  We define the derived category of a concrete category in a way which extends
the usual definition of the derived category of a ring, and we prove that the
bounded-below derived category of $\Spec \mathbb{M}_0$ (an approximation, used
by e.g. Connes and Consani, to "$\Spec$ of the field with one element") is the
stable homotopy category of connective spectra. We also describe some basic
features of Grothendieck duality for the map from $\Spec \mathbb{Z}$ to $\Spec
\mathbb{M}_0$, or, what comes to the same thing, the map from $\Spec
\mathbb{Z}$ to $\Spec$ of the sphere spectrum; these basic features include a
computation of the homology of the dualizing complex $f^!(S)$ of abelian groups
associated to the sphere spectrum.
