Algebras of endofunctors #
This file defines (co)algebras of an endofunctor, and provides the category instance for them.
It also defines the forgetful functor from the category of (co)algebras. It is shown that the
structure map of the initial algebra of an endofunctor is an isomorphism. Furthermore, it is shown
that for an adjunction F ⊣ G the category of algebras over F is equivalent to the category of
coalgebras over G.
TODO #
- Prove the dual result about the structure map of the terminal coalgebra of an endofunctor.
- Prove that if the countable infinite product over the powers of the endofunctor exists, then algebras over the endofunctor coincide with algebras over the free monad on the endofunctor.
structure
CategoryTheory.Endofunctor.Algebra
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Type (max u v)
An algebra of an endofunctor; str stands for "structure morphism"
- a : C
carrier of the algebra
- str : F.obj self.a ⟶ self.a
structure morphism of the algebra
Instances For
instance
CategoryTheory.Endofunctor.instInhabitedAlgebraId
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[Inhabited C]
:
Equations
- CategoryTheory.Endofunctor.instInhabitedAlgebraId = { default := { a := default, str := CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id C).obj default) } }
structure
CategoryTheory.Endofunctor.Algebra.Hom
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(A₀ A₁ : CategoryTheory.Endofunctor.Algebra F)
:
Type v
A morphism between algebras of endofunctor F
- f : A₀.a ⟶ A₁.a
underlying morphism between the carriers
- h : CategoryTheory.CategoryStruct.comp (F.map self.f) A₁.str = CategoryTheory.CategoryStruct.comp A₀.str self.f
compatibility condition
Instances For
theorem
CategoryTheory.Endofunctor.Algebra.Hom.ext
{C : Type u}
{inst✝ : CategoryTheory.Category.{v, u} C}
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
{x y : A₀.Hom A₁}
(f : x.f = y.f)
:
x = y
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.Hom.h_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
(self : A₀.Hom A₁)
{Z : C}
(h : A₁.a ⟶ Z)
:
CategoryTheory.CategoryStruct.comp (F.map self.f) (CategoryTheory.CategoryStruct.comp A₁.str h) = CategoryTheory.CategoryStruct.comp A₀.str (CategoryTheory.CategoryStruct.comp self.f h)
def
CategoryTheory.Endofunctor.Algebra.Hom.id
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(A : CategoryTheory.Endofunctor.Algebra F)
:
A.Hom A
The identity morphism of an algebra of endofunctor F
Equations
- CategoryTheory.Endofunctor.Algebra.Hom.id A = { f := CategoryTheory.CategoryStruct.id A.a, h := ⋯ }
Instances For
instance
CategoryTheory.Endofunctor.Algebra.Hom.instInhabited
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(A : CategoryTheory.Endofunctor.Algebra F)
:
Inhabited (A.Hom A)
Equations
- CategoryTheory.Endofunctor.Algebra.Hom.instInhabited A = { default := { f := CategoryTheory.CategoryStruct.id A.a, h := ⋯ } }
def
CategoryTheory.Endofunctor.Algebra.Hom.comp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ A₂ : CategoryTheory.Endofunctor.Algebra F}
(f : A₀.Hom A₁)
(g : A₁.Hom A₂)
:
A₀.Hom A₂
The composition of morphisms between algebras of endofunctor F
Equations
- f.comp g = { f := CategoryTheory.CategoryStruct.comp f.f g.f, h := ⋯ }
Instances For
instance
CategoryTheory.Endofunctor.Algebra.instCategoryStruct
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Equations
- CategoryTheory.Endofunctor.Algebra.instCategoryStruct F = CategoryTheory.CategoryStruct.mk CategoryTheory.Endofunctor.Algebra.Hom.id (@CategoryTheory.Endofunctor.Algebra.Hom.comp C inst F)
theorem
CategoryTheory.Endofunctor.Algebra.ext
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A B : CategoryTheory.Endofunctor.Algebra F}
{f g : A ⟶ B}
(w : f.f = g.f := by aesop_cat)
:
f = g
@[simp]
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.id_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(A : CategoryTheory.Endofunctor.Algebra F)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.comp_eq_comp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ A₂ : CategoryTheory.Endofunctor.Algebra F}
(f : A₀ ⟶ A₁)
(g : A₁ ⟶ A₂)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.comp_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ A₂ : CategoryTheory.Endofunctor.Algebra F}
(f : A₀ ⟶ A₁)
(g : A₁ ⟶ A₂)
:
(CategoryTheory.CategoryStruct.comp f g).f = CategoryTheory.CategoryStruct.comp f.f g.f
instance
CategoryTheory.Endofunctor.Algebra.instCategory
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Algebras of an endofunctor F form a category
Equations
def
CategoryTheory.Endofunctor.Algebra.isoMk
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
(h : A₀.a ≅ A₁.a)
(w : CategoryTheory.CategoryStruct.comp (F.map h.hom) A₁.str = CategoryTheory.CategoryStruct.comp A₀.str h.hom := by
aesop_cat)
:
A₀ ≅ A₁
To construct an isomorphism of algebras, it suffices to give an isomorphism of the As which commutes with the structure morphisms.
