| import Mathlib.Algebra.Homology.Embedding.Extend |
| import Mathlib.Algebra.Homology.HomologicalComplexAbelian |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| namespace ArExplicit |
| universe u v |
| variable {C : Type u} [Category.{v} C] [Abelian C] |
| variable {ι ι' : Type*} {c : ComplexShape ι} {c' : ComplexShape ι'} |
|
|
| theorem extend_shortExact (e : c.Embedding c') |
| (S : ShortComplex (HomologicalComplex C c)) (h : S.ShortExact) : |
| (S.map (e.extendFunctor C)).ShortExact := by |
| have := h.mono_f |
| have := h.epi_g |
| apply HomologicalComplex.shortExact_of_degreewise_shortExact |
| intro j |
| by_cases hj : ∃ i, e.f i = j |
| · obtain ⟨i, hi⟩ := hj |
| let E : ((S.map (e.extendFunctor C)).map (HomologicalComplex.eval C c' j)) ≅ |
| S.map (HomologicalComplex.eval C c i) := |
| ShortComplex.isoMk (S.X₁.extendXIso e hi) (S.X₂.extendXIso e hi) |
| (S.X₃.extendXIso e hi) |
| (by |
| change (S.X₁.extendXIso e hi).hom ≫ S.f.f i = |
| (HomologicalComplex.extendMap S.f e).f j ≫ (S.X₂.extendXIso e hi).hom |
| rw [HomologicalComplex.extendMap_f _ _ hi] |
| simp) |
| (by |
| change (S.X₂.extendXIso e hi).hom ≫ S.g.f i = |
| (HomologicalComplex.extendMap S.g e).f j ≫ (S.X₃.extendXIso e hi).hom |
| rw [HomologicalComplex.extendMap_f _ _ hi] |
| simp) |
| exact ShortComplex.shortExact_of_iso E.symm (h.map (HomologicalComplex.eval C c i)) |
| · have hj' : ∀ i, e.f i ≠ j := by simpa using hj |
| have h₁ := S.X₁.isZero_extend_X e j hj' |
| have h₂ := S.X₂.isZero_extend_X e j hj' |
| have h₃ := S.X₃.isZero_extend_X e j hj' |
| exact { exact := ShortComplex.exact_of_isZero_X₂ _ h₂ |
| mono_f := h₁.mono _ |
| epi_g := h₃.epi _ } |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
