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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | 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
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