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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 | 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
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