| import OAI.Algebra.Finitistic.Ordinary.OrdinaryProductProjective |
| import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.ProductModules |
| universe u |
| variable (L R : Type u) [Ring L] [Ring R] |
| |
| instance firstFunctor_additive : (firstFunctor.{u,u} L R).Additive where |
| map_add := by intros; rfl |
|
|
| / |
| theorem firstFunctor_mapsExact (K : ShortComplex (ModuleCat.{u} (L × R))) (hK : K.Exact) : |
| (K.map (firstFunctor L R)).Exact := by |
| rw [ShortComplex.moduleCat_exact_iff] at hK ⊢ |
| intro y hy |
| obtain ⟨x,hx⟩ := hK y.val (congrArg Subtype.val hy) |
| refine ⟨⟨((1,0) : L × R) • x,?_⟩,?_⟩ |
| · change ((1,0) : L × R) • (((1,0) : L × R) • x) = ((1,0) : L × R) • x |
| rw [← mul_smul] |
| have h : ((1,0) : L × R) * (1,0) = (1,0) := by ext <;> simp |
| rw [h] |
| · apply Subtype.ext |
| change K.f.hom (((1,0) : L × R) • x) = y.val |
| rw [K.f.hom.map_smul,hx,y.property] |
| |
| instance firstFunctor_preservesHomology : (firstFunctor.{u,u} L R).PreservesHomology := |
| Functor.preservesHomology_of_map_exact _ (firstFunctor_mapsExact L R) |
| |
| instance firstFunctor_preservesFiniteLimits : PreservesFiniteLimits (firstFunctor.{u,u} L R) := |
| Functor.preservesFiniteLimits_of_preservesHomology _ |
| |
| instance firstFunctor_preservesFiniteColimits : PreservesFiniteColimits (firstFunctor.{u,u} L R) := |
| Functor.preservesFiniteColimits_of_preservesHomology _ |
| |
| / |
| def firstInflationComponentEquiv (M : ModuleCat.{u} L) : |
| firstObject L R ((firstInflation L R).obj M) ≃ₗ[L] M where |
| toFun x := x.val |
| invFun x := ⟨x, one_smul L x⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| / |
| def firstInflationComponentIso : firstInflation.{u,u} L R ⋙ firstFunctor L R ≅ 𝟭 (ModuleCat.{u} L) := |
| NatIso.ofComponents (fun M => (firstInflationComponentEquiv L R M).toModuleIso) |
| (by intros; rfl) |
| end LittleFinitistic.ProductModules |
|
|
| end |
|
|
| end OAI |
|
|
