  We present a robust categorical foundation for the duality theory introduced
by Eisenbud and Schreyer to prove the Boij-S\"oderberg conjectures describing
numerical invariants of syzygies. The new foundation allows us to extend the
reach of the theory substantially.
  More explicitly, we construct a pairing between derived categories that
simultaneously categorifies all the functionals used by Eisenbud and Schreyer.
With this new tool, we describe the cone of Betti tables of finite, minimal
free complexes having homology modules of specified dimensions over a
polynomial ring, and we treat many examples beyond polynomial rings. We also
construct an analogue of our pairing between derived categories on a toric
variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer
functionals.
