English
The forward naturality: the homomorphism part of the restriction–homology iso composed with homologyι j′ equals the homologyι j composed with the restriction–opcycles iso.
Русский
Прямая натуральность: отображение гомологии через ограничение–изоморфизм состоит с homologyι(j′) равно отображению через homologyι(j) после ограничения–opcycles.
LaTeX
$$$ (K_{|e}\\mathrm{RestrictionHomologyIso}\\,e\\,i\\,j\\,k).hom \\circ K_{\\mathrm{homology\\iota}}(j') = K_{|e}\\mathrm{homology\\iota}(j) \\circ (K_{\\mathrm restrictionOpcyclesIso}})^{-1} $$$
Lean4
/-- Compatibility of `nullHomotopicMap'` with the precomposition by a morphism
of complexes. -/
theorem comp_nullHomotopicMap' (f : C ⟶ D) (hom : ∀ i j, c.Rel j i → (D.X i ⟶ E.X j)) :
f ≫ nullHomotopicMap' hom = nullHomotopicMap' fun i j hij => f.f i ≫ hom i j hij :=
by
rw [nullHomotopicMap', comp_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [comp_zero]
