Monads

We construct the categories of monads and comonads, and their forgetful functors to endofunctors.

(Note that these are the category theorist's monads, not the programmers monads. For the translation, see the file CategoryTheory.Monad.Types.)

For the fact that monads are "just" monoids in the category of endofunctors, see the file CategoryTheory.Monad.EquivMon.

structure CategoryTheory.Monad (C : Type u₁) [Category.{v₁, u₁} C] extends CategoryTheory.Functor C C : Type (max u₁ v₁)

The data of a monad on C consists of an endofunctor T together with natural transformations η : 𝟭 C ⟶ T and μ : T ⋙ T ⟶ T satisfying three equations:

• T μ_X ≫ μ_X = μ_(TX) ≫ μ_X (associativity)
• η_(TX) ≫ μ_X = 1_X (left unit)
• Tη_X ≫ μ_X = 1_X (right unit)

• obj : C → C
• map {X Y : C} : (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
• map_id (X : C) : self.map (CategoryStruct.id X) = CategoryStruct.id (self.obj X)
• map_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : self.map (CategoryStruct.comp f g) = CategoryStruct.comp (self.map f) (self.map g)
• η : Functor.id C ⟶ self.toFunctor

  The unit for the monad.

• μ : self.comp self.toFunctor ⟶ self.toFunctor

  The multiplication for the monad.

• assoc (X : C) : CategoryStruct.comp (self.map (self.μ.app X)) (self.μ.app X) = CategoryStruct.comp (self.μ.app (self.obj X)) (self.μ.app X)
• left_unit (X : C) : CategoryStruct.comp (self.η.app (self.obj X)) (self.μ.app X) = CategoryStruct.id ((Functor.id C).obj (self.obj X))
• right_unit (X : C) : CategoryStruct.comp (self.map (self.η.app X)) (self.μ.app X) = CategoryStruct.id (self.obj ((Functor.id C).obj X))

theorem CategoryTheory.Monad.unit_naturality (C : Type u₁) [Category.{v₁, u₁} C] (T : Monad C) ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp f (T.η.app Y) = CategoryStruct.comp (T.η.app X) (T.map f)

theorem CategoryTheory.Monad.unit_naturality_assoc (C : Type u₁) [Category.{v₁, u₁} C] (T : Monad C) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : C} (h : T.obj Y ⟶ Z) : CategoryStruct.comp f (CategoryStruct.comp (T.η.app Y) h) = CategoryStruct.comp (T.η.app X) (CategoryStruct.comp (T.map f) h)

theorem CategoryTheory.Monad.mu_naturality (C : Type u₁) [Category.{v₁, u₁} C] (T : Monad C) ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp (T.map (T.map f)) (T.μ.app Y) = CategoryStruct.comp (T.μ.app X) (T.map f)

theorem CategoryTheory.Monad.mu_naturality_assoc (C : Type u₁) [Category.{v₁, u₁} C] (T : Monad C) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : C} (h : T.obj Y ⟶ Z) : CategoryStruct.comp (T.map (T.map f)) (CategoryStruct.comp (T.μ.app Y) h) = CategoryStruct.comp (T.μ.app X) (CategoryStruct.comp (T.map f) h)

structure CategoryTheory.Comonad (C : Type u₁) [Category.{v₁, u₁} C] extends CategoryTheory.Functor C C : Type (max u₁ v₁)

The data of a comonad on C consists of an endofunctor G together with natural transformations ε : G ⟶ 𝟭 C and δ : G ⟶ G ⋙ G satisfying three equations:

• δ_X ≫ G δ_X = δ_X ≫ δ_(GX) (coassociativity)
• δ_X ≫ ε_(GX) = 1_X (left counit)
• δ_X ≫ G ε_X = 1_X (right counit)

• obj : C → C
• map {X Y : C} : (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
• map_id (X : C) : self.map (CategoryStruct.id X) = CategoryStruct.id (self.obj X)
• map_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : self.map (CategoryStruct.comp f g) = CategoryStruct.comp (self.map f) (self.map g)
• ε : self.toFunctor ⟶ Functor.id C

  The counit for the comonad.

• δ : self.toFunctor ⟶ self.comp self.toFunctor

  The comultiplication for the comonad.

