| import Mathlib.Algebra.Category.ModuleCat.Basic |
| import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.AdditiveSplitImage |
| universe u v |
| variable {C : Type*} [Category C] |
| variable {R : Type u} [Ring R] |
| variable (F : C ⥤ ModuleCat.{v} R) |
| variable {V M : C} (i : V ⟶ M) (p : M ⟶ V) (hip : i ≫ p = 𝟙 V) |
|
|
| def fixed : Submodule R (F.obj M) := ((F.map (p ≫ i)).hom - LinearMap.id).ker |
|
|
| @[simp] theorem mem_fixed (x : F.obj M) : x ∈ fixed F i p ↔ (F.map (p ≫ i)).hom x = x := by |
| simp [fixed,sub_eq_zero] |
|
|
| def forward : F.obj V →ₗ[R] fixed F i p := |
| (F.map i).hom.codRestrict _ (by |
| intro x |
| rw [mem_fixed] |
| have he : F.map i ≫ F.map (p ≫ i) = F.map i := by |
| rw [← F.map_comp,← Category.assoc,hip,Category.id_comp] |
| exact congrArg (fun a : F.obj V ⟶ F.obj M => a.hom x) he) |
|
|
| def inverse : fixed F i p →ₗ[R] F.obj V := |
| (F.map p).hom.comp (fixed F i p).subtype |
|
|
| /-- Any additive module-valued functor preserves the actual image of a split idempotent. -/ |
| def equiv : F.obj V ≃ₗ[R] fixed F i p where |
| __ := forward F i p hip |
| invFun := inverse F i p |
| left_inv x := by |
| have he : F.map i ≫ F.map p = 𝟙 (F.obj V) := by rw [← F.map_comp,hip,F.map_id] |
| exact congrArg (fun a : F.obj V ⟶ F.obj V => a.hom x) he |
| right_inv x := by |
| apply Subtype.ext |
| change (F.map i).hom ((F.map p).hom x.val) = x.val |
| rw [show (F.map i).hom ((F.map p).hom x.val) = (F.map (p ≫ i)).hom x.val from |
| congrArg (fun a : F.obj M ⟶ F.obj M => a.hom x.val) (F.map_comp p i).symm] |
| exact (mem_fixed F i p x.val).1 x.property |
|
|
| @[simp] theorem equiv_apply_val (x : F.obj V) : (equiv F i p hip x).val = (F.map i).hom x := rfl |
| @[simp] theorem equiv_symm_apply (x : fixed F i p) : (equiv F i p hip).symm x = (F.map p).hom x.val := rfl |
| end LittleFinitistic.AdditiveSplitImage |
|
|
| end |
|
|
| end OAI |
|
|
