File size: 2,261 Bytes
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 | 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
|