Sheaves over complexes of groups and developability

TL;DR

This paper introduces a new concept of sheaves over complexes of groups and provides a criterion to determine when such complexes are developable, especially when related to morphisms into general linear groups.

Contribution

It defines sheaves over complexes of groups and offers a developability criterion, including a characterization when developability is witnessed by a linear representation.

Findings

A new notion of sheaves over complexes of groups is established.

A developability criterion for complexes of groups is provided.

Characterization of developability via morphisms to GL(V) is achieved.

Abstract

We define the notion of a sheaf over a complex of groups. As an application, we give a criterion for the developability of a complex of groups. When the developability is witnessed by a morphism to $\mathrm{GL}(V)$ for some $V$, our criterion is a characterization of developability.