%
\label{def:tau-site}
The \textbf{$\tau$-site} is the pair
$(\mathrm{Cat}_\tau, J_\tau)$,
where $J_\tau$ is the
\textbf{primorial coverage}
defined as follows.

For each object $X$ in $\mathrm{Cat}_\tau$
and each primorial stage $k \geq 1$,
the Chinese Remainder Theorem
(Section~\ref{subsec:ch30-crt}, I.D29)
gives:
\[
    \mathbb{Z}/M_k\mathbb{Z}
    \;\cong\;
    \mathbb{Z}/p_1\mathbb{Z}
    \;\times\;
    \cdots
    \;\times\;
    \mathbb{Z}/p_k\mathbb{Z},
\]
where $M_k = p_1 \cdots p_k$ is the $k$-th primorial.
The \textbf{primorial covering family}
of $X$ at stage $k$ is:
\[
    \boxed{%
    \mathcal{U}_k(X)
    \;:=\;
    \bigl\{\,
    \phi_i : X_i \to X
    \;\bigm|\;
    i = 1, \ldots, k
    \,\bigr\},}
\]
where $X_i := X \bmod p_i$
is the residue at the $i$-th prime,
viewed as an object of $\mathrm{Cat}_\tau$
via the canonical CRT inclusion,
and $\phi_i$ is the CRT projection arrow.

The Grothendieck topology $J_\tau$
is generated by these families:
a sieve $S$ on $X$ is a \emph{covering sieve}
if and only if
$\mathcal{U}_k(X) \subseteq S$
for some $k \geq 1$.