English
Reiterates the two-out-of-three property of quasi-isomorphisms under retracts.
Русский
Повторяется свойство тригономического «два из трех» квазиизоморфизмов относительно retracts.
LaTeX
$$$\text{quasiIso}.HasTwoOutOfThreeProperty$$$
Lean4
/-- The explicit left homology data of a short complex of abelian group that is
given by a kernel and a quotient given by the `AddMonoidHom` API. -/
@[simps]
def abLeftHomologyData : S.LeftHomologyData
where
K := AddCommGrpCat.of (AddMonoidHom.ker S.g.hom)
H := AddCommGrpCat.of ((AddMonoidHom.ker S.g.hom) ⧸ AddMonoidHom.range S.abToCycles)
i := AddCommGrpCat.ofHom <| (AddMonoidHom.ker S.g.hom).subtype
π := AddCommGrpCat.ofHom <| QuotientAddGroup.mk' _
wi := by
ext ⟨_, hx⟩
exact hx
hi := AddCommGrpCat.kernelIsLimit _
wπ := by
ext (x : S.X₁)
dsimp
rw [QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]
apply exists_apply_eq_apply
hπ := AddCommGrpCat.cokernelIsColimit (AddCommGrpCat.ofHom S.abToCycles)
