English
Inverse-homologyι naturality across restriction–iso and opcycles inverse diagrams.
Русский
Естественность обращения гомологии и ограничение через обратные диаграммы opcycles.
LaTeX
$$$ (K_{|e}\\mathrm{RestrictionHomologyIso}\\,e\\,i\\,j\\,k).inv \\circ (K_{|e}\\mathrm{homology\\iota}) = K_{\\mathrm{homology\\iota}}(j) \\circ (K_{|e}\\mathrm{restrictionOpcyclesIso}\\,e\\,i\\,j).inv $$$
Lean4
/-- Compatibility of `nullHomotopicMap` with the precomposition by a morphism
of complexes. -/
theorem comp_nullHomotopicMap (f : C ⟶ D) (hom : ∀ i j, D.X i ⟶ E.X j) :
f ≫ nullHomotopicMap hom = nullHomotopicMap fun i j => f.f i ≫ hom i j :=
by
ext n
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
simp only [Preadditive.comp_add, assoc, f.comm_assoc]
