openAI-math-dataset / lean /OAI /Algebra /Finitistic /Localization /LocalizedEvaluationStructure.lean
| import OAI.Algebra.Finitistic.Localization.BoundedQuotientEvaluation | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits | |
| namespace LittleFinitistic.QuadraticEncoding | |
| open ThickEvaluation | |
| universe u v | |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) | |
| /-- The constructed quotient carries its canonical triangulated structure. -/ | |
| theorem boundedQ_triangulated : IsTriangulated (BoundedQ P) := inferInstance | |
| variable {P} {Y : Type v} [AddCommGroup Y] [Module k Y] (X : Representation P Y) | |
| instance quotientEvaluationAdditive : (boundedQuotientEvaluation X).Additive := by | |
| rw [Localization.functor_additive_iff (boundedDenominators P).Q (boundedDenominators P)] | |
| exact Functor.additive_of_iso (boundedQuotientEvaluationFactors X).symm | |
| instance quotientEvaluationLifting : Localization.Lifting | |
| (boundedDenominators P).Q (boundedDenominators P) | |
| (boundedEvaluation X) (boundedQuotientEvaluation X) := | |
| ⟨boundedQuotientEvaluationFactors X⟩ | |
| instance quotientEvaluationShift : (boundedQuotientEvaluation X).CommShift ℤ := | |
| Functor.commShiftOfLocalization (boundedDenominators P).Q (boundedDenominators P) ℤ | |
| (boundedEvaluation X) (boundedQuotientEvaluation X) | |
| instance quotientEvaluationFactorsShift : | |
| NatTrans.CommShift (boundedQuotientEvaluationFactors X).hom ℤ := | |
| inferInstanceAs (NatTrans.CommShift | |
| (Localization.Lifting.iso (boundedDenominators P).Q (boundedDenominators P) | |
| (boundedEvaluation X) (boundedQuotientEvaluation X)).hom ℤ) | |
| instance quotientEvaluationTriangulated : (boundedQuotientEvaluation X).IsTriangulated := by | |
| have : (boundedDenominators P).Q.mapArrow.EssSurj := | |
| Localization.essSurj_mapArrow (boundedDenominators P).Q (boundedDenominators P) | |
| exact Functor.isTriangulated_of_precomp_iso (boundedQuotientEvaluationFactors X) | |
| end LittleFinitistic.QuadraticEncoding | |
| end | |
| end OAI | |