Castelnuovo-Mumford regularity of representations of certain product categories

Abstract: We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories $\mathrm{FI}^m$ and $\mathrm{FI}_G^m$.