category_theory.linear.linear_functor
⟷
Mathlib.CategoryTheory.Linear.LinearFunctor
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
-import Mathbin.CategoryTheory.Preadditive.AdditiveFunctor
-import Mathbin.CategoryTheory.Linear.Basic
+import CategoryTheory.Preadditive.AdditiveFunctor
+import CategoryTheory.Linear.Basic
#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.linear.linear_functor
-! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Preadditive.AdditiveFunctor
import Mathbin.CategoryTheory.Linear.Basic
+#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
+
/-!
# Linear Functors
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -50,10 +50,12 @@ section
variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
[CategoryTheory.Linear R C] [CategoryTheory.Linear R D] (F : C ⥤ D) [Additive F] [Linear R F]
+#print CategoryTheory.Functor.map_smul /-
@[simp]
theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=
Functor.Linear.map_smul'
#align category_theory.functor.map_smul CategoryTheory.Functor.map_smul
+-/
instance : Linear R (𝟭 C) where
@@ -62,15 +64,19 @@ instance {E : Type _} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (
variable (R)
+#print CategoryTheory.Functor.mapLinearMap /-
/-- `F.map_linear_map` is an `R`-linear map whose underlying function is `F.map`. -/
@[simps]
def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
{ F.mapAddHom with map_smul' := fun r f => F.map_smul r f }
#align category_theory.functor.map_linear_map CategoryTheory.Functor.mapLinearMap
+-/
+#print CategoryTheory.Functor.coe_mapLinearMap /-
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMap
+-/
end
@@ -86,9 +92,11 @@ instance inducedFunctorLinear : Functor.Linear R (inducedFunctor F) where
end InducedCategory
+#print CategoryTheory.Functor.fullSubcategoryInclusionLinear /-
instance fullSubcategoryInclusionLinear {C : Type _} [Category C] [Preadditive C]
[CategoryTheory.Linear R C] (Z : C → Prop) : (fullSubcategoryInclusion Z).Linear R where
#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinear
+-/
section
@@ -100,9 +108,11 @@ instance natLinear : F.Linear ℕ where map_smul' X Y f r := F.mapAddHom.map_nsm
#align category_theory.functor.nat_linear CategoryTheory.Functor.natLinear
-/
+#print CategoryTheory.Functor.intLinear /-
instance intLinear : F.Linear ℤ
where map_smul' X Y f r := (F.mapAddHom : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)).map_zsmul f r
#align category_theory.functor.int_linear CategoryTheory.Functor.intLinear
+-/
variable [CategoryTheory.Linear ℚ C] [CategoryTheory.Linear ℚ D]
@@ -120,9 +130,11 @@ namespace Equivalence
variable {C D : Type _} [Category C] [Category D] [Preadditive C] [Linear R C] [Preadditive D]
[Linear R D]
+#print CategoryTheory.Equivalence.inverseLinear /-
instance inverseLinear (e : C ≌ D) [e.Functor.Additive] [e.Functor.Linear R] : e.inverse.Linear R
where map_smul' X Y r f := by apply e.functor.map_injective; simp
#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinear
+-/
end Equivalence
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -36,7 +36,7 @@ variable (R : Type _) [Semiring R]
#print CategoryTheory.Functor.Linear /-
/-- An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. -/
class Functor.Linear {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
- [Linear R C] [Linear R D] (F : C ⥤ D) [F.Additive] : Prop where
+ [Linear R C] [Linear R D] (F : C ⥤ D) [F.Additive] : Prop where
map_smul' : ∀ {X Y : C} {f : X ⟶ Y} {r : R}, F.map (r • f) = r • F.map f := by obviously
#align category_theory.functor.linear CategoryTheory.Functor.Linear
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -50,9 +50,6 @@ section
variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
[CategoryTheory.Linear R C] [CategoryTheory.Linear R D] (F : C ⥤ D) [Additive F] [Linear R F]
-/- warning: category_theory.functor.map_smul -> CategoryTheory.Functor.map_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_smul CategoryTheory.Functor.map_smulₓ'. -/
@[simp]
theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=
Functor.Linear.map_smul'
@@ -65,21 +62,12 @@ instance {E : Type _} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (
variable (R)
-/- warning: category_theory.functor.map_linear_map -> CategoryTheory.Functor.mapLinearMap is a dubious translation:
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/-- `F.map_linear_map` is an `R`-linear map whose underlying function is `F.map`. -/
@[simps]
def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