Equations
- CategoryTheory.Endofunctor.Algebra.isoMk h w = { hom := { f := h.hom, h := ⋯ }, inv := { f := h.inv, h := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.isoMk_inv_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
(h : A₀.a ≅ A₁.a)
(w : CategoryTheory.CategoryStruct.comp (F.map h.hom) A₁.str = CategoryTheory.CategoryStruct.comp A₀.str h.hom := by
aesop_cat)
:
(CategoryTheory.Endofunctor.Algebra.isoMk h w).inv.f = h.inv
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.isoMk_hom_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
(h : A₀.a ≅ A₁.a)
(w : CategoryTheory.CategoryStruct.comp (F.map h.hom) A₁.str = CategoryTheory.CategoryStruct.comp A₀.str h.hom := by
aesop_cat)
:
(CategoryTheory.Endofunctor.Algebra.isoMk h w).hom.f = h.hom
def
CategoryTheory.Endofunctor.Algebra.forget
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
The forgetful functor from the category of algebras, forgetting the algebraic structure.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.forget_obj
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
(A : CategoryTheory.Endofunctor.Algebra F)
:
(CategoryTheory.Endofunctor.Algebra.forget F).obj A = A.a
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.forget_map
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
{X✝ Y✝ : CategoryTheory.Endofunctor.Algebra F}
(self : X✝.Hom Y✝)
:
(CategoryTheory.Endofunctor.Algebra.forget F).map self = self.f
theorem
CategoryTheory.Endofunctor.Algebra.iso_of_iso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A₀ A₁ : CategoryTheory.Endofunctor.Algebra F}
(f : A₀ ⟶ A₁)
[CategoryTheory.IsIso f.f]
:
An algebra morphism with an underlying isomorphism hom in C is an algebra isomorphism.
instance
CategoryTheory.Endofunctor.Algebra.forget_reflects_iso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
(CategoryTheory.Endofunctor.Algebra.forget F).ReflectsIsomorphisms
Equations
- ⋯ = ⋯
instance
CategoryTheory.Endofunctor.Algebra.forget_faithful
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
(CategoryTheory.Endofunctor.Algebra.forget F).Faithful
Equations
- ⋯ = ⋯
theorem
CategoryTheory.Endofunctor.Algebra.epi_of_epi
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{X Y : CategoryTheory.Endofunctor.Algebra F}
(f : X ⟶ Y)
[h : CategoryTheory.Epi f.f]
:
An algebra morphism with an underlying epimorphism hom in C is an algebra epimorphism.
theorem
CategoryTheory.Endofunctor.Algebra.mono_of_mono
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{X Y : CategoryTheory.Endofunctor.Algebra F}
(f : X ⟶ Y)
[h : CategoryTheory.Mono f.f]
:
An algebra morphism with an underlying monomorphism hom in C is an algebra monomorphism.