• coassoc (X : C) : CategoryStruct.comp (self.δ.app X) (self.map (self.δ.app X)) = CategoryStruct.comp (self.δ.app X) (self.δ.app (self.obj X))
• left_counit (X : C) : CategoryStruct.comp (self.δ.app X) (self.ε.app (self.obj X)) = CategoryStruct.id (self.obj X)
• right_counit (X : C) : CategoryStruct.comp (self.δ.app X) (self.map (self.ε.app X)) = CategoryStruct.id (self.obj X)

theorem CategoryTheory.Comonad.counit_naturality (C : Type u₁) [Category.{v₁, u₁} C] (T : Comonad C) ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp (T.map f) (T.ε.app Y) = CategoryStruct.comp (T.ε.app X) f

theorem CategoryTheory.Comonad.counit_naturality_assoc (C : Type u₁) [Category.{v₁, u₁} C] (T : Comonad C) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : C} (h : Y ⟶ Z) : CategoryStruct.comp (T.map f) (CategoryStruct.comp (T.ε.app Y) h) = CategoryStruct.comp (T.ε.app X) (CategoryStruct.comp f h)

theorem CategoryTheory.Comonad.delta_naturality (C : Type u₁) [Category.{v₁, u₁} C] (T : Comonad C) ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp (T.map f) (T.δ.app Y) = CategoryStruct.comp (T.δ.app X) (T.map (T.map f))

theorem CategoryTheory.Comonad.delta_naturality_assoc (C : Type u₁) [Category.{v₁, u₁} C] (T : Comonad C) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : C} (h : T.obj (T.obj Y) ⟶ Z) : CategoryStruct.comp (T.map f) (CategoryStruct.comp (T.δ.app Y) h) = CategoryStruct.comp (T.δ.app X) (CategoryStruct.comp (T.map (T.map f)) h)

instance CategoryTheory.coeMonad {C : Type u₁} [Category.{v₁, u₁} C] : Coe (Monad C) (Functor C C)

instance CategoryTheory.coeComonad {C : Type u₁} [Category.{v₁, u₁} C] : Coe (Comonad C) (Functor C C)

@[simp]

theorem CategoryTheory.Monad.left_unit_assoc {C : Type u₁} [Category.{v₁, u₁} C] (self : Monad C) (X : C) {Z : C} (h : self.obj X ⟶ Z) : CategoryStruct.comp (self.η.app (self.obj X)) (CategoryStruct.comp (self.μ.app X) h) = h

@[simp]

theorem CategoryTheory.Monad.right_unit_assoc {C : Type u₁} [Category.{v₁, u₁} C] (self : Monad C) (X : C) {Z : C} (h : self.obj X ⟶ Z) : CategoryStruct.comp (self.map (self.η.app X)) (CategoryStruct.comp (self.μ.app X) h) = h

@[simp]

theorem CategoryTheory.Comonad.coassoc_assoc {C : Type u₁} [Category.{v₁, u₁} C] (self : Comonad C) (X : C) {Z : C} (h : self.obj (self.obj (self.obj X)) ⟶ Z) : CategoryStruct.comp (self.δ.app X) (CategoryStruct.comp (self.map (self.δ.app X)) h) = CategoryStruct.comp (self.δ.app X) (CategoryStruct.comp (self.δ.app (self.obj X)) h)

@[simp]

theorem CategoryTheory.Comonad.right_counit_assoc {C : Type u₁} [Category.{v₁, u₁} C] (self : Comonad C) (X : C) {Z : C} (h : self.obj X ⟶ Z) : CategoryStruct.comp (self.δ.app X) (CategoryStruct.comp (self.map (self.ε.app X)) h) = h

@[simp]

theorem CategoryTheory.Comonad.left_counit_assoc {C : Type u₁} [Category.{v₁, u₁} C] (self : Comonad C) (X : C) {Z : C} (h : self.obj X ⟶ Z) : CategoryStruct.comp (self.δ.app X) (CategoryStruct.comp (self.ε.app (self.obj X)) h) = h

structure CategoryTheory.MonadHom {C : Type u₁} [Category.{v₁, u₁} C] (T₁ T₂ : Monad C) extends CategoryTheory.NatTrans T₁.toFunctor T₂.toFunctor : Type (max u₁ v₁)

A morphism of monads is a natural transformation compatible with η and μ.

