A family of acyclic functors

TL;DR

This paper characterizes a family of acyclic functors from posets to abelian groups, using spectral sequences and duality, with applications to various graded posets like simplicial complexes.

Contribution

It introduces a new family of acyclic functors and characterizes projective and injective functors on graded posets, expanding understanding of higher limits.

Findings

Higher direct limits vanish on the identified functors.

Spectral sequences are used to analyze functor properties.

Dual results for inverse limits are also established.

Abstract

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.