Higher Cech theory

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.