• app (X : C) : T₁.obj X ⟶ T₂.obj X
• naturality ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp (T₁.map f) (self.app Y) = CategoryStruct.comp (self.app X) (T₂.map f)
• app_η (X : C) : CategoryStruct.comp (T₁.η.app X) (self.app X) = T₂.η.app X
• app_μ (X : C) : CategoryStruct.comp (T₁.μ.app X) (self.app X) = CategoryStruct.comp (CategoryStruct.comp (T₁.map (self.app X)) (self.app (T₂.obj X))) (T₂.μ.app X)

theorem CategoryTheory.MonadHom.ext {C : Type u₁} {inst✝ : Category.{v₁, u₁} C} {T₁ T₂ : Monad C} {x y : MonadHom T₁ T₂} (app : x.app = y.app) : x = y

theorem CategoryTheory.MonadHom.ext_iff {C : Type u₁} {inst✝ : Category.{v₁, u₁} C} {T₁ T₂ : Monad C} {x y : MonadHom T₁ T₂} : x = y ↔ x.app = y.app

structure CategoryTheory.ComonadHom {C : Type u₁} [Category.{v₁, u₁} C] (M N : Comonad C) extends CategoryTheory.NatTrans M.toFunctor N.toFunctor : Type (max u₁ v₁)

A morphism of comonads is a natural transformation compatible with ε and δ.

• app (X : C) : M.obj X ⟶ N.obj X
• naturality ⦃X Y : C⦄ (f : X ⟶ Y) : CategoryStruct.comp (M.map f) (self.app Y) = CategoryStruct.comp (self.app X) (N.map f)
• app_ε (X : C) : CategoryStruct.comp (self.app X) (N.ε.app X) = M.ε.app X
• app_δ (X : C) : CategoryStruct.comp (self.app X) (N.δ.app X) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (self.app (M.obj X)) (N.map (self.app X)))

theorem CategoryTheory.ComonadHom.ext {C : Type u₁} {inst✝ : Category.{v₁, u₁} C} {M N : Comonad C} {x y : ComonadHom M N} (app : x.app = y.app) : x = y

theorem CategoryTheory.ComonadHom.ext_iff {C : Type u₁} {inst✝ : Category.{v₁, u₁} C} {M N : Comonad C} {x y : ComonadHom M N} : x = y ↔ x.app = y.app

@[simp]

theorem CategoryTheory.MonadHom.app_μ_assoc {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Monad C} (self : MonadHom T₁ T₂) (X : C) {Z : C} (h : T₂.obj X ⟶ Z) : CategoryStruct.comp (T₁.μ.app X) (CategoryStruct.comp (self.app X) h) = CategoryStruct.comp (T₁.map (self.app X)) (CategoryStruct.comp (self.app (T₂.obj X)) (CategoryStruct.comp (T₂.μ.app X) h))

@[simp]

theorem CategoryTheory.MonadHom.app_η_assoc {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Monad C} (self : MonadHom T₁ T₂) (X : C) {Z : C} (h : T₂.obj X ⟶ Z) : CategoryStruct.comp (T₁.η.app X) (CategoryStruct.comp (self.app X) h) = CategoryStruct.comp (T₂.η.app X) h

@[simp]

theorem CategoryTheory.ComonadHom.app_δ_assoc {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (self : ComonadHom M N) (X : C) {Z : C} (h : N.obj (N.obj X) ⟶ Z) : CategoryStruct.comp (self.app X) (CategoryStruct.comp (N.δ.app X) h) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (self.app (M.obj X)) (CategoryStruct.comp (N.map (self.app X)) h))

