File size: 2,255 Bytes
216fb31 | 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 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
|