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ae00908 | 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 | import Mathlib
import OAI.Algebra.Finitistic.Basic
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic
universe u
variable (A : Type u) [Ring A]
def bigFinitisticDimension : WithBot ℕ∞ :=
⨆ (M : ModuleCat.{u} A) (_ : projectiveDimension M < ⊤), projectiveDimension M
def InjectivesGenerate : Prop :=
letI := HasDerivedCategory.standard (ModuleCat.{u} A)
∀ (P : ObjectProperty (DerivedCategory (ModuleCat.{u} A))),
P.IsClosedUnderIsomorphisms → P.IsTriangulated →
(∀ I : Type u, P.IsClosedUnderColimitsOfShape (Discrete I)) →
(∀ (M : ModuleCat.{u} A), Injective M →
P ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj M)) →
∀ X, P X
end LittleFinitistic
end
end OAI
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