File size: 1,625 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 | import Mathlib.Algebra.Category.ModuleCat.Biproducts
import OAI.Algebra.Finitistic.Rectification.TensorComplex
import OAI.Algebra.Finitistic.Encoding.RelationBasisCoefficients
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.ArrowBasisIso
universe u
variable (k : Type u) [Field k] (d : ℕ)
abbrev family : Fin (d+1) → ModuleCat.{u} k := fun _ => ModuleCat.of k k
/-- The adjacent arrow space is precisely the direct sum of its scalar arrow columns. -/
def iso : (⨁ family k d) ≅ ModuleCat.of k (QuadraticEncoding.Arrows k d) :=
ModuleCat.biproductIsoPi (family k d)
theorem hom_proj (j : Fin (d+1)) :
(iso 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,Iso.hom_inv_id_assoc]
/-- The coproduct injection sends a scalar to that scalar times its original arrow basis. -/
theorem injection_apply (i : Fin (d+1)) (c : k) :
((biproduct.ι (family k d) i ≫ (iso k d).hom).hom) c =
c • QuadraticEncoding.basis i := by
funext j
have h := congrArg (fun f : (family k d i) ⟶ ModuleCat.of k k => f.hom c)
(congrArg (fun f => biproduct.ι (family k d) i ≫ f) (hom_proj k d j))
have h' : ((biproduct.ι (family k d) i ≫ (iso k d).hom).hom c) j =
(biproduct.ι (family k d) i ≫ biproduct.π (family k d) j).hom c := h
rw [h']
classical
by_cases hij : i=j
· subst j
simp [QuadraticEncoding.basis]
· simp [hij,QuadraticEncoding.basis,Ne.symm hij]
end LittleFinitistic.ArrowBasisIso
end
end OAI
|