File size: 3,499 Bytes
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 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Algebra.Finitistic.Tensor.BalancedVertexImage
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.BalancedVertexFunctor
open ExternalTensor.Section4
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
variable (e : B) (he : e*e=e)
variable (N : Type u) [AddCommGroup N] [Module k N] [Module B N]
abbrev Space (M : ModuleCat.{u} (Enveloping k B)) :=
IdempotentTensor.leftSpace (k := k) e (BimoduleBalancedTensor.Tensor k B M N)
def spaceMap {M M' : ModuleCat.{u} (Enveloping k B)} (f : M ⟶ M') :
Space k B e N M →ₗ[k] Space k B e N M' :=
(((BimoduleBalancedTensor.map k B N f).restrictScalars k).comp
(Space k B e N M).subtype).codRestrict _ (by
intro x
change e • (BimoduleBalancedTensor.map k B N f x.val) =
BimoduleBalancedTensor.map k B N f x.val
rw [← (BimoduleBalancedTensor.map k B N f).map_smul,x.property])
/-- The actual left vertex of the balanced tensor, with its original induced maps. -/
def functor : ModuleCat.{u} (Enveloping k B) ⥤ ModuleCat.{u} k where
obj M := ModuleCat.of k (Space k B e N M)
map f := ModuleCat.ofHom (spaceMap k B e N f)
map_id M := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
apply Subtype.ext
exact congrArg (fun f : (BimoduleBalancedTensor.functor k B N).obj M ⟶
(BimoduleBalancedTensor.functor k B N).obj M => f.hom x.val)
((BimoduleBalancedTensor.functor k B N).map_id M)
map_comp f g := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
apply Subtype.ext
exact congrArg (fun f => ModuleCat.Hom.hom f x.val)
((BimoduleBalancedTensor.functor k B N).map_comp f g)
instance additive : (functor k B e N).Additive where
map_add := by
intros M M' f g
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
apply Subtype.ext
exact congrArg (fun a : (BimoduleBalancedTensor.functor k B N).obj M ⟶
(BimoduleBalancedTensor.functor k B N).obj M' => a.hom x.val)
((BimoduleBalancedTensor.functor k B N).map_add (f := f) (g := g))
/-- Vertex extraction commutes naturally with the original balanced tensor functor. -/
def comparison : BimoduleVertex.functor k B e ⋙ BalancedTensor.tensorFunctor k B N ≅
functor k B e N :=
NatIso.ofComponents (fun M => (BalancedVertexImage.equiv k B M e he N).toModuleIso) (by
intro M M' f
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
apply Subtype.ext
have hn : (BimoduleVertex.functor k B e).map f ≫ BalancedVertexImage.inclusion k B M' e =
BalancedVertexImage.inclusion k B M e ≫ (forgetRight k B).map f := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro y
rfl
have hF := congrArg (fun a => (BalancedTensor.tensorFunctor k B N).map a) hn
simp only [Functor.map_comp] at hF
exact congrArg (fun a => ModuleCat.Hom.hom a x) hF)
/-- The natural comparison gives an actual chain isomorphism on every bimodule complex. -/
def complexIso (P : BiComplex k B) :
((BimoduleVertex.functor k B e ⋙ BalancedTensor.tensorFunctor k B N).mapHomologicalComplex (.up ℤ)).obj P ≅
((functor k B e N).mapHomologicalComplex (.up ℤ)).obj P :=
HomologicalComplex.Hom.isoOfComponents (fun n => (comparison k B e he N).app (P.X n))
(fun n m _ => ((comparison k B e he N).hom.naturality (P.d n m)).symm)
end LittleFinitistic.BalancedVertexFunctor
end
end OAI
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