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\label{def:primorial-presheaf}
The \textbf{primorial category} $\mathbf{Prim}$
(sketched in Remark~\ref{rem:naturality}) is the category:
\begin{enumerate}
    \item \textbf{Objects}: natural numbers $d \geq 1$
          (primorial depths).
    \item \textbf{Morphisms}: for each $k \leq \ell$,
          a unique morphism $\pi_{\ell \to k} : \ell \to k$
          (the primorial reduction map,
          reduction modulo $M_k$).
    \item \textbf{Composition}:
          $\pi_{\ell \to k} \circ \pi_{m \to \ell}
          = \pi_{m \to k}$.
\end{enumerate}
$\mathbf{Prim}$ is a subcategory of $\mathrm{Cat}_\tau$
(Definition~\ref{def:cat-tau}, I.D51).

Three presheaves on $\mathbf{Prim}$:
\begin{enumerate}
    \item The \textbf{source presheaf}
          $\mathcal{F} : \mathbf{Prim}^{\mathrm{op}} \to \mathbf{Ring}$,
          sending $d \mapsto \mathbb{Z}/M_d\mathbb{Z}$
          with restriction maps $\pi_{\ell \to k}$.
    \item The \textbf{split-complex presheaf}
          $\mathcal{F}_{\jj} : \mathbf{Prim}^{\mathrm{op}} \to \mathbf{Ring}$,
          sending $d \mapsto \mathbb{Z}/M_d\mathbb{Z}[\jj]$
          with restriction maps extended $\mathbb{Z}[\jj]$-linearly.
    \item The \textbf{spectral presheaf pair}
          $(\mathcal{F}_B, \mathcal{F}_C)$:
          both copies of $\mathcal{F}$
          (as set-valued presheaves),
          obtained from $\mathcal{F}_{\jj}$
          via the characters $\chi_+$ and $\chi_-$
          (Definition~\ref{def:lemniscate-characters}, I.D37).
\end{enumerate}