| import OAI.Algebra.Finitistic.Rectification.CornerColumnComplex |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace LittleFinitistic.CornerColumnPaths |
| universe u |
| variable {k V W U : Type u} [Field k] [AddCommGroup V] [Module k V] |
| [AddCommGroup W] [Module k W] [AddCommGroup U] [Module k U] |
| variable (μ : W →ₗ[k] V →ₗ[k] U) |
| local notation "T" => ThreeVertexAlgebra μ |
| attribute [local instance] ExternalTensor.rightModule ExternalTensor.bimodule |
|
|
| /-- Right multiplication by the original path on the actual vertex corner. -/ |
| def cornerMap (v i j : Fin 3) (a : ThreeVertexAlgebra μ) |
| (ha : a * ThreeVertexAlgebra.vertex μ i = a) : |
| CornerSpaces.corner μ v j →ₗ[k] CornerSpaces.corner μ v i where |
| toFun x := ⟨x.val*a,by |
| change ThreeVertexAlgebra.vertex μ v * (x.val*a) * ThreeVertexAlgebra.vertex μ i = x.val*a |
| calc |
| _ = (ThreeVertexAlgebra.vertex μ v * x.val) * |
| (a * ThreeVertexAlgebra.vertex μ i) := by simp only [mul_assoc] |
| _ = x.val*a := by rw [CornerProjection.corner_left_fixed,ha]⟩ |
| map_add' x y := Subtype.ext (add_mul _ _ _) |
| map_smul' c x := Subtype.ext (smul_mul_assoc _ _ _) |
|
|
| /-- The corner-tensor identification carries the actual induced path map to right multiplication. -/ |
| theorem path_naturality (v i j : Fin 3) (a : ThreeVertexAlgebra μ) |
| (ha : a * ThreeVertexAlgebra.vertex μ i = a) |
| (D : CochainComplex (ModuleCat.{u} (ThreeVertexAlgebra μ)ᵐᵒᵖ) ℤ) : |
| TensorComplex.complexMap D (ModuleCat.ofHom (cornerMap μ v i j a ha)) ≫ |
| (CornerColumnComplex.complexIso μ v i D).hom = |
| (CornerColumnComplex.complexIso μ v j D).hom ≫ |
| ((BimoduleVertex.functor k T (ThreeVertexAlgebra.vertex μ v)).mapHomologicalComplex (.up ℤ)).map |
| (ExternalTensor.Section4.path k T (ThreeVertexAlgebra.vertex μ i) |
| (ThreeVertexAlgebra.vertex μ j) a ha D) := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using TensorProduct.inductionOn with |
| | tmul c z => rfl |
| | add x y hx hy => |
| exact (map_add _ _ _).trans |
| ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm) |
|
|
| end LittleFinitistic.CornerColumnPaths |
|
|
| end |
|
|
| end OAI |
|
|
