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ae00908 | 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 52 | import OAI.Algebra.Finitistic.Encoding.ArrowBasisIso
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.TensorArrowSum
universe u
variable (k : Type u) [Field k] (d : ℕ)
def insertion (a : Fin (d+1)) : ModuleCat.of k k ⟶ ModuleCat.of k (QuadraticEncoding.Arrows k d) :=
biproduct.ι (ArrowBasisIso.family k d) a ≫ (ArrowBasisIso.iso k d).hom
@[simp] theorem insertion_apply (a : Fin (d+1)) (c : k) :
(insertion k d a).hom c = c • QuadraticEncoding.basis a :=
ArrowBasisIso.injection_apply k d a c
/-- An additive realization of the actual arrow-basis sum is an isomorphism. -/
theorem desc_isIso {C : Type*} [Category C] [Preadditive C] [HasFiniteBiproducts C]
(F : ModuleCat.{u} k ⥤ C) [F.Additive] :
IsIso (biproduct.desc (fun a : Fin (d+1) => F.map (insertion k d a))) := by
have he : biproduct.desc (fun a : Fin (d+1) => F.map (insertion k d a)) =
(F.mapBiproduct (ArrowBasisIso.family k d)).inv ≫ F.map (ArrowBasisIso.iso k d).hom := by
apply biproduct.hom_ext'
intro a
simp only [insertion, Functor.map_comp, Functor.mapBiproduct_inv]
erw [biproduct.ι_desc, biproduct.ι_desc_assoc]
rw [he]
infer_instance
/-- Coordinate isomorphisms identify an actual arrow family with the realized basis. -/
theorem desc_isIso_of_coordinates {C : Type*} [Category C] [Preadditive C]
[HasFiniteBiproducts C] (F : ModuleCat.{u} k ⥤ C) [F.Additive]
(X : Fin (d+1) → C) (Y : C) (α : ∀ a, X a ≅ F.obj (ModuleCat.of k k))
(β : Y ≅ F.obj (ModuleCat.of k (QuadraticEncoding.Arrows k d)))
(f : ∀ a, X a ⟶ Y)
(hf : ∀ a, (α a).hom ≫ F.map (insertion k d a) = f a ≫ β.hom) :
IsIso (biproduct.desc f) := by
have he : biproduct.desc f ≫ β.hom =
(biproduct.mapIso α).hom ≫ biproduct.desc (fun a => F.map (insertion k d a)) := by
apply biproduct.hom_ext'
intro a
simp only [biproduct.mapIso_hom,biproduct.ι_desc_assoc,biproduct.ι_map_assoc,biproduct.ι_desc]
exact (hf a).symm
let := desc_isIso k d F
rw [← isIso_comp_right_iff _ β.hom,he]
infer_instance
end LittleFinitistic.TensorArrowSum
end
end OAI
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