{ F.mapAddHom with map_smul' := fun r f => F.map_smul r f }
#align category_theory.functor.map_linear_map CategoryTheory.Functor.mapLinearMap
-/- warning: category_theory.functor.coe_map_linear_map -> CategoryTheory.Functor.coe_mapLinearMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMapₓ'. -/
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMap
@@ -98,12 +86,6 @@ instance inducedFunctorLinear : Functor.Linear R (inducedFunctor F) where
end InducedCategory
-/- warning: category_theory.functor.full_subcategory_inclusion_linear -> CategoryTheory.Functor.fullSubcategoryInclusionLinear is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinearₓ'. -/
instance fullSubcategoryInclusionLinear {C : Type _} [Category C] [Preadditive C]
[CategoryTheory.Linear R C] (Z : C → Prop) : (fullSubcategoryInclusion Z).Linear R where
#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinear
@@ -118,12 +100,6 @@ instance natLinear : F.Linear ℕ where map_smul' X Y f r := F.mapAddHom.map_nsm
#align category_theory.functor.nat_linear CategoryTheory.Functor.natLinear
-/
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- forall {C : Type.{u1}} {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] [_inst_3 : CategoryTheory.Category.{u4, u2} D] [_inst_4 : CategoryTheory.Preadditive.{u3, u1} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u4, u2} D _inst_3] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_6 : CategoryTheory.Functor.Additive.{u1, u2, u3, u4} C D _inst_2 _inst_3 _inst_4 _inst_5 F], CategoryTheory.Functor.Linear.{0, u1, u2, u3, u4} Int Int.semiring C D _inst_2 _inst_3 _inst_4 _inst_5 (CategoryTheory.Linear.preadditiveIntLinear.{u3, u1} C _inst_2 _inst_4) (CategoryTheory.Linear.preadditiveIntLinear.{u4, u2} D _inst_3 _inst_5) F _inst_6
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- forall {C : Type.{u1}} {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] [_inst_3 : CategoryTheory.Category.{u4, u2} D] [_inst_4 : CategoryTheory.Preadditive.{u3, u1} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u4, u2} D _inst_3] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_6 : CategoryTheory.Functor.Additive.{u1, u2, u3, u4} C D _inst_2 _inst_3 _inst_4 _inst_5 F], CategoryTheory.Functor.Linear.{0, u1, u2, u3, u4} Int Int.instSemiringInt C D _inst_2 _inst_3 _inst_4 _inst_5 (CategoryTheory.Linear.preadditiveIntLinear.{u3, u1} C _inst_2 _inst_4) (CategoryTheory.Linear.preadditiveIntLinear.{u4, u2} D _inst_3 _inst_5) F _inst_6
-Case conversion may be inaccurate. Consider using '#align category_theory.functor.int_linear CategoryTheory.Functor.intLinearₓ'. -/
instance intLinear : F.Linear ℤ
where map_smul' X Y f r := (F.mapAddHom : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)).map_zsmul f r
#align category_theory.functor.int_linear CategoryTheory.Functor.intLinear
@@ -144,12 +120,6 @@ namespace Equivalence
variable {C D : Type _} [Category C] [Category D] [Preadditive C] [Linear R C] [Preadditive D]
[Linear R D]
-/- warning: category_theory.equivalence.inverse_linear -> CategoryTheory.Equivalence.inverseLinear is a dubious translation:
-lean 3 declaration is
- forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_6 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_6] (e : CategoryTheory.Equivalence.{u4, u5, u2, u3} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C _inst_2 D _inst_3 e)] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_6 _inst_5 _inst_7 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C _inst_2 D _inst_3 e) _inst_8], CategoryTheory.Functor.Linear.{u1, u3, u2, u5, u4} R _inst_1 D C _inst_3 _inst_2 _inst_6 _inst_4 _inst_7 _inst_5 (CategoryTheory.Equivalence.inverse.{u4, u5, u2, u3} C _inst_2 D _inst_3 e) (CategoryTheory.Equivalence.inverse_additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 e _inst_8)
-but is expected to have type
- forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_6 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_6] (e : CategoryTheory.Equivalence.{u4, u5, u2, u3} C D _inst_2 _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e)] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_6 _inst_5 _inst_7 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) _inst_8], CategoryTheory.Functor.Linear.{u1, u3, u2, u5, u4} R _inst_1 D C _inst_3 _inst_2 _inst_6 _inst_4 _inst_7 _inst_5 (CategoryTheory.Equivalence.inverse.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) (CategoryTheory.Equivalence.inverse_additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 e _inst_8)
