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We give a geometric interpretation of sheaf cohomology for higher degrees nin terms of torsors on the member of degree d=n-1 in hypercoverings of typer=n-2, endowed with an additional data, the so-called rigidification. Thisgeneralizes the fact that cohomology in degree one is the group of isomorphismclasses of torsors, where the rigidification becomes vacuous, and thatcohomology in degree two can be expressed in terms of bundle gerbes, where therigidification becomes an associativity constraint.

Rigidified torsor cocycles, hypercoverings and bundle gerbes