| 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 |
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| end OAI |
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