| import Mathlib.Algebra.Category.ModuleCat.Biproducts |
| import OAI.Algebra.Finitistic.Encoding.RelationBasisCoefficients |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.FiniteBasisIso |
| universe u |
| variable {k V : Type u} [Field k] [AddCommGroup V] [Module k V] |
| variable {I : Type} [Fintype I] (b : Module.Basis I k V) |
| abbrev family : I → ModuleCat.{u} k := fun _ => ModuleCat.of k k |
|
|
| /-- An actual finite basis gives the scalar-column isomorphism, including the empty basis. -/ |
| def iso : (⨁ (family (k := k) : I → _)) ≅ ModuleCat.of k V := |
| ModuleCat.biproductIsoPi _ ≪≫ b.equivFun.symm.toModuleIso |
|
|
| theorem pi_hom_proj (j : I) : |
| (ModuleCat.biproductIsoPi (family (k := k) : I → _)).hom ≫ |
| ModuleCat.ofHom (LinearMap.proj j) = biproduct.π _ j := by |
| rw [← ModuleCat.biproductIsoPi_inv_comp_π (family (k := k) : I → _) j] |
| simp only [Iso.hom_inv_id_assoc] |
|
|
| theorem injection_apply (i : I) (c : k) : |
| (biproduct.ι (family (k := k) : I → _) i ≫ (iso b).hom).hom c = c • b i := by |
| apply b.equivFun.injective |
| change b.equivFun (b.equivFun.symm ((ModuleCat.Hom.hom |
| (biproduct.ι (family (k := k) : I → _) i ≫ (ModuleCat.biproductIsoPi _).hom)) c)) = |
| b.equivFun (c • b i) |
| rw [LinearEquiv.apply_symm_apply] |
| funext j |
| have h := congrArg (fun f : ModuleCat.of k k ⟶ ModuleCat.of k k => f.hom c) |
| (congrArg (fun f => biproduct.ι (family (k := k) : I → _) i ≫ f) |
| (pi_hom_proj (k := k) j)) |
| rw [show ((biproduct.ι (family (k := k) : I → _) i ≫ |
| (ModuleCat.biproductIsoPi _).hom).hom c) j = |
| (biproduct.ι (family (k := k) : I → _) i ≫ biproduct.π _ j).hom c from h] |
| classical |
| by_cases hij : i=j |
| · subst j; simp |
| · simp [hij,Ne.symm hij] |
| end LittleFinitistic.FiniteBasisIso |
|
|
| end |
|
|
| end OAI |
| |
