Multisets of finite intervals and a universal category of poset representations

For any finite totally ordered set, the multisets of intervals form an abelian category.

Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences, including Catalan and large Schröder numbers.

Surprisingly, in some cases new integer sequences arise.

The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients.

For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.