Abstract

Grothendieck’s plus construction sends a presheaf F to the presheaf of equivalence classes of “matching families” of elements of F; two iterations yield the associated sheaf of F (n + 2 iterations are generally required for n-stacks). This construction involves colimits over covers. If we instead take colimits over hypercovers, only one iteration is required (for all n); this corresponds to taking equivalence classes of “locally matching famillies” of elements. A similar construction appears in Verdier’s hypercovering theorem on abelian sheaf cohomology.

In this talk I will give an account of hypercovers within the higher categorical theory of non-abelian cohomology introduced in my previous talk. I will define (n-)hypercovers over a site to be the indexed n-functors which are locally k-surjective for all k in { 0, 1, ..., n }. These maps form the left class of a factorisation system which will allow us to see the plus construction over hypercovers as another generalised Lawvere construction; for the presheaf case the relevant hypercovers are the locally eso and locally full functors. I will also compare this approach to the simplicial theory of hypercovers.