Linear categories

An R-linear category is a category in which X ⟶ Y is an R-module in such a way that composition of morphisms is R-linear in both variables.

Note that sometimes in the literature a "linear category" is further required to be abelian.

Implementation

Corresponding to the fact that we need to have an AddCommGroup X structure in place to talk about a Module R X structure, we need Preadditive C as a prerequisite typeclass for Linear R C. This makes for longer signatures than would be ideal.

Future work

It would be nice to have a usable framework of enriched categories in which this just became a category enriched in Module R.

class CategoryTheory.Linear (R : Type w) [Semiring R] (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : Type (max (max u v) w)

A category is called R-linear if P ⟶ Q is an R-module such that composition is R-linear in both variables.

• homModule : (X Y : C) → Module R (X ⟶ Y)
• smul_comp : ∀ (X Y Z : C) (r : R) (f : X ⟶ Y) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp (r • f) g = r • CategoryTheory.CategoryStruct.comp f g

  compatibility of the scalar multiplication with the post-composition

• comp_smul : ∀ (X Y Z : C) (f : X ⟶ Y) (r : R) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f (r • g) = r • CategoryTheory.CategoryStruct.comp f g

  compatibility of the scalar multiplication with the pre-composition

@[simp]

theorem CategoryTheory.Linear.smul_comp {R : Type w} : ∀ {inst : Semiring R} {C : Type u} {inst_1 : CategoryTheory.Category.{v, u} C} {inst_2 : CategoryTheory.Preadditive C} [self : CategoryTheory.Linear R C] (X Y Z : C) (r : R) (f : X ⟶ Y) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp (r • f) g = r • CategoryTheory.CategoryStruct.comp f g

compatibility of the scalar multiplication with the post-composition

@[simp]

theorem CategoryTheory.Linear.comp_smul {R : Type w} : ∀ {inst : Semiring R} {C : Type u} {inst_1 : CategoryTheory.Category.{v, u} C} {inst_2 : CategoryTheory.Preadditive C} [self : CategoryTheory.Linear R C] (X Y Z : C) (f : X ⟶ Y) (r : R) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f (r • g) = r • CategoryTheory.CategoryStruct.comp f g

compatibility of the scalar multiplication with the pre-composition

instance CategoryTheory.Linear.preadditiveNatLinear {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : CategoryTheory.Linear ℕ C

Equations

• CategoryTheory.Linear.preadditiveNatLinear = { homModule := inferInstance, smul_comp := ⋯, comp_smul := ⋯ }

instance CategoryTheory.Linear.preadditiveIntLinear {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] : CategoryTheory.Linear ℤ C

Equations

• CategoryTheory.Linear.preadditiveIntLinear = { homModule := inferInstance, smul_comp := ⋯, comp_smul := ⋯ }

instance CategoryTheory.Linear.instModuleEnd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [Semiring R] [CategoryTheory.Linear R C] (X : C) : Module R (CategoryTheory.End X)

Equations

• CategoryTheory.Linear.instModuleEnd X = id inferInstance

instance CategoryTheory.Linear.instAlgebraEnd {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [CommSemiring R] [CategoryTheory.Linear R C] (X : C) : Algebra R (CategoryTheory.End X)

Equations

• CategoryTheory.Linear.instAlgebraEnd X = Algebra.ofModule ⋯ ⋯

instance CategoryTheory.Linear.inducedCategory {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [Semiring R] [CategoryTheory.Linear R C] {D : Type u'} (F : D → C) : CategoryTheory.Linear R (CategoryTheory.InducedCategory C F)

Equations

• CategoryTheory.Linear.inducedCategory F = { homModule := fun (X Y : CategoryTheory.InducedCategory C F) => CategoryTheory.Linear.homModule (F X) (F Y), smul_comp := ⋯, comp_smul := ⋯ }

instance CategoryTheory.Linear.fullSubcategory {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [Semiring R] [CategoryTheory.Linear R C] (Z : C → Prop) : CategoryTheory.Linear R (CategoryTheory.FullSubcategory Z)

Equations

• CategoryTheory.Linear.fullSubcategory Z = { homModule := fun (X Y : CategoryTheory.FullSubcategory Z) => CategoryTheory.Linear.homModule X.obj Y.obj, smul_comp := ⋯, comp_smul := ⋯ }

@[simp]

theorem CategoryTheory.Linear.leftComp_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} (Z : C) (f : X ⟶ Y) (g : Y ⟶ Z) : (CategoryTheory.Linear.leftComp R Z f) g = CategoryTheory.CategoryStruct.comp f g

def CategoryTheory.Linear.leftComp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} (Z : C) (f : X ⟶ Y) : (Y ⟶ Z) →ₗ[R] X ⟶ Z

Composition by a fixed left argument as an R-linear map.