def
CategoryTheory.Endofunctor.Algebra.functorOfNatTrans
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : G ⟶ F)
:
From a natural transformation α : G → F we get a functor from
algebras of F to algebras of G.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTrans_map_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : G ⟶ F)
{X✝ Y✝ : CategoryTheory.Endofunctor.Algebra F}
(f : X✝ ⟶ Y✝)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTrans α).map f).f = f.f
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTrans_obj_str
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : G ⟶ F)
(A : CategoryTheory.Endofunctor.Algebra F)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTrans α).obj A).str = CategoryTheory.CategoryStruct.comp (α.app A.a) A.str
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTrans_obj_a
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : G ⟶ F)
(A : CategoryTheory.Endofunctor.Algebra F)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTrans α).obj A).a = A.a
def
CategoryTheory.Endofunctor.Algebra.functorOfNatTransId
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
The identity transformation induces the identity endofunctor on the category of algebras.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransId_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(X : CategoryTheory.Endofunctor.Algebra F)
:
(CategoryTheory.Endofunctor.Algebra.functorOfNatTransId.inv.app X).f = CategoryTheory.CategoryStruct.id X.a
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransId_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(X : CategoryTheory.Endofunctor.Algebra F)
:
(CategoryTheory.Endofunctor.Algebra.functorOfNatTransId.hom.app X).f = CategoryTheory.CategoryStruct.id X.a
def
CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
:
A composition of natural transformations gives the composition of corresponding functors.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
(X : CategoryTheory.Endofunctor.Algebra F₂)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp α β).hom.app X).f = CategoryTheory.CategoryStruct.id X.a
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
(X : CategoryTheory.Endofunctor.Algebra F₂)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp α β).inv.app X).f = CategoryTheory.CategoryStruct.id X.a
def
CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
:
If α and β are two equal natural transformations, then the functors of algebras induced by them
are isomorphic.
We define it like this as opposed to using eq_to_iso so that the components are nicer to prove
lemmas about.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
(X : CategoryTheory.Endofunctor.Algebra G)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq h).hom.app X).f = CategoryTheory.CategoryStruct.id X.a
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
(X : CategoryTheory.Endofunctor.Algebra G)
:
((CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq h).inv.app X).f = CategoryTheory.CategoryStruct.id X.a
def
CategoryTheory.Endofunctor.Algebra.equivOfNatIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
Naturally isomorphic endofunctors give equivalent categories of algebras.
Furthermore, they are equivalent as categories over C, that is,
we have equiv_of_nat_iso h ⋙ forget = forget.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.equivOfNatIso_counitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
(CategoryTheory.Endofunctor.Algebra.equivOfNatIso α).counitIso = (CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp α.inv α.hom).symm ≪≫ CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq ⋯ ≪≫ CategoryTheory.Endofunctor.Algebra.functorOfNatTransId
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.equivOfNatIso_inverse
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.equivOfNatIso_functor
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Algebra.equivOfNatIso_unitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
(CategoryTheory.Endofunctor.Algebra.equivOfNatIso α).unitIso = CategoryTheory.Endofunctor.Algebra.functorOfNatTransId.symm ≪≫ CategoryTheory.Endofunctor.Algebra.functorOfNatTransEq ⋯ ≪≫ CategoryTheory.Endofunctor.Algebra.functorOfNatTransComp α.hom α.inv
def
CategoryTheory.Endofunctor.Algebra.Initial.strInv
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A : CategoryTheory.Endofunctor.Algebra F}
(h : CategoryTheory.Limits.IsInitial A)
:
A.a ⟶ F.obj A.a
The inverse of the structure map of an initial algebra
Equations
- CategoryTheory.Endofunctor.Algebra.Initial.strInv h = (h.to { a := F.obj A.a, str := F.map A.str }).f
Instances For
theorem
CategoryTheory.Endofunctor.Algebra.Initial.left_inv'
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A : CategoryTheory.Endofunctor.Algebra F}
(h : CategoryTheory.Limits.IsInitial A)
:
{ f := CategoryTheory.CategoryStruct.comp (CategoryTheory.Endofunctor.Algebra.Initial.strInv h) A.str, h := ⋯ } = CategoryTheory.CategoryStruct.id A
