The Stabilization and Idempotent Completion of a Left Triangulated Category

Abstract:

      Let $({\mathscr {C}},\Omega,\Delta)$ be a left triangulated category with a fully faithful endofunctor $\Omega$. We show a triangle-equivalence $({S(\overline{\mathscr{C}})},{\widetilde{\overline {\Omega}}},{\widetilde{\overline{\Delta}}})$ $\cong$ $({\overline{S(\mathscr{C})}},{\overline{\widetilde {\Omega}}}, {\overline{\widetilde{\Delta}}})$, where $({S(\overline{\mathscr {C}})}, {\widetilde{\overline{\Omega}}},{\widetilde{\overline{\Delta}}})$ denotes the stabilization of the idempotent completion of $({\mathscr {C}},\Omega,\Delta)$ and $({\overline{S(\mathscr {C})}},{\overline{\widetilde {\Omega}}}, {\overline{\widetilde{\Delta}}})$ denotes the idempotent completion of the stabilization of $({\mathscr {C}},\Omega, \Delta)$.