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Mathlib/Algebra/Category/AlgebraCat/Basic.lean

Lines changed: 5 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -66,7 +66,7 @@ structure Hom (A B : AlgebraCat.{v} R) where
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instance : Category (AlgebraCat.{v} R) where
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Hom A B := Hom A B
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id A := ⟨AlgHom.id R A⟩
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comp f g := ⟨g.algHom.comp f.algHom
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comp f g := ⟨g.hom.comp f.hom
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instance {M N : AlgebraCat.{v} R} : CoeFun (M ⟶ N) (fun _ ↦ M → N) where
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coe f := f.hom
@@ -130,7 +130,7 @@ instance : Inhabited (AlgebraCat R) :=
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instance : ConcreteCategory.{v} (AlgebraCat.{v} R) where
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forget :=
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{ obj := fun R => R
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map := fun f => f.algHom }
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map := fun f => f.hom }
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forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
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lemma forget_obj {A : AlgebraCat.{v} R} : (forget (AlgebraCat.{v} R)).obj A = A := rfl
@@ -148,12 +148,12 @@ instance {S : AlgebraCat.{v} R} : Algebra R ((forget (AlgebraCat R)).obj S) :=
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instance hasForgetToRing : HasForget₂ (AlgebraCat.{v} R) RingCat.{v} where
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forget₂ :=
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{ obj := fun A => RingCat.of A
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map := fun f => RingCat.ofHom f.algHom.toRingHom }
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map := fun f => RingCat.ofHom f.hom.toRingHom }
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instance hasForgetToModule : HasForget₂ (AlgebraCat.{v} R) (ModuleCat.{v} R) where
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forget₂ :=
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{ obj := fun M => ModuleCat.of R M
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map := fun f => ModuleCat.asHom f.algHom.toLinearMap }
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map := fun f => ModuleCat.asHom f.hom.toLinearMap }
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@[simp]
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lemma forget₂_module_obj (X : AlgebraCat.{v} R) :
@@ -183,7 +183,7 @@ def free : Type u ⥤ AlgebraCat.{u} R where
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def adj : free.{u} R ⊣ forget (AlgebraCat.{u} R) :=
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Adjunction.mkOfHomEquiv
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{ homEquiv := fun _ _ =>
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{ toFun := fun f ↦ (FreeAlgebra.lift _).symm f.algHom
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{ toFun := fun f ↦ (FreeAlgebra.lift _).symm f.hom
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invFun := fun f ↦ ofHom <| (FreeAlgebra.lift _) f
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left_inv := fun f ↦ by aesop
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right_inv := fun f ↦ by simp [forget_obj, forget_map] } }

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