  We use supernatural bundles to build GL-equivariant resolutions supported on
the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the
diagonal. We thus obtain results about supernatural bundles that largely
parallel known results about exceptional collections. We apply this
construction to Boij-S\"oderberg decompositions of cohomology tables of vector
bundles, yielding a proof of concept for the idea that those positive rational
decompositions should admit meaningful categorifications.
