed.hom.ext_iff {X Y : Bipointed} (x y : X.hom Y) :
x = y x.to_fun = y.to_fun
@[ext]
structure Bipointed.hom (X Y : Bipointed) :
Type u

Morphisms in Bipointed.

Instances for Bipointed.hom
def Bipointed.hom.id (X : Bipointed) :
X.hom X

The identity morphism of X : Bipointed.

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@[simp]
theorem Bipointed.hom.id_to_fun (X : Bipointed) (a : X) :
@[protected, instance]
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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.

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@[simp]
theorem Bipointed.hom.comp_to_fun {X Y Z : Bipointed} (f : X.hom Y) (g : Y.hom Z) (ᾰ : X) :
(f.comp g).to_fun = (g.to_fun f.to_fun)
@[protected, instance]
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@[protected, instance]
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Swaps the pointed elements of a bipointed type. prod.swap as a functor.

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@[simp]
theorem Bipointed.swap_map_to_fun (X Y : Bipointed) (f : X Y) (ᾰ : X) :

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

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@[simp]
@[simp]

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

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

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The functor from Pointed to Bipointed which bipoints the point.

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

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

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