-Case conversion may be inaccurate. Consider using '#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinearₓ'. -/
instance inverseLinear (e : C ≌ D) [e.Functor.Additive] [e.Functor.Linear R] : e.inverse.Linear R
where map_smul' X Y r f := by apply e.functor.map_injective; simp
#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinear
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -151,9 +151,7 @@ but is expected to have type
forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_6 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_6] (e : CategoryTheory.Equivalence.{u4, u5, u2, u3} C D _inst_2 _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e)] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_6 _inst_5 _inst_7 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) _inst_8], CategoryTheory.Functor.Linear.{u1, u3, u2, u5, u4} R _inst_1 D C _inst_3 _inst_2 _inst_6 _inst_4 _inst_7 _inst_5 (CategoryTheory.Equivalence.inverse.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) (CategoryTheory.Equivalence.inverse_additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 e _inst_8)
Case conversion may be inaccurate. Consider using '#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinearₓ'. -/
instance inverseLinear (e : C ≌ D) [e.Functor.Additive] [e.Functor.Linear R] : e.inverse.Linear R
- where map_smul' X Y r f := by
- apply e.functor.map_injective
- simp
+ where map_smul' X Y r f := by apply e.functor.map_injective; simp
#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinear
end Equivalence
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -51,10 +51,7 @@ variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditi
[CategoryTheory.Linear R C] [CategoryTheory.Linear R D] (F : C ⥤ D) [Additive F] [Linear R F]
/- warning: category_theory.functor.map_smul -> CategoryTheory.Functor.map_smul is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_smul CategoryTheory.Functor.map_smulₓ'. -/
@[simp]
theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=
@@ -81,10 +78,7 @@ def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
#align category_theory.functor.map_linear_map CategoryTheory.Functor.mapLinearMap
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMapₓ'. -/
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -84,7 +84,7 @@ def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
lean 3 declaration is
forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_6 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_5] (F : CategoryTheory.Functor.{u4, u5, u2, u3} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_5 F] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8] {X : C} {Y : C}, Eq.{max (succ u4) (succ u5)} ((Quiver.Hom.{succ u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} C (CategoryTheory.Category.toCategoryStruct.{u4, u2} C _inst_2)) X Y) -> (Quiver.Hom.{succ u5, u3} D 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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMapₓ'. -/
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e
@@ -84,7 +84,7 @@ def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
lean 3 declaration is
forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_6 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_5] (F : CategoryTheory.Functor.{u4, u5, u2, u3} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_5 F] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8] {X : C} {Y : C}, Eq.{max (succ u4) (succ u5)} ((Quiver.Hom.{succ u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} C (CategoryTheory.Category.toCategoryStruct.{u4, u2} C _inst_2)) X Y) -> (Quiver.Hom.{succ u5, u3} D 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but is expected to have type
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_inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (CategoryTheory.Functor.mapLinearMap.{u1, u3, u2, u5, u4} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8 _inst_9 X Y)) (Prefunctor.map.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X Y)
Case conversion may be inaccurate. Consider using '#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMapₓ'. -/
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
! This file was ported from Lean 3 source module category_theory.linear.linear_functor
-! leanprover-community/mathlib commit 829895f162a1f29d0133f4b3538f4cd1fb5bffd3
+! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.CategoryTheory.Linear.Basic
/-!
