English
The class epiWithInjectiveKernel is closed under identities and composition (multiplicative).
Русский
Класс epiWithInjectiveKernel замкнут относительно тождественных и композиции (множимо).
LaTeX
$$$\\mathrm{epiWithInjectiveKernel}$ is multiplicative$$
Lean4
/-- A functor preserving zero morphisms, monos, and cokernels preserves homology. -/
theorem preservesHomology_of_preservesMonos_and_cokernels [PreservesZeroMorphisms L] [PreservesMonomorphisms L]
[∀ {X Y} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) L] : PreservesHomology L :=
by
apply preservesHomology_of_map_exact
intro S hS
let φ : (ShortComplex.mk _ _ (Abelian.comp_coimage_π_eq_zero S.zero)).map L ⟶ S.map L :=
{ τ₁ := 𝟙 _
τ₂ := 𝟙 _
τ₃ := L.map (Abelian.factorThruCoimage S.g)
comm₂₃ := by
dsimp
rw [Category.id_comp, ← L.map_comp, cokernel.π_desc] }
apply (ShortComplex.exact_iff_of_epi_of_isIso_of_mono φ).1
apply ShortComplex.exact_of_g_is_cokernel
exact CokernelCofork.mapIsColimit _ ((S.exact_iff_exact_coimage_π).1 hS).gIsCokernel L
