Idempotent completions of $n$-exangulated categories

Abstract

Suppose (C, E, s) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of C are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of C into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from (C, E, s) to (resp. weakly) idempotent complete n-exangulated categories. 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 nexangulated isomorphism of n-exangulated categories.Acknowledgements. The authors would like to thank Theo Bühler, Ruben Henrard and Adam-Christiaan van Roosmalen for useful email communications, and Andrew Brooke-Taylor and Peter Jørgensen for helpful discussions.

Pre-$(n+2)$-angulated categories

“…see [HLN1,Proposition 4.5 and Proposition 4.34]. However, there are some other examples of n-exangulated categories which are neither (n + 2)-angulated nor n-exact, see [HLN1,HLN2,LZ,HZZ,HZZ1,HHZ,KMS].…”