# Linear Functors
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
An additive functor between two `R`-linear categories is called *linear*
if the induced map on hom types is a morphism of `R`-modules.
mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931
@@ -30,11 +30,13 @@ namespace CategoryTheory
variable (R : Type _) [Semiring R]
+#print CategoryTheory.Functor.Linear /-
/-- An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. -/
class Functor.Linear {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
[Linear R C] [Linear R D] (F : C ⥤ D) [F.Additive] : Prop where
map_smul' : ∀ {X Y : C} {f : X ⟶ Y} {r : R}, F.map (r • f) = r • F.map f := by obviously
#align category_theory.functor.linear CategoryTheory.Functor.Linear
+-/
section Linear
@@ -45,6 +47,12 @@ section
variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
[CategoryTheory.Linear R C] [CategoryTheory.Linear R D] (F : C ⥤ D) [Additive F] [Linear R F]
+/- warning: category_theory.functor.map_smul -> CategoryTheory.Functor.map_smul is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_6 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_5] (F : CategoryTheory.Functor.{u4, u5, u2, u3} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_5 F] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8] {X : C} {Y : C} (r : R) (f : Quiver.Hom.{succ u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} C (CategoryTheory.Category.toCategoryStruct.{u4, u2} C _inst_2)) X Y), Eq.{succ u5} (Quiver.Hom.{succ u5, u3} D 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+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {C : Type.{u4}} {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u5, u4} C] [_inst_3 : CategoryTheory.Category.{u3, u2} D] [_inst_4 : CategoryTheory.Preadditive.{u5, u4} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u3, u2} D _inst_3] [_inst_6 : CategoryTheory.Linear.{u1, u5, u4} R _inst_1 C _inst_2 _inst_4] [_inst_7 : CategoryTheory.Linear.{u1, u3, u2} R _inst_1 D _inst_3 _inst_5] (F : CategoryTheory.Functor.{u5, u3, u4, u2} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u4, u2, u5, u3} C D _inst_2 _inst_3 _inst_4 _inst_5 F] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u4, u2, u5, u3} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8] {X : C} {Y : C} (r : R) (f : Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y), Eq.{succ u3} (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (Prefunctor.map.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X Y (HSMul.hSMul.{u1, u5, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (instHSMul.{u1, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (SMulZeroClass.toSMul.{u1, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u5, u4} C _inst_2 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u5, u4} C _inst_2 _inst_4) X Y) (SMulWithZero.toSMulZeroClass.{u1, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u5, u4} C _inst_2 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u5, u4} C _inst_2 _inst_4) X Y) (MulActionWithZero.toSMulWithZero.{u1, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (Semiring.toMonoidWithZero.{u1} R _inst_1) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u5, u4} C _inst_2 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u5, u4} C _inst_2 _inst_4) X Y) (Module.toMulActionWithZero.{u1, u5} R (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) _inst_1 (AddCommGroup.toAddCommMonoid.{u5} (Quiver.Hom.{succ u5, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) X Y) (CategoryTheory.Preadditive.homGroup.{u5, u4} C _inst_2 _inst_4 X Y)) (CategoryTheory.Linear.homModule.{u1, u5, u4} R _inst_1 C _inst_2 _inst_4 _inst_6 X Y)))))) r f)) (HSMul.hSMul.{u1, u3, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (instHSMul.{u1, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (SMulZeroClass.toSMul.