Idempotent Completions of n-Exangulated Categories

Abstract

Suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of \(\mathcal {C}\) are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of \(\mathcal {C}\) into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and \((n+2)\)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.