@[simp]

theorem CategoryTheory.ComonadHom.app_ε_assoc {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (self : ComonadHom M N) (X : C) {Z : C} (h : X ⟶ Z) : CategoryStruct.comp (self.app X) (CategoryStruct.comp (N.ε.app X) h) = CategoryStruct.comp (M.ε.app X) h

instance CategoryTheory.instQuiverMonad {C : Type u₁} [Category.{v₁, u₁} C] : Quiver (Monad C)

instance CategoryTheory.instQuiverComonad {C : Type u₁} [Category.{v₁, u₁} C] : Quiver (Comonad C)

theorem CategoryTheory.MonadHom.ext' {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Monad C} (f g : T₁ ⟶ T₂) (h : f.app = g.app) : f = g

theorem CategoryTheory.MonadHom.ext'_iff {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Monad C} {f g : T₁ ⟶ T₂} : f = g ↔ f.app = g.app

theorem CategoryTheory.ComonadHom.ext' {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Comonad C} (f g : T₁ ⟶ T₂) (h : f.app = g.app) : f = g

theorem CategoryTheory.ComonadHom.ext'_iff {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ : Comonad C} {f g : T₁ ⟶ T₂} : f = g ↔ f.app = g.app

instance CategoryTheory.instCategoryMonad {C : Type u₁} [Category.{v₁, u₁} C] : Category.{max u₁ v₁, max u₁ v₁} (Monad C)

instance CategoryTheory.instCategoryComonad {C : Type u₁} [Category.{v₁, u₁} C] : Category.{max u₁ v₁, max u₁ v₁} (Comonad C)

instance CategoryTheory.instInhabitedMonadHom {C : Type u₁} [Category.{v₁, u₁} C] {T : Monad C} : Inhabited (MonadHom T T)

@[simp]

theorem CategoryTheory.MonadHom.id_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] (T : Monad C) : (CategoryStruct.id T).toNatTrans = CategoryStruct.id T.toFunctor

@[simp]

theorem CategoryTheory.MonadHom.comp_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ T₃ : Monad C} (f : T₁ ⟶ T₂) (g : T₂ ⟶ T₃) : (CategoryStruct.comp f g).toNatTrans = CategoryStruct.comp f.toNatTrans g.toNatTrans

instance CategoryTheory.instInhabitedComonadHom {C : Type u₁} [Category.{v₁, u₁} C] {G : Comonad C} : Inhabited (ComonadHom G G)

@[simp]

theorem CategoryTheory.ComonadHom.id_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] (T : Comonad C) : (CategoryStruct.id T).toNatTrans = CategoryStruct.id T.toFunctor

@[simp]

theorem CategoryTheory.comp_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {T₁ T₂ T₃ : Comonad C} (f : T₁ ⟶ T₂) (g : T₂ ⟶ T₃) : (CategoryStruct.comp f g).toNatTrans = CategoryStruct.comp f.toNatTrans g.toNatTrans

def CategoryTheory.MonadIso.mk {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (f : M.toFunctor ≅ N.toFunctor) (f_η : ∀ (X : C), CategoryStruct.comp (M.η.app X) (f.hom.app X) = N.η.app X := by cat_disch) (f_μ : ∀ (X : C), CategoryStruct.comp (M.μ.app X) (f.hom.app X) = CategoryStruct.comp (CategoryStruct.comp (M.map (f.hom.app X)) (f.hom.app (N.obj X))) (N.μ.app X) := by cat_disch) : M ≅ N

Construct a monad isomorphism from a natural isomorphism of functors where the forward direction is a monad morphism.

@[simp]

theorem CategoryTheory.MonadIso.mk_hom_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (f : M.toFunctor ≅ N.toFunctor) (f_η : ∀ (X : C), CategoryStruct.comp (M.η.app X) (f.hom.app X) = N.η.app X := by cat_disch) (f_μ : ∀ (X : C), CategoryStruct.comp (M.μ.app X) (f.hom.app X) = CategoryStruct.comp (CategoryStruct.comp (M.map (f.hom.app X)) (f.hom.app (N.obj X))) (N.μ.app X) := by cat_disch) : (mk f f_η f_μ).hom.toNatTrans = f.hom

@[simp]

theorem CategoryTheory.MonadIso.mk_inv_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (f : M.toFunctor ≅ N.toFunctor) (f_η : ∀ (X : C), CategoryStruct.comp (M.η.app X) (f.hom.app X) = N.η.app X := by cat_disch) (f_μ : ∀ (X : C), CategoryStruct.comp (M.μ.app X) (f.hom.app X) = CategoryStruct.comp (CategoryStruct.comp (M.map (f.hom.app X)) (f.hom.app (N.obj X))) (N.μ.app X) := by cat_disch) : (mk f f_η f_μ).inv.toNatTrans = f.inv

def CategoryTheory.ComonadIso.mk {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (f : M.toFunctor ≅ N.toFunctor) (f_ε : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.ε.app X) = M.ε.app X := by cat_disch) (f_δ : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.δ.app X) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (f.hom.app (M.obj X)) (N.map (f.hom.app X))) := by cat_disch) : M ≅ N

Construct a comonad isomorphism from a natural isomorphism of functors where the forward direction is a comonad morphism.