{u1, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u3, u2} D _inst_3 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u3, u2} D _inst_3 _inst_5) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (SMulWithZero.toSMulZeroClass.{u1, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u3, u2} D _inst_3 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u3, u2} D _inst_3 _inst_5) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (MulActionWithZero.toSMulWithZero.{u1, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (Semiring.toMonoidWithZero.{u1} R _inst_1) (CategoryTheory.Limits.HasZeroMorphisms.Zero.{u3, u2} D _inst_3 (CategoryTheory.Preadditive.preadditiveHasZeroMorphisms.{u3, u2} D _inst_3 _inst_5) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (Module.toMulActionWithZero.{u1, u3} R (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) _inst_1 (AddCommGroup.toAddCommMonoid.{u3} (Quiver.Hom.{succ u3, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y)) (CategoryTheory.Preadditive.homGroup.{u3, u2} D _inst_3 _inst_5 (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y))) (CategoryTheory.Linear.homModule.{u1, u3, u2} R _inst_1 D _inst_3 _inst_5 _inst_7 (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) Y))))))) r (Prefunctor.map.{succ u5, succ u3, u4, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u4} C (CategoryTheory.Category.toCategoryStruct.{u5, u4} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} D (CategoryTheory.Category.toCategoryStruct.{u3, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u3, u4, u2} C _inst_2 D _inst_3 F) X Y f))
+Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_smul CategoryTheory.Functor.map_smulₓ'. -/
@[simp]
theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=
Functor.Linear.map_smul'
@@ -57,12 +65,24 @@ instance {E : Type _} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (
variable (R)
+/- warning: category_theory.functor.map_linear_map -> CategoryTheory.Functor.mapLinearMap is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.functor.map_linear_map CategoryTheory.Functor.mapLinearMapₓ'. -/
/-- `F.map_linear_map` is an `R`-linear map whose underlying function is `F.map`. -/
@[simps]
def mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=
{ F.mapAddHom with map_smul' := fun r f => F.map_smul r f }
#align category_theory.functor.map_linear_map CategoryTheory.Functor.mapLinearMap
+/- warning: category_theory.functor.coe_map_linear_map -> CategoryTheory.Functor.coe_mapLinearMap is a dubious translation:
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D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y))) (Quiver.Hom.{succ u5, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) X Y) (fun (_x : Quiver.Hom.{succ u5, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) X Y) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Quiver.Hom.{succ u5, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) X Y) => Quiver.Hom.{succ u4, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y)) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u5, u4} R R (Quiver.Hom.{succ u5, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) X Y) (Quiver.Hom.{succ u4, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y)) _inst_1 _inst_1 (AddCommGroup.toAddCommMonoid.{u5} (Quiver.Hom.{succ u5, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) X Y) (CategoryTheory.Preadditive.homGroup.{u5, u3} C _inst_2 _inst_4 X Y)) (AddCommGroup.toAddCommMonoid.{u4} (Quiver.Hom.{succ u4, u2} D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y)) (CategoryTheory.Preadditive.homGroup.{u4, u2} D _inst_3 _inst_5 (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y))) (CategoryTheory.Linear.homModule.{u1, u5, u3} R _inst_1 C _inst_2 _inst_4 _inst_6 X Y) (CategoryTheory.Linear.homModule.{u1, u4, u2} R _inst_1 D _inst_3 _inst_5 _inst_7 (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X) (Prefunctor.obj.