Idempotent Completions of n-Exangulated Categories

Abstract

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