theorem
CategoryTheory.Endofunctor.Algebra.Initial.left_inv
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A : CategoryTheory.Endofunctor.Algebra F}
(h : CategoryTheory.Limits.IsInitial A)
:
theorem
CategoryTheory.Endofunctor.Algebra.Initial.right_inv
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A : CategoryTheory.Endofunctor.Algebra F}
(h : CategoryTheory.Limits.IsInitial A)
:
theorem
CategoryTheory.Endofunctor.Algebra.Initial.str_isIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A : CategoryTheory.Endofunctor.Algebra F}
(h : CategoryTheory.Limits.IsInitial A)
:
CategoryTheory.IsIso A.str
The structure map of the initial algebra is an isomorphism, hence endofunctors preserve their initial algebras
structure
CategoryTheory.Endofunctor.Coalgebra
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Type (max u v)
A coalgebra of an endofunctor; str stands for "structure morphism"
- V : C
carrier of the coalgebra
- str : self.V ⟶ F.obj self.V
structure morphism of the coalgebra
Instances For
instance
CategoryTheory.Endofunctor.instInhabitedCoalgebraId
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[Inhabited C]
:
Equations
- CategoryTheory.Endofunctor.instInhabitedCoalgebraId = { default := { V := default, str := CategoryTheory.CategoryStruct.id default } }
structure
CategoryTheory.Endofunctor.Coalgebra.Hom
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F)
:
Type v
A morphism between coalgebras of an endofunctor F
- f : V₀.V ⟶ V₁.V
underlying morphism between two carriers
- h : CategoryTheory.CategoryStruct.comp V₀.str (F.map self.f) = CategoryTheory.CategoryStruct.comp self.f V₁.str
compatibility condition
Instances For
theorem
CategoryTheory.Endofunctor.Coalgebra.Hom.ext
{C : Type u}
{inst✝ : CategoryTheory.Category.{v, u} C}
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
{x y : V₀.Hom V₁}
(f : x.f = y.f)
:
x = y
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.Hom.h_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
(self : V₀.Hom V₁)
{Z : C}
(h : F.obj V₁.V ⟶ Z)
:
CategoryTheory.CategoryStruct.comp V₀.str (CategoryTheory.CategoryStruct.comp (F.map self.f) h) = CategoryTheory.CategoryStruct.comp self.f (CategoryTheory.CategoryStruct.comp V₁.str h)
def
CategoryTheory.Endofunctor.Coalgebra.Hom.id
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(V : CategoryTheory.Endofunctor.Coalgebra F)
:
V.Hom V
The identity morphism of an algebra of endofunctor F
Equations
- CategoryTheory.Endofunctor.Coalgebra.Hom.id V = { f := CategoryTheory.CategoryStruct.id V.V, h := ⋯ }
Instances For
instance
CategoryTheory.Endofunctor.Coalgebra.Hom.instInhabited
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(V : CategoryTheory.Endofunctor.Coalgebra F)
:
Inhabited (V.Hom V)
Equations
- CategoryTheory.Endofunctor.Coalgebra.Hom.instInhabited V = { default := { f := CategoryTheory.CategoryStruct.id V.V, h := ⋯ } }
def
CategoryTheory.Endofunctor.Coalgebra.Hom.comp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ V₂ : CategoryTheory.Endofunctor.Coalgebra F}
(f : V₀.Hom V₁)
(g : V₁.Hom V₂)
:
V₀.Hom V₂
The composition of morphisms between algebras of endofunctor F
Equations
- f.comp g = { f := CategoryTheory.CategoryStruct.comp f.f g.f, h := ⋯ }
Instances For
instance
CategoryTheory.Endofunctor.Coalgebra.instCategoryStruct
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Equations
- One or more equations did not get rendered due to their size.
theorem
CategoryTheory.Endofunctor.Coalgebra.ext
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{A B : CategoryTheory.Endofunctor.Coalgebra F}
{f g : A ⟶ B}
(w : f.f = g.f := by aesop_cat)
:
f = g
@[simp]
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.id_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(V : CategoryTheory.Endofunctor.Coalgebra F)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.comp_eq_comp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ V₂ : CategoryTheory.Endofunctor.Coalgebra F}
(f : V₀ ⟶ V₁)
(g : V₁ ⟶ V₂)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.comp_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ V₂ : CategoryTheory.Endofunctor.Coalgebra F}
(f : V₀ ⟶ V₁)
(g : V₁ ⟶ V₂)
:
(CategoryTheory.CategoryStruct.comp f g).f = CategoryTheory.CategoryStruct.comp f.f g.f
instance
CategoryTheory.Endofunctor.Coalgebra.instCategory
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
Coalgebras of an endofunctor F form a category
Equations
def
CategoryTheory.Endofunctor.Coalgebra.isoMk
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
(h : V₀.V ≅ V₁.V)
(w : CategoryTheory.CategoryStruct.comp V₀.str (F.map h.hom) = CategoryTheory.CategoryStruct.comp h.hom V₁.str := by
aesop_cat)
:
V₀ ≅ V₁
To construct an isomorphism of coalgebras, it suffices to give an isomorphism of the Vs which commutes with the structure morphisms.
