Higher Čech Theory

Abstract

We introduce a notion of ‘cover of level n ’ for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n -covers biject with global homotopy classes when the target is an n -type. When the target is an Eilenberg–MacLane sheaf, this specializes to computing derived functor cohomology, up to degree n , via simplicial homotopy classes taken along n -covers. Our approach is purely simplicial and combinatorial.