@[simp]

theorem CategoryTheory.ComonadIso.mk_inv_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (f : M.toFunctor ≅ N.toFunctor) (f_ε : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.ε.app X) = M.ε.app X := by cat_disch) (f_δ : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.δ.app X) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (f.hom.app (M.obj X)) (N.map (f.hom.app X))) := by cat_disch) : (mk f f_ε f_δ).inv.toNatTrans = f.inv

@[simp]

theorem CategoryTheory.ComonadIso.mk_hom_toNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (f : M.toFunctor ≅ N.toFunctor) (f_ε : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.ε.app X) = M.ε.app X := by cat_disch) (f_δ : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.δ.app X) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (f.hom.app (M.obj X)) (N.map (f.hom.app X))) := by cat_disch) : (mk f f_ε f_δ).hom.toNatTrans = f.hom

def CategoryTheory.monadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : Functor (Monad C) (Functor C C)

The forgetful functor from the category of monads to the category of endofunctors.

@[simp]

theorem CategoryTheory.monadToFunctor_obj (C : Type u₁) [Category.{v₁, u₁} C] (T : Monad C) : (monadToFunctor C).obj T = T.toFunctor

@[simp]

theorem CategoryTheory.monadToFunctor_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : Monad C} (f : X✝ ⟶ Y✝) : (monadToFunctor C).map f = f.toNatTrans

instance CategoryTheory.instFaithfulMonadFunctorMonadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : (monadToFunctor C).Faithful

theorem CategoryTheory.monadToFunctor_mapIso_monad_iso_mk (C : Type u₁) [Category.{v₁, u₁} C] {M N : Monad C} (f : M.toFunctor ≅ N.toFunctor) (f_η : ∀ (X : C), CategoryStruct.comp (M.η.app X) (f.hom.app X) = N.η.app X) (f_μ : ∀ (X : C), CategoryStruct.comp (M.μ.app X) (f.hom.app X) = CategoryStruct.comp (CategoryStruct.comp (M.map (f.hom.app X)) (f.hom.app (N.obj X))) (N.μ.app X)) : (monadToFunctor C).mapIso (MonadIso.mk f f_η f_μ) = f

instance CategoryTheory.instReflectsIsomorphismsMonadFunctorMonadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : (monadToFunctor C).ReflectsIsomorphisms

def CategoryTheory.comonadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : Functor (Comonad C) (Functor C C)

The forgetful functor from the category of comonads to the category of endofunctors.

@[simp]

theorem CategoryTheory.comonadToFunctor_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : Comonad C} (f : X✝ ⟶ Y✝) : (comonadToFunctor C).map f = f.toNatTrans

@[simp]

theorem CategoryTheory.comonadToFunctor_obj (C : Type u₁) [Category.{v₁, u₁} C] (G : Comonad C) : (comonadToFunctor C).obj G = G.toFunctor

instance CategoryTheory.instFaithfulComonadFunctorComonadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : (comonadToFunctor C).Faithful

theorem CategoryTheory.comonadToFunctor_mapIso_comonad_iso_mk (C : Type u₁) [Category.{v₁, u₁} C] {M N : Comonad C} (f : M.toFunctor ≅ N.toFunctor) (f_ε : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.ε.app X) = M.ε.app X) (f_δ : ∀ (X : C), CategoryStruct.comp (f.hom.app X) (N.δ.app X) = CategoryStruct.comp (M.δ.app X) (CategoryStruct.comp (f.hom.app (M.obj X)) (N.map (f.hom.app X)))) : (comonadToFunctor C).mapIso (ComonadIso.mk f f_ε f_δ) = f

instance CategoryTheory.instReflectsIsomorphismsComonadFunctorComonadToFunctor (C : Type u₁) [Category.{v₁, u₁} C] : (comonadToFunctor C).ReflectsIsomorphisms

def CategoryTheory.MonadIso.toNatIso {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (h : M ≅ N) : M.toFunctor ≅ N.toFunctor

An isomorphism of monads gives a natural isomorphism of the underlying functors.