Equations
- CategoryTheory.Endofunctor.Coalgebra.isoMk h w = { hom := { f := h.hom, h := ⋯ }, inv := { f := h.inv, h := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.isoMk_hom_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
(h : V₀.V ≅ V₁.V)
(w : CategoryTheory.CategoryStruct.comp V₀.str (F.map h.hom) = CategoryTheory.CategoryStruct.comp h.hom V₁.str := by
aesop_cat)
:
(CategoryTheory.Endofunctor.Coalgebra.isoMk h w).hom.f = h.hom
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.isoMk_inv_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
(h : V₀.V ≅ V₁.V)
(w : CategoryTheory.CategoryStruct.comp V₀.str (F.map h.hom) = CategoryTheory.CategoryStruct.comp h.hom V₁.str := by
aesop_cat)
:
(CategoryTheory.Endofunctor.Coalgebra.isoMk h w).inv.f = h.inv
def
CategoryTheory.Endofunctor.Coalgebra.forget
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
:
The forgetful functor from the category of coalgebras, forgetting the coalgebraic structure.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.forget_map
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
{X✝ Y✝ : CategoryTheory.Endofunctor.Coalgebra F}
(f : X✝ ⟶ Y✝)
:
(CategoryTheory.Endofunctor.Coalgebra.forget F).map f = f.f
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.forget_obj
{C : Type u}
[CategoryTheory.Category.{v, u} C]
(F : CategoryTheory.Functor C C)
(A : CategoryTheory.Endofunctor.Coalgebra F)
:
(CategoryTheory.Endofunctor.Coalgebra.forget F).obj A = A.V
theorem
CategoryTheory.Endofunctor.Coalgebra.iso_of_iso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{V₀ V₁ : CategoryTheory.Endofunctor.Coalgebra F}
(f : V₀ ⟶ V₁)
[CategoryTheory.IsIso f.f]
:
A coalgebra morphism with an underlying isomorphism hom in C is a coalgebra isomorphism.
instance
CategoryTheory.Endofunctor.Coalgebra.forget_reflects_iso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
(CategoryTheory.Endofunctor.Coalgebra.forget F).ReflectsIsomorphisms
Equations
- ⋯ = ⋯
instance
CategoryTheory.Endofunctor.Coalgebra.forget_faithful
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
(CategoryTheory.Endofunctor.Coalgebra.forget F).Faithful
Equations
- ⋯ = ⋯
theorem
CategoryTheory.Endofunctor.Coalgebra.epi_of_epi
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{X Y : CategoryTheory.Endofunctor.Coalgebra F}
(f : X ⟶ Y)
[h : CategoryTheory.Epi f.f]
:
An algebra morphism with an underlying epimorphism hom in C is an algebra epimorphism.
theorem
CategoryTheory.Endofunctor.Coalgebra.mono_of_mono
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
{X Y : CategoryTheory.Endofunctor.Coalgebra F}
(f : X ⟶ Y)
[h : CategoryTheory.Mono f.f]
:
An algebra morphism with an underlying monomorphism hom in C is an algebra monomorphism.
def
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ⟶ G)
:
From a natural transformation α : F → G we get a functor from
coalgebras of F to coalgebras of G.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans_obj_V
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ⟶ G)
(V : CategoryTheory.Endofunctor.Coalgebra F)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans α).obj V).V = V.V
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans_obj_str
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ⟶ G)
(V : CategoryTheory.Endofunctor.Coalgebra F)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans α).obj V).str = CategoryTheory.CategoryStruct.comp V.str (α.app V.V)
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans_map_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ⟶ G)
{X✝ Y✝ : CategoryTheory.Endofunctor.Coalgebra F}
(f : X✝ ⟶ Y✝)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTrans α).map f).f = f.f
def
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
:
The identity transformation induces the identity endofunctor on the category of coalgebras.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(X : CategoryTheory.Endofunctor.Coalgebra F)
:
(CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId.hom.app X).f = CategoryTheory.CategoryStruct.id X.V
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C C}
(X : CategoryTheory.Endofunctor.Coalgebra F)
:
(CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId.inv.app X).f = CategoryTheory.CategoryStruct.id X.V
def
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
:
A composition of natural transformations gives the composition of corresponding functors.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
(X : CategoryTheory.Endofunctor.Coalgebra F₀)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp α β).inv.app X).f = CategoryTheory.CategoryStruct.id X.V
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F₀ F₁ F₂ : CategoryTheory.Functor C C}
(α : F₀ ⟶ F₁)
(β : F₁ ⟶ F₂)
(X : CategoryTheory.Endofunctor.Coalgebra F₀)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp α β).hom.app X).f = CategoryTheory.CategoryStruct.id X.V
def
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
:
If α and β are two equal natural transformations, then the functors of coalgebras induced by
them are isomorphic.
