Grothendieck Topologies and Sheaves on Čech closure spaces

Abstract: We introduce a new approach to the construction of sheaf theory on discrete and combinatorial spaces by constructing sheaves on \v{C}ech closure spaces, a category which contains the major classes of spaces of most interest to applications: topological spaces, graphs, and metric spaces with a priviledged scale. In the process, we construct a new Grothendieck topology on \v{C}ech closure spaces, and we show that sheaves on {C}ech closure spaces are isomorphic to the sheaf of sections of an appropriately defined \'etale space.