We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.

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.

The modular Catalan numbers C k,n , introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of x0 * x1 * • • • * xn where * is a binary k-associative operation and k is a positive integer. The classical notion of associativity is just 1-associativity, in which case C1,n = 1 and the size of the unique class is given by the Catalan number Cn. In this paper we introduce modular Fuss-Catalan numbers C m k,n which count equivalence classes of parenthesizations of x0 * x1 * • • • * xn where * is an m-ary k-associative operation for m ≥ 2. Our main results are a closed formula for C m k,n and a characterisation of k-associativity.

We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.

A Correction to this paper has been published: 10.1007/s10485-021-09664-8