Idempotent Completion of n-Angulated Categories

Abstract

Let ${\mathcal {C}}$ be an n-angulated category. We prove that its idempotent completion $\widetilde{{\mathcal {C}}}$ admits a unique n-angulated structure such that the canonical functor $\iota : {\mathcal {C}}\rightarrow \widetilde{{\mathcal {C}}}$ is n-angulated. Moreover, the functor $\iota$ induces an equivalence $Hom _{n-ang }(\widetilde{{\mathcal {C}}},{\mathcal {D}})\cong Hom _{n-ang }({\mathcal {C}},{\mathcal {D}})$ for any idempotent complete n-angulated category ${\mathcal {D}}$, where $Hom _{n-ang }$ denotes the category of n-angulated functors.