  The zeta function of a hyperelliptic curve $C$ over a finite field factors
into a product of $L$-functions, one of which is the $L$-function of $C$. We
categorify this formula using objective linear algebra in the abstract
incidence algebra of the poset of effective $0$-cycles of $C$. As an
application, we prove a collection of combinatorial formulas relating the
number of ramified, split and inert points on $C$ to the overall point count of
$C$.