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) Y)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (CategoryTheory.Functor.mapLinearMap.{u1, u3, u2, u5, u4} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 F _inst_8 _inst_9 X Y)) (Prefunctor.map.{succ u5, succ u4, u3, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u3} C (CategoryTheory.Category.toCategoryStruct.{u5, u3} C _inst_2)) D (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} D (CategoryTheory.Category.toCategoryStruct.{u4, u2} D _inst_3)) (CategoryTheory.Functor.toPrefunctor.{u5, u4, u3, u2} C _inst_2 D _inst_3 F) X Y)
+Case conversion may be inaccurate. Consider using '#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMapₓ'. -/
theorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y :=
rfl
#align category_theory.functor.coe_map_linear_map CategoryTheory.Functor.coe_mapLinearMap
@@ -74,11 +94,19 @@ section InducedCategory
variable {C : Type _} {D : Type _} [Category D] [Preadditive D] [CategoryTheory.Linear R D]
(F : C → D)
+#print CategoryTheory.Functor.inducedFunctorLinear /-
instance inducedFunctorLinear : Functor.Linear R (inducedFunctor F) where
#align category_theory.functor.induced_functor_linear CategoryTheory.Functor.inducedFunctorLinear
+-/
end InducedCategory
+/- warning: category_theory.functor.full_subcategory_inclusion_linear -> CategoryTheory.Functor.fullSubcategoryInclusionLinear is a dubious translation:
+lean 3 declaration is
+ forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u2} C] [_inst_3 : CategoryTheory.Preadditive.{u3, u2} C _inst_2] [_inst_4 : CategoryTheory.Linear.{u1, u3, u2} R _inst_1 C _inst_2 _inst_3] (Z : C -> Prop), CategoryTheory.Functor.Linear.{u1, u2, u2, u3, u3} R _inst_1 (CategoryTheory.FullSubcategoryₓ.{u3, u2} C _inst_2 Z) C (CategoryTheory.InducedCategory.category.{u3, u2, u2} (CategoryTheory.FullSubcategoryₓ.{u3, u2} C _inst_2 Z) C _inst_2 (CategoryTheory.FullSubcategoryₓ.obj.{u3, u2} C _inst_2 Z)) _inst_2 (CategoryTheory.Preadditive.fullSubcategory.{u3, u2} C _inst_2 _inst_3 Z) _inst_3 (CategoryTheory.Linear.fullSubcategory.{u1, u3, u2} C _inst_2 _inst_3 R _inst_1 _inst_4 Z) _inst_4 (CategoryTheory.fullSubcategoryInclusion.{u3, u2} C _inst_2 Z) (CategoryTheory.Functor.fullSubcategoryInclusion_additive.{u2, u3} C _inst_2 _inst_3 Z)
+but is expected to have type
+ forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u2} C] [_inst_3 : CategoryTheory.Preadditive.{u3, u2} C _inst_2] [_inst_4 : CategoryTheory.Linear.{u1, u3, u2} R _inst_1 C _inst_2 _inst_3] (Z : C -> Prop), CategoryTheory.Functor.Linear.{u1, u2, u2, u3, u3} R _inst_1 (CategoryTheory.FullSubcategory.{u2} C Z) C (CategoryTheory.FullSubcategory.category.{u3, u2} C _inst_2 Z) _inst_2 (CategoryTheory.Preadditive.fullSubcategory.{u3, u2} C _inst_2 _inst_3 Z) _inst_3 (CategoryTheory.Linear.fullSubcategory.{u1, u3, u2} C _inst_2 _inst_3 R _inst_1 _inst_4 Z) _inst_4 (CategoryTheory.fullSubcategoryInclusion.{u3, u2} C _inst_2 Z) (CategoryTheory.Functor.fullSubcategoryInclusion_additive.{u2, u3} C _inst_2 _inst_3 Z)
+Case conversion may be inaccurate. Consider using '#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinearₓ'. -/
instance fullSubcategoryInclusionLinear {C : Type _} [Category C] [Preadditive C]
[CategoryTheory.Linear R C] (Z : C → Prop) : (fullSubcategoryInclusion Z).Linear R where
#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinear
@@ -88,17 +116,27 @@ section
variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D] (F : C ⥤ D)
[Additive F]
+#print CategoryTheory.Functor.natLinear /-
instance natLinear : F.Linear ℕ where map_smul' X Y f r := F.mapAddHom.map_nsmul f r
#align category_theory.functor.nat_linear CategoryTheory.Functor.natLinear
+-/
