The Clowder Project

The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines functors¹

where

• Action on Objects. For each $A\in \text{Obj}\webleft (\mathrm{Rel}\webright )$, we have
  \begin{align*} \mathcal{P}_{*}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{-1}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}^{-1}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{!}\webleft (A\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright );\end{align*}

• Action on Morphisms. For each morphism $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ of $\mathrm{Rel}$, the images
  \begin{align*} \mathcal{P}_{*}\webleft (R\webright ) & \colon \mathcal{P}\webleft (A\webright ) \to \mathcal{P}\webleft (B\webright ),\\ \mathcal{P}_{-1}\webleft (R\webright ) & \colon \mathcal{P}\webleft (B\webright ) \to \mathcal{P}\webleft (A\webright ),\\ \mathcal{P}^{-1}\webleft (R\webright ) & \colon \mathcal{P}\webleft (B\webright ) \to \mathcal{P}\webleft (A\webright ),\\ \mathcal{P}_{!}\webleft (R\webright ) & \colon \mathcal{P}\webleft (A\webright ) \to \mathcal{P}\webleft (B\webright )\end{align*}

  of $R$ by $\mathcal{P}_{*}$, $\mathcal{P}_{-1}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{!}$ are defined by

  \begin{align*} \mathcal{P}_{*}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{*},\\ \mathcal{P}_{-1}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1},\\ \mathcal{P}^{-1}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{-1},\\ \mathcal{P}_{!}\webleft (R\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{!},\end{align*}

  as in Definition 7.4.1.1.1, Definition 7.4.2.1.1, Definition 7.4.3.1.1, and Definition 7.4.4.1.1.