Equations

• CategoryTheory.Linear.leftComp R Z f = { toFun := fun (g : Y ⟶ Z) => CategoryTheory.CategoryStruct.comp f g, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem CategoryTheory.Linear.rightComp_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] (X : C) {Y : C} {Z : C} (g : Y ⟶ Z) (f : X ⟶ Y) : (CategoryTheory.Linear.rightComp R X g) f = CategoryTheory.CategoryStruct.comp f g

def CategoryTheory.Linear.rightComp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] (X : C) {Y : C} {Z : C} (g : Y ⟶ Z) : (X ⟶ Y) →ₗ[R] X ⟶ Z

Composition by a fixed right argument as an R-linear map.

Equations

• CategoryTheory.Linear.rightComp R X g = { toFun := fun (f : X ⟶ Y) => CategoryTheory.CategoryStruct.comp f g, map_add' := ⋯, map_smul' := ⋯ }

instance CategoryTheory.Linear.instEpiHSMulHomOfInvertible {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] (r : R) [Invertible r] : CategoryTheory.Epi (r • f)

Equations

• ⋯ = ⋯

instance CategoryTheory.Linear.instMonoHSMulHomOfInvertible {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (R : Type w) [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] (r : R) [Invertible r] : CategoryTheory.Mono (r • f)

Equations

• ⋯ = ⋯

def CategoryTheory.Linear.homCongr (k : Type u_1) {C : Type u_2} [CategoryTheory.Category.{u_3, u_2} C] [Semiring k] [CategoryTheory.Preadditive C] [CategoryTheory.Linear k C] {X : C} {Y : C} {W : C} {Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) : (X ⟶ W) ≃ₗ[k] Y ⟶ Z

Given isomorphic objects X ≅ Y, W ≅ Z in a k-linear category, we have a k-linear isomorphism between Hom(X, W) and Hom(Y, Z).

Equations

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

theorem CategoryTheory.Linear.homCongr_apply (k : Type u_1) {C : Type u_2} [CategoryTheory.Category.{u_3, u_2} C] [Semiring k] [CategoryTheory.Preadditive C] [CategoryTheory.Linear k C] {X : C} {Y : C} {W : C} {Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : X ⟶ W) : (CategoryTheory.Linear.homCongr k f₁ f₂) f = CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f₁.inv f) f₂.hom

theorem CategoryTheory.Linear.homCongr_symm_apply (k : Type u_1) {C : Type u_2} [CategoryTheory.Category.{u_3, u_2} C] [Semiring k] [CategoryTheory.Preadditive C] [CategoryTheory.Linear k C] {X : C} {Y : C} {W : C} {Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : Y ⟶ Z) : (CategoryTheory.Linear.homCongr k f₁ f₂).symm f = CategoryTheory.CategoryStruct.comp f₁.hom (CategoryTheory.CategoryStruct.comp f f₂.inv)

@[simp]

theorem CategoryTheory.Linear.units_smul_comp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} {Z : C} (r : Rˣ) (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (r • f) g = r • CategoryTheory.CategoryStruct.comp f g

@[simp]

theorem CategoryTheory.Linear.comp_units_smul {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {R : Type w} [Semiring R] [CategoryTheory.Linear R C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (r : Rˣ) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp f (r • g) = r • CategoryTheory.CategoryStruct.comp f g

@[simp]

theorem CategoryTheory.Linear.comp_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {S : Type w} [CommSemiring S] [CategoryTheory.Linear S C] (X : C) (Y : C) (Z : C) (f : X ⟶ Y) : (CategoryTheory.Linear.comp X Y Z) f = CategoryTheory.Linear.leftComp S Z f

def CategoryTheory.Linear.comp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {S : Type w} [CommSemiring S] [CategoryTheory.Linear S C] (X : C) (Y : C) (Z : C) : (X ⟶ Y) →ₗ[S] (Y ⟶ Z) →ₗ[S] X ⟶ Z

Composition as a bilinear map.

Equations

• CategoryTheory.Linear.comp X Y Z = { toFun := fun (f : X ⟶ Y) => CategoryTheory.Linear.leftComp S Z f, map_add' := ⋯, map_smul' := ⋯ }