| import OAI.Algebra.Finitistic.Bimodule.BimoduleTotalTerms |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.BimoduleTotal |
| open ExternalTensor.Section4 |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] |
| variable (M : ModuleCat.{u} (Enveloping k B)) |
|
|
| abbrev InputTensor (N : ModuleCat.{u} B) := |
| BimoduleBalancedTensor.Tensor k B M (BalancedTensor.ScalarLeftModule k B N) |
|
|
| def inputModuleMap {N N' : ModuleCat.{u} B} (f : N ⟶ N') : |
| InputTensor k B M N →ₗ[B] InputTensor k B M N' where |
| __ := (@BalancedTensor.mapRightTo k B (BimoduleBalancedTensor.Input k B M) |
| (BalancedTensor.ScalarLeftModule k B N) (BalancedTensor.ScalarLeftModule k B N') |
| _ _ _ _ _ _ _ _ _ (BalancedTensor.scalarLeftTower k B N) |
| _ _ _ (BalancedTensor.scalarLeftTower k B N') f.hom).toAddMonoidHom |
| map_smul' b x := by |
| induction x using BalancedTensor.induction_on with |
| | hz => simp |
| | ht m n => rfl |
| | ha x y hx hy => simp_all [smul_add] |
|
|
| /-- Tensor by one genuine bimodule, varying the left-module input and retaining the outside action. -/ |
| def inputFunctor : ModuleCat.{u} B ⥤ ModuleCat.{u} B where |
| obj N := ModuleCat.of B (InputTensor k B M N) |
| map f := ModuleCat.ofHom (inputModuleMap k B M f) |
| map_id N := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using BalancedTensor.induction_on with |
| | hz => simp |
| | ht m n => rfl |
| | ha x y hx hy => |
| exact (map_add _ _ _).trans |
| ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm) |
| map_comp f g := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using BalancedTensor.induction_on with |
| | hz => simp |
| | ht m n => rfl |
| | ha x y hx hy => |
| exact (map_add _ _ _).trans |
| ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm) |
|
|
| instance inputFunctor_additive : (inputFunctor k B M).Additive where |
| map_add {N N'} {f g} := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| induction x using BalancedTensor.induction_on with |
| | hz => exact (map_zero _).trans (map_zero _).symm |
| | ht m n => exact (BalancedTensor.mk k B _ _ m).map_add _ _ |
| | ha x y hx hy => |
| exact (map_add _ _ _).trans |
| ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm) |
|
|
| abbrev columnFunctor (P : BiComplex k B) (p : ℤ) : |
| CochainComplex (ModuleCat.{u} B) ℤ ⥤ CochainComplex (ModuleCat.{u} B) ℤ := |
| (inputFunctor k B (P.X p)).mapHomologicalComplex (.up ℤ) ⋙ shiftFunctor _ (-p) |
|
|
| instance columnFunctor_additive (P : BiComplex k B) (p : ℤ) : |
| (columnFunctor k B P p).Additive := by unfold columnFunctor; infer_instance |
|
|
| end LittleFinitistic.BimoduleTotal |
|
|
| end |
|
|
| end OAI |
|
|
