Linear structure on functor categories

If C and D are categories and D is R-linear, then C ⥤ D is also R-linear.

@[instance_reducible]

instance CategoryTheory.functorCategoryLinear {R : Type u_1} [Semiring R] {C : Type u_2} {D : Type u_3} [Category.{v_1, u_2} C] [Category.{v_2, u_3} D] [Preadditive D] [Linear R D] : Linear R (Functor C D)

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.NatTrans.appLinearMap {R : Type u_1} [Semiring R] {C : Type u_2} {D : Type u_3} [Category.{v_1, u_2} C] [Category.{v_2, u_3} D] [Preadditive D] [Linear R D] {F G : Functor C D} (X : C) : (F ⟶ G) →ₗ[R] F.obj X ⟶ G.obj X

Application of a natural transformation at a fixed object, as group homomorphism

Equations

• CategoryTheory.NatTrans.appLinearMap X = { toFun := fun (α : F ⟶ G) => α.app X, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem CategoryTheory.NatTrans.appLinearMap_apply {R : Type u_1} [Semiring R] {C : Type u_2} {D : Type u_3} [Category.{v_1, u_2} C] [Category.{v_2, u_3} D] [Preadditive D] [Linear R D] {F G : Functor C D} (X : C) (α : F ⟶ G) : (appLinearMap X) α = α.app X

@[simp]

theorem CategoryTheory.NatTrans.app_smul {R : Type u_1} [Semiring R] {C : Type u_2} {D : Type u_3} [Category.{v_1, u_2} C] [Category.{v_2, u_3} D] [Preadditive D] [Linear R D] {F G : Functor C D} (X : C) (r : R) (α : F ⟶ G) : (r • α).app X = r • α.app X