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d7f9a33 | 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 | 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
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