@[simp]

theorem CategoryTheory.MonadIso.toNatIso_hom {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (h : M ≅ N) : (toNatIso h).hom = h.hom.toNatTrans

@[simp]

theorem CategoryTheory.MonadIso.toNatIso_inv {C : Type u₁} [Category.{v₁, u₁} C] {M N : Monad C} (h : M ≅ N) : (toNatIso h).inv = h.inv.toNatTrans

def CategoryTheory.ComonadIso.toNatIso {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (h : M ≅ N) : M.toFunctor ≅ N.toFunctor

An isomorphism of comonads gives a natural isomorphism of the underlying functors.

@[simp]

theorem CategoryTheory.ComonadIso.toNatIso_hom {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (h : M ≅ N) : (toNatIso h).hom = h.hom.toNatTrans

@[simp]

theorem CategoryTheory.ComonadIso.toNatIso_inv {C : Type u₁} [Category.{v₁, u₁} C] {M N : Comonad C} (h : M ≅ N) : (toNatIso h).inv = h.inv.toNatTrans

def CategoryTheory.Monad.id (C : Type u₁) [Category.{v₁, u₁} C] : Monad C

The identity monad.

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theorem CategoryTheory.Monad.id_obj (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).obj X = X

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theorem CategoryTheory.Monad.id_η_app (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).η.app X = CategoryStruct.id X

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theorem CategoryTheory.Monad.id_μ_app (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).μ.app X = CategoryStruct.id X

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theorem CategoryTheory.Monad.id_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : C} (f : X✝ ⟶ Y✝) : (id C).map f = f

instance CategoryTheory.Monad.instInhabited (C : Type u₁) [Category.{v₁, u₁} C] : Inhabited (Monad C)

def CategoryTheory.Comonad.id (C : Type u₁) [Category.{v₁, u₁} C] : Comonad C

The identity comonad.

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theorem CategoryTheory.Comonad.id_ε_app (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).ε.app X = CategoryStruct.id X

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theorem CategoryTheory.Comonad.id_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : C} (f : X✝ ⟶ Y✝) : (id C).map f = f

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theorem CategoryTheory.Comonad.id_obj (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).obj X = X

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theorem CategoryTheory.Comonad.id_δ_app (C : Type u₁) [Category.{v₁, u₁} C] (X : C) : (id C).δ.app X = CategoryStruct.id X

instance CategoryTheory.Comonad.instInhabited (C : Type u₁) [Category.{v₁, u₁} C] : Inhabited (Comonad C)

def CategoryTheory.Monad.transport {C : Type u₁} [Category.{v₁, u₁} C] {F : Functor C C} (T : Monad C) (i : T.toFunctor ≅ F) : Monad C

Transport a monad structure on a functor along an isomorphism of functors.

def CategoryTheory.Comonad.transport {C : Type u₁} [Category.{v₁, u₁} C] {F : Functor C C} (T : Comonad C) (i : T.toFunctor ≅ F) : Comonad C

Transport a comonad structure on a functor along an isomorphism of functors.

theorem CategoryTheory.Monad.map_unit_app {C : Type u₁} [Category.{v₁, u₁} C] (T : Monad C) (X : C) [IsIso T.μ] : T.map (T.η.app X) = T.η.app (T.obj X)

theorem CategoryTheory.Monad.isSplitMono_iff_isIso_unit {C : Type u₁} [Category.{v₁, u₁} C] (T : Monad C) (X : C) [IsIso T.μ] : IsSplitMono (T.η.app X) ↔ IsIso (T.η.app X)

theorem CategoryTheory.Comonad.map_counit_app {C : Type u₁} [Category.{v₁, u₁} C] (T : Comonad C) (X : C) [IsIso T.δ] : T.map (T.ε.app X) = T.ε.app (T.obj X)

theorem CategoryTheory.Comonad.isSplitEpi_iff_isIso_counit {C : Type u₁} [Category.{v₁, u₁} C] (T : Comonad C) (X : C) [IsIso T.δ] : IsSplitEpi (T.ε.app X) ↔ IsIso (T.ε.app X)