+/- warning: category_theory.functor.int_linear -> CategoryTheory.Functor.intLinear is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] [_inst_3 : CategoryTheory.Category.{u4, u2} D] [_inst_4 : CategoryTheory.Preadditive.{u3, u1} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u4, u2} D _inst_3] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_6 : CategoryTheory.Functor.Additive.{u1, u2, u3, u4} C D _inst_2 _inst_3 _inst_4 _inst_5 F], CategoryTheory.Functor.Linear.{0, u1, u2, u3, u4} Int Int.semiring C D _inst_2 _inst_3 _inst_4 _inst_5 (CategoryTheory.Linear.preadditiveIntLinear.{u3, u1} C _inst_2 _inst_4) (CategoryTheory.Linear.preadditiveIntLinear.{u4, u2} D _inst_3 _inst_5) F _inst_6
+but is expected to have type
+ forall {C : Type.{u1}} {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] [_inst_3 : CategoryTheory.Category.{u4, u2} D] [_inst_4 : CategoryTheory.Preadditive.{u3, u1} C _inst_2] [_inst_5 : CategoryTheory.Preadditive.{u4, u2} D _inst_3] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_6 : CategoryTheory.Functor.Additive.{u1, u2, u3, u4} C D _inst_2 _inst_3 _inst_4 _inst_5 F], CategoryTheory.Functor.Linear.{0, u1, u2, u3, u4} Int Int.instSemiringInt C D _inst_2 _inst_3 _inst_4 _inst_5 (CategoryTheory.Linear.preadditiveIntLinear.{u3, u1} C _inst_2 _inst_4) (CategoryTheory.Linear.preadditiveIntLinear.{u4, u2} D _inst_3 _inst_5) F _inst_6
+Case conversion may be inaccurate. Consider using '#align category_theory.functor.int_linear CategoryTheory.Functor.intLinearₓ'. -/
instance intLinear : F.Linear ℤ
where map_smul' X Y f r := (F.mapAddHom : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)).map_zsmul f r
#align category_theory.functor.int_linear CategoryTheory.Functor.intLinear
variable [CategoryTheory.Linear ℚ C] [CategoryTheory.Linear ℚ D]
+#print CategoryTheory.Functor.ratLinear /-
instance ratLinear : F.Linear ℚ where map_smul' X Y f r := F.mapAddHom.toRatLinearMap.map_smul r f
#align category_theory.functor.rat_linear CategoryTheory.Functor.ratLinear
+-/
end
@@ -109,6 +147,12 @@ namespace Equivalence
variable {C D : Type _} [Category C] [Category D] [Preadditive C] [Linear R C] [Preadditive D]
[Linear R D]
+/- warning: category_theory.equivalence.inverse_linear -> CategoryTheory.Equivalence.inverseLinear is a dubious translation:
+lean 3 declaration is
+ forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_6 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_6] (e : CategoryTheory.Equivalence.{u4, u5, u2, u3} C _inst_2 D _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C _inst_2 D _inst_3 e)] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_6 _inst_5 _inst_7 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C _inst_2 D _inst_3 e) _inst_8], CategoryTheory.Functor.Linear.{u1, u3, u2, u5, u4} R _inst_1 D C _inst_3 _inst_2 _inst_6 _inst_4 _inst_7 _inst_5 (CategoryTheory.Equivalence.inverse.{u4, u5, u2, u3} C _inst_2 D _inst_3 e) (CategoryTheory.Equivalence.inverse_additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 e _inst_8)
+but is expected to have type
+ forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {C : Type.{u2}} {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u2} C] [_inst_3 : CategoryTheory.Category.{u5, u3} D] [_inst_4 : CategoryTheory.Preadditive.{u4, u2} C _inst_2] [_inst_5 : CategoryTheory.Linear.{u1, u4, u2} R _inst_1 C _inst_2 _inst_4] [_inst_6 : CategoryTheory.Preadditive.{u5, u3} D _inst_3] [_inst_7 : CategoryTheory.Linear.{u1, u5, u3} R _inst_1 D _inst_3 _inst_6] (e : CategoryTheory.Equivalence.{u4, u5, u2, u3} C D _inst_2 _inst_3) [_inst_8 : CategoryTheory.Functor.Additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e)] [_inst_9 : CategoryTheory.Functor.Linear.{u1, u2, u3, u4, u5} R _inst_1 C D _inst_2 _inst_3 _inst_4 _inst_6 _inst_5 _inst_7 (CategoryTheory.Equivalence.functor.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) _inst_8], CategoryTheory.Functor.Linear.{u1, u3, u2, u5, u4} R _inst_1 D C _inst_3 _inst_2 _inst_6 _inst_4 _inst_7 _inst_5 (CategoryTheory.Equivalence.inverse.{u4, u5, u2, u3} C D _inst_2 _inst_3 e) (CategoryTheory.Equivalence.inverse_additive.{u2, u3, u4, u5} C D _inst_2 _inst_3 _inst_4 _inst_6 e _inst_8)
