A family of acyclic functors

We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets.

💡 Research Summary

The paper investigates functors from a poset P to the category of abelian groups and determines a broad class of such functors for which all higher derived limits vanish. The authors begin by giving a complete characterization of projective functors in this setting. They show that a functor F is projective precisely when it can be expressed as a direct sum of “free” functors associated to the graded pieces of P. Concretely, if P carries a grading P₀, P₁,…, P_r, then for each level i the free functor ℤ