| import OAI.Algebra.Finitistic.Localization.LocalizedEvaluationStructure | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits | |
| namespace LittleFinitistic.QuadraticEncoding | |
| open FiniteProjectiveHomotopy ThickEvaluation | |
| universe u v | |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} | |
| variable {Y : Type v} [AddCommGroup Y] [Module k Y] [FiniteDimensional k Y] | |
| variable (X : Representation P Y) | |
| /-- The concrete bounded homotopy category of finite-dimensional vector spaces. -/ | |
| abbrev FiniteVectorHomotopy := K.{u,max u v} k | |
| theorem finiteEvaluation_property (T : BoundedK P) : | |
| objects k ((boundedEvaluation X).obj T) := by | |
| obtain ⟨hb,hf⟩ := boundedEvaluation_finite X T | |
| refine ⟨hb,fun i => ⟨hf i,?_⟩⟩ | |
| exact ModuleCat.projective_of_categoryTheory_projective _ | |
| /-- Evaluation with bounded finite-dimensional vector-space complexes as its actual target. -/ | |
| def finiteEvaluation : BoundedK P ⥤ FiniteVectorHomotopy.{u,v} (k := k) := | |
| (objects k).lift (boundedEvaluation X) (finiteEvaluation_property X) | |
| instance finiteEvaluationAdditive : (finiteEvaluation X).Additive := by | |
| dsimp [finiteEvaluation] | |
| infer_instance | |
| instance finiteEvaluationShift : (finiteEvaluation X).CommShift ℤ := by | |
| dsimp [finiteEvaluation] | |
| infer_instance | |
| instance finiteEvaluationTriangulated : (finiteEvaluation X).IsTriangulated := by | |
| dsimp [finiteEvaluation] | |
| infer_instance | |
| lemma finiteEvaluation_kills_generators : ∀ T, boundedConeGenerators P T → | |
| IsZero ((finiteEvaluation X).obj T) := by | |
| intro T hT | |
| apply IsZero.of_full_of_faithful_of_isZero (inclusion k) | |
| exact boundedEvaluation_kills_generators X T hT | |
| /-- The fixed quotient Q evaluates directly into bounded finite-dimensional complexes. -/ | |
| def finiteQuotientEvaluation : BoundedQ P ⥤ FiniteVectorHomotopy.{u,v} (k := k) := | |
| quotientEvaluation (boundedConeGenerators P) (finiteEvaluation X) | |
| (finiteEvaluation_kills_generators X) | |
| def finiteQuotientEvaluationFactors : | |
| (boundedDenominators P).Q ⋙ finiteQuotientEvaluation X ≅ finiteEvaluation X := | |
| quotientEvaluationFactors (boundedConeGenerators P) (finiteEvaluation X) | |
| (finiteEvaluation_kills_generators X) | |
| end LittleFinitistic.QuadraticEncoding | |
| end | |
| end OAI | |