Quasi-isomorphisms of periodic complexes #
A morphism of cyclic cochain complexes is a quasi-isomorphism exactly when its mapping cone is acyclic. This identifies the homology-based weak equivalences with the cone criterion used for the Verdier quotient of the periodic homotopy category by acyclic complexes.
No positivity hypothesis on the period is needed; at period zero this applies to integer-indexed cochain complexes.
References #
- T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199--236, Section 3.
theorem
TauCeti.PeriodicComplex.quasiIso_iff_acyclic_homotopyCofiber
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Abelian C]
{n : ℕ}
{X Y : CochainComplex C (ZMod n)}
(f : X ⟶ Y)
:
A morphism of cyclic cochain complexes is a quasi-isomorphism if and only if its mapping cone is acyclic.