Powers of homological complex endomorphisms #
Degreewise recursion for powers of a complex endomorphism. This only needs a category with zero morphisms, so it is available independently of additive structure and chain homotopies.
theorem
HomologicalComplex.pow_f_succ
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Limits.HasZeroMorphisms C]
{ι : Type u_1}
{c : ComplexShape ι}
{K : HomologicalComplex C c}
{s : K ⟶ K}
(m : ℕ)
(i : ι)
:
(CategoryTheory.End.of s ^ (m + 1)).f i = CategoryTheory.CategoryStruct.comp ((CategoryTheory.End.of s ^ m).f i) (s.f i)
Degreewise recursion for the powers of a chain endomorphism.