| import OAI.Algebra.Finitistic.Bar.StrictSupportExtension |
| import Mathlib.Algebra.Homology.HomologicalComplexLimits |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| noncomputable section |
| namespace DerivedBar |
| universe u v w w' z z' |
| variable {C : Type u} [Category.{v} C] [HasZeroMorphisms C] [HasZeroObject C] |
| {ι : Type w} {ι' : Type w'} {c : ComplexShape ι} {c' : ComplexShape ι'} |
| (e : c.Embedding c') |
|
|
| /-- Evaluation of extension at an index in its image is the original evaluation, |
| naturally on all complexes and chain maps. -/ |
| def extensionEvalIso (i : ι) (j : ι') (h : e.f i=j) : |
| e.extendFunctor C ⋙ HomologicalComplex.eval C c' j ≅ HomologicalComplex.eval C c i := |
| NatIso.ofComponents (fun K => K.extendXIso e h) (by |
| intro K L f |
| change (HomologicalComplex.extendMap f e).f j ≫ (L.extendXIso e h).hom = |
| (K.extendXIso e h).hom ≫ f.f i |
| rw [HomologicalComplex.extendMap_f f e h] |
| simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]) |
|
|
| /-- Extension by zero preserves the actual colimit, proved degreewise including |
| the zero degrees. In particular it preserves the count coproduct. -/ |
| theorem extensionPreservesColimit {J : Type z} [Category.{z'} J] [HasColimitsOfShape J C] |
| (F : J ⥤ HomologicalComplex C c) : PreservesColimit F (e.extendFunctor C) := by |
| classical |
| apply preservesColimit_of_preserves_colimit_cocone (colimit.isColimit F) |
| apply HomologicalComplex.isColimitOfEval |
| intro j |
| by_cases hj : ∃ i, e.f i=j |
| · let i := hj.choose |
| have hi : e.f i=j := hj.choose_spec |
| letI : PreservesColimit F (e.extendFunctor C ⋙ HomologicalComplex.eval C c' j) := |
| preservesColimit_of_natIso F (extensionEvalIso e i j hi).symm |
| exact isColimitOfPreserves (e.extendFunctor C ⋙ HomologicalComplex.eval C c' j) (colimit.isColimit F) |
| · refine { desc := fun t => 0, fac := ?_, uniq := ?_ } |
| · intro t obj |
| exact ((F.obj obj).isZero_extend_X e j (not_exists.mp hj)).eq_of_src _ _ |
| · intro t m hm |
| exact ((colimit F).isZero_extend_X e j (not_exists.mp hj)).eq_of_src _ _ |
| end DerivedBar |
|
|
| end |
|
|
| end OAI |
|
|