We define it like this as opposed to using eq_to_iso so that the components are nicer to prove
lemmas about.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq_inv_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
(X : CategoryTheory.Endofunctor.Coalgebra F)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq h).inv.app X).f = CategoryTheory.CategoryStruct.id X.V
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq_hom_app_f
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
{α β : F ⟶ G}
(h : α = β)
(X : CategoryTheory.Endofunctor.Coalgebra F)
:
((CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq h).hom.app X).f = CategoryTheory.CategoryStruct.id X.V
def
CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
Naturally isomorphic endofunctors give equivalent categories of coalgebras.
Furthermore, they are equivalent as categories over C, that is,
we have equiv_of_nat_iso h ⋙ forget = forget.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_unitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
(CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso α).unitIso = CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId.symm ≪≫ CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq ⋯ ≪≫ CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp α.hom α.inv
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_functor
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_inverse
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
@[simp]
theorem
CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso_counitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(α : F ≅ G)
:
(CategoryTheory.Endofunctor.Coalgebra.equivOfNatIso α).counitIso = (CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransComp α.inv α.hom).symm ≪≫ CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransEq ⋯ ≪≫ CategoryTheory.Endofunctor.Coalgebra.functorOfNatTransId
theorem
CategoryTheory.Endofunctor.Adjunction.Algebra.homEquiv_naturality_str
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
(A₁ A₂ : CategoryTheory.Endofunctor.Algebra F)
(f : A₁ ⟶ A₂)
:
CategoryTheory.CategoryStruct.comp ((adj.homEquiv A₁.a A₁.a) A₁.str) (G.map f.f) = CategoryTheory.CategoryStruct.comp f.f ((adj.homEquiv A₂.a A₂.a) A₂.str)
theorem
CategoryTheory.Endofunctor.Adjunction.Coalgebra.homEquiv_naturality_str_symm
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
(V₁ V₂ : CategoryTheory.Endofunctor.Coalgebra G)
(f : V₁ ⟶ V₂)
:
CategoryTheory.CategoryStruct.comp (F.map f.f) ((adj.homEquiv V₂.V V₂.V).symm V₂.str) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv V₁.V V₁.V).symm V₁.str) f.f
def
CategoryTheory.Endofunctor.Adjunction.Algebra.toCoalgebraOf
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
:
Given an adjunction F ⊣ G, the functor that associates to an algebra over F a
coalgebra over G defined via adjunction applied to the structure map.
Equations
- One or more equations did not get rendered due to their size.
Instances For
def
CategoryTheory.Endofunctor.Adjunction.Coalgebra.toAlgebraOf
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
:
Given an adjunction F ⊣ G, the functor that associates to a coalgebra over G an algebra over
F defined via adjunction applied to the structure map.
Equations
- One or more equations did not get rendered due to their size.
Instances For
def
CategoryTheory.Endofunctor.Adjunction.AlgCoalgEquiv.unitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
:
Given an adjunction, assigning to an algebra over the left adjoint a coalgebra over its right adjoint and going back is isomorphic to the identity functor.
Equations
- One or more equations did not get rendered due to their size.
Instances For
def
CategoryTheory.Endofunctor.Adjunction.AlgCoalgEquiv.counitIso
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
:
Given an adjunction, assigning to a coalgebra over the right adjoint an algebra over the left adjoint and going back is isomorphic to the identity functor.
Equations
- One or more equations did not get rendered due to their size.
Instances For
def
CategoryTheory.Endofunctor.Adjunction.algebraCoalgebraEquiv
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{F G : CategoryTheory.Functor C C}
(adj : F ⊣ G)
:
If F is left adjoint to G, then the category of algebras over F is equivalent to the
category of coalgebras over G.
Equations
- One or more equations did not get rendered due to their size.