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d7f9a33 | 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 | import OAI.Algebra.AuslanderReiten.Ext.ExternalFreeEvaluation
namespace OAI
noncomputable section
namespace ArExplicit.ExternalBalanced
open CategoryTheory
open scoped TensorProduct
private theorem replace_product {C : Type*} [Category C]
{P Q R S T : C} (d : P ⟶ Q) (p : Q ⟶ R) (q : R ⟶ S)
(r : S ⟶ T) (s : Q ⟶ S) (h : s = p ≫ q) :
d ≫ p ≫ q ≫ r = d ≫ s ≫ r := by
rw [h, Category.assoc]
universe u
variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B]
attribute [local instance 2000] moduleCatBaseModule moduleCatBaseTower
variable (M : ModuleCat.{u} (A ⊗[k] Aᵐᵒᵖ)) (N : ModuleCat.{u} (B ⊗[k] Bᵐᵒᵖ))
variable (X : ModuleCat.{u} A) (Y : ModuleCat.{u} B)
variable (U : ModuleCat.{u} A) (V : ModuleCat.{u} B)
variable (a : (EnvelopingBimodule.evaluationFunctor k A X).obj M ⟶ U)
(b : (EnvelopingBimodule.evaluationFunctor k B Y).obj N ⟶ V)
variable (f : U ⟶ ModuleCat.of A A) (g : V ⟶ ModuleCat.of B B)
theorem freeEvaluation_factorization :
(shuffle k A B M N X Y).toModuleIso.hom ≫ ExternalTensor.mapHom k A B a b ≫
ExternalTensor.mapHom k A B f g ≫
(ExternalTensor.canonicalTensorIso k A B A B).hom ≫
(ExternalTensor.regularEquiv k A B).toModuleIso.hom =
freeEvaluation k A B M N X Y (a ≫ f) (b ≫ g) :=
replace_product (shuffle k A B M N X Y).toModuleIso.hom
(ExternalTensor.mapHom k A B a b) (ExternalTensor.mapHom k A B f g)
((ExternalTensor.canonicalTensorIso k A B A B).hom ≫
(ExternalTensor.regularEquiv k A B).toModuleIso.hom)
(ExternalTensor.mapHom k A B (a ≫ f) (b ≫ g))
(ExternalTensor.mapHom_comp k A B a f b g)
end ArExplicit.ExternalBalanced
end
end OAI
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