+Case conversion may be inaccurate. Consider using '#align category_theory.equivalence.inverse_linear CategoryTheory.Equivalence.inverseLinearₓ'. -/
instance inverseLinear (e : C ≌ D) [e.Functor.Additive] [e.Functor.Linear R] : e.inverse.Linear R
where map_smul' X Y r f := by
apply e.functor.map_injective
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -49,6 +49,10 @@ theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map
Functor.Linear.map_smul _ _
#align category_theory.functor.map_smul CategoryTheory.Functor.map_smul
+@[simp]
+theorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by
+ apply map_smul
+
instance : Linear R (𝟭 C) where
instance {E : Type*} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (G : D ⥤ E)
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -25,10 +25,10 @@ for every two objects `X` and `Y`, the map
namespace CategoryTheory
-variable (R : Type _) [Semiring R]
+variable (R : Type*) [Semiring R]
/-- An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. -/
-class Functor.Linear {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
+class Functor.Linear {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D]
[Linear R C] [Linear R D] (F : C ⥤ D) [F.Additive] : Prop where
/-- the functor induces a linear map on morphisms -/
map_smul : ∀ {X Y : C} (f : X ⟶ Y) (r : R), F.map (r • f) = r • F.map f := by aesop_cat
@@ -41,7 +41,7 @@ namespace Functor
section
variable {R}
-variable {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D]
+variable {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D]
[CategoryTheory.Linear R C] [CategoryTheory.Linear R D] (F : C ⥤ D) [Additive F] [Linear R F]
@[simp]
@@ -51,7 +51,7 @@ theorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map
instance : Linear R (𝟭 C) where
-instance {E : Type _} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (G : D ⥤ E)
+instance {E : Type*} [Category E] [Preadditive E] [CategoryTheory.Linear R E] (G : D ⥤ E)
[Additive G] [Linear R G] : Linear R (F ⋙ G) where
variable (R)
@@ -69,7 +69,7 @@ end
section InducedCategory
-variable {C : Type _} {D : Type _} [Category D] [Preadditive D] [CategoryTheory.Linear R D]
+variable {C : Type*} {D : Type*} [Category D] [Preadditive D] [CategoryTheory.Linear R D]
(F : C → D)
instance inducedFunctorLinear : Functor.Linear R (inducedFunctor F) where
@@ -77,13 +77,13 @@ instance inducedFunctorLinear : Functor.Linear R (inducedFunctor F) where
end InducedCategory
-instance fullSubcategoryInclusionLinear {C : Type _} [Category C] [Preadditive C]
+instance fullSubcategoryInclusionLinear {C : Type*} [Category C] [Preadditive C]
[CategoryTheory.Linear R C] (Z : C → Prop) : (fullSubcategoryInclusion Z).Linear R where
#align category_theory.functor.full_subcategory_inclusion_linear CategoryTheory.Functor.fullSubcategoryInclusionLinear
section
-variable {R} {C D : Type _} [Category C] [Category D] [Preadditive C] [Preadditive D] (F : C ⥤ D)
+variable {R} {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D] (F : C ⥤ D)
[Additive F]
instance natLinear : F.Linear ℕ where
@@ -106,7 +106,7 @@ end Functor
namespace Equivalence
-variable {C D : Type _} [Category C] [Category D] [Preadditive C] [Linear R C] [Preadditive D]
+variable {C D : Type*} [Category C] [Category D] [Preadditive C] [Linear R C] [Preadditive D]
[Linear R D]
instance inverseLinear (e : C ≌ D) [e.functor.Additive] [e.functor.Linear R] :
@@ -2,15 +2,12 @@
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.linear.linear_functor
-! leanprover-community/mathlib commit 829895f162a1f29d0133f4b3538f4cd1fb5bffd3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Linear.Basic
+#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
+
/-!
# Linear Functors
The unported dependencies are