左三角范畴的稳定化和幂等完备

The Stabilization and Idempotent Completion of a Left Triangulated Category

中文关键词:  左三角范畴  幂等完备化  稳定化  三角等价.

英文关键词:left triangulated category  idempotent completion  stabilization  triangle-equivalence.

中文摘要:

      设$({\mathscr {C}},\Omega,\Delta)$是具有满忠实自函子$\Omega$的左三角范畴. 本文证明存在三角等价$({S(\overline{\mathscr{C}})},{\widetilde{\overline {\Omega}}},{\widetilde{\overline{\Delta}}})$ $\cong$ $({\overline{S(\mathscr{C})}},{\overline{\widetilde {\Omega}}}, {\overline{\widetilde{\Delta}}})$, 其中 $({S(\overline{\mathscr {C}})}, {\widetilde{\overline{\Omega}}},{\widetilde{\overline{\Delta}}})$ 表示$({\mathscr {C}},\Omega,\Delta)$先幂等完备化后稳定化范畴，而$({\overline{S(\mathscr {C})}},{\overline{\widetilde {\Omega}}},{\overline{\widetilde{\Delta}}})$表示$({\mathscr {C}},\Omega,\Delta)$先稳定化后幂等完备化范畴.

英文摘要:

      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 $({\mathscr {C}},\Omega, \Delta)$.