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ae00908 | 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 50 51 52 53 54 | 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
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