| import OAI.Algebra.Finitistic.Encoding.ArrowBasisIso |
| import Mathlib.LinearAlgebra.TensorProduct.Pi |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct |
| namespace LittleFinitistic.PathBasisIso |
| universe u |
| variable (k : Type u) [Field k] (d : ℕ) |
| abbrev family : Fin (d+1) → ModuleCat.{u} k := fun _ => ModuleCat.of k (QuadraticEncoding.Arrows k d) |
|
|
| /-- Group paths by their first traversed arrow; the later arrow is the first tensor factor. -/ |
| def iso : (⨁ family k d) ≅ ModuleCat.of k (QuadraticEncoding.Paths k d) := |
| ModuleCat.biproductIsoPi (family k d) ≪≫ |
| (TensorProduct.piScalarRight k k (QuadraticEncoding.Arrows k d) (Fin (d+1))).symm.toModuleIso |
|
|
| theorem pi_hom_proj (j : Fin (d+1)) : |
| (ModuleCat.biproductIsoPi (family k d)).hom ≫ ModuleCat.ofHom (LinearMap.proj j) = |
| biproduct.π (family k d) j := by |
| rw [← ModuleCat.biproductIsoPi_inv_comp_π (family k d) j] |
| simp only [Iso.hom_inv_id_assoc] |
|
|
| theorem pi_injection_apply (a : Fin (d+1)) (w : QuadraticEncoding.Arrows k d) : |
| (biproduct.ι (family k d) a ≫ (ModuleCat.biproductIsoPi (family k d)).hom).hom w = Pi.single a w := by |
| funext j |
| have h := congrArg (fun f : (family k d a) ⟶ family k d j => f.hom w) |
| (congrArg (fun f => biproduct.ι (family k d) a ≫ f) (pi_hom_proj k d j)) |
| change ((biproduct.ι (family k d) a ≫ (ModuleCat.biproductIsoPi (family k d)).hom).hom w) j = _ |
| rw [show ((biproduct.ι (family k d) a ≫ (ModuleCat.biproductIsoPi (family k d)).hom).hom w) j = |
| (biproduct.ι (family k d) a ≫ biproduct.π (family k d) j).hom w from h] |
| classical |
| by_cases haj : a=j |
| · subst j; simp |
| · simp [haj,Ne.symm haj] |
| |
| /-- A summand w at a maps to the actual tensor path w tensor a. -/ |
| theorem injection_apply (a : Fin (d+1)) (w : QuadraticEncoding.Arrows k d) : |
| (biproduct.ι (family k d) a ≫ (iso k d).hom).hom w = w ⊗ₜ[k] QuadraticEncoding.basis a := by |
| change (TensorProduct.piScalarRight k k (QuadraticEncoding.Arrows k d) (Fin (d+1))).symm |
| ((biproduct.ι (family k d) a ≫ (ModuleCat.biproductIsoPi (family k d)).hom).hom w) = _ |
| rw [pi_injection_apply,TensorProduct.piScalarRight_symm_single] |
| rfl |
| end LittleFinitistic.PathBasisIso |
|
|
| end |
|
|
| end OAI |
|
|
