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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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Algebra.Finitistic.Bimodule.BimoduleTensorVertexBridge
import OAI.Algebra.Finitistic.Localization.NativeArrowBridge
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.TensorArrowComparison
open ExternalTensor.Section4
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
variable (N : Type u) [AddCommGroup N] [Module k N] [Module B N]
/-- Tensor the actual right vertex, then identify it with the vertex of the full
left B-module tensor, preserving the original balanced tensor classes. -/
def component (e : B) (he : e*e=e) (M : ModuleCat.{u} (Enveloping k B)) :
(BalancedTensor.tensorFunctor k B N).obj ((BimoduleVertex.functor k B e).obj M) ≅
(LeftVertexHomology.vertex k B e).obj ((BimoduleBalancedTensor.functor k B N).obj M) :=
(BalancedVertexImage.equiv k B M e he N).toModuleIso ≪≫
(BimoduleTensorVertexBridge.equiv k B N e M).symm.toModuleIso
/-- The tensor/vertex comparison retains left multiplication by every original
supported arrow, and does not require the evaluation module to be flat. -/
theorem arrow_square (ei ej a : B) (hi : ei*ei=ei) (hj : ej*ej=ej) (ha : ej*a=a)
(M : ModuleCat.{u} (Enveloping k B)) :
(BalancedTensor.tensorFunctor k B N).map
((ArrowVertex.arrow k B ei ej a ha).app M) ≫ (component k B N ej hj M).hom =
(component k B N ei hi M).hom ≫
(LeftVertexHomology.arrow k B ei ej a ha).app ((BimoduleBalancedTensor.functor k B N).obj M) := by
let l := ((BalancedTensor.tensorFunctor k B N).map
((ArrowVertex.arrow k B ei ej a ha).app M) ≫ (component k B N ej hj M).hom).hom
let r := ((component k B N ei hi M).hom ≫
(LeftVertexHomology.arrow k B ei ej a ha).app
((BimoduleBalancedTensor.functor k B N).obj M)).hom
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
change l x = r x
induction x using BalancedTensor.induction_on with
| hz => exact l.map_zero.trans r.map_zero.symm
| ht m n => rfl
| ha x y hx hy =>
exact (l.map_add x y).trans ((congrArg₂ (· + ·) hx hy).trans (r.map_add x y).symm)
/-- Natural comparison for the full B-valued tensor functor. -/
def comparison (e : B) (he : e*e=e) :
BimoduleVertex.functor k B e ⋙ BalancedTensor.tensorFunctor k B N ≅
BimoduleBalancedTensor.functor k B N ⋙ LeftVertexHomology.vertex k B e :=
BalancedVertexFunctor.comparison k B e he N ≪≫
(BimoduleTensorVertexBridge.comparison k B N e).symm
def complexIso (e : B) (he : e*e=e) (P : BiComplex k B) :
(((BalancedTensor.tensorFunctor k B N).mapHomologicalComplex (.up ℤ)).obj
(((BimoduleVertex.functor k B e).mapHomologicalComplex (.up ℤ)).obj P)) ≅
(((LeftVertexHomology.vertex k B e).mapHomologicalComplex (.up ℤ)).obj
(((BimoduleBalancedTensor.functor k B N).mapHomologicalComplex (.up ℤ)).obj P)) :=
HomologicalComplex.Hom.isoOfComponents (fun n => component k B N e he (P.X n))
(fun n m _ => ((comparison k B N e he).hom.naturality (P.d n m)).symm)
/-- The full chain comparison respects the original left multiplication arrows. -/
theorem complex_arrow_square (ei ej a : B) (hi : ei*ei=ei) (hj : ej*ej=ej) (ha : ej*a=a)
(P : BiComplex k B) :
((BalancedTensor.tensorFunctor k B N).mapHomologicalComplex (.up ℤ)).map
(ArrowVertex.arrowComplex k B ei ej a ha P) ≫ (complexIso k B N ej hj P).hom =
(complexIso k B N ei hi P).hom ≫
((LeftVertexHomology.arrow k B ei ej a ha).mapHomologicalComplex (.up ℤ)).app
(((BimoduleBalancedTensor.functor k B N).mapHomologicalComplex (.up ℤ)).obj P) := by
apply HomologicalComplex.Hom.ext
funext n
exact arrow_square k B N ei ej a hi hj ha (P.X n)
end LittleFinitistic.TensorArrowComparison
end
end OAI
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