The Clowder Project

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

where

• •

  Action on Objects. For each $X\in \operatorname {\mathrm{Obj}}(\mathrm{Rel})$, we have

  \begin{align*} \mathcal{P}_{!}(X) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(X),\\ \mathcal{P}_{-1}(X) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(X),\\ \mathcal{P}^{-1}(X) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(X),\\ \mathcal{P}_{*}(X) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(X).\end{align*}
• •

  Action on Morphisms. For each morphism $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y$ of $\mathrm{Rel}$, the images

  \begin{align*} \mathcal{P}_{!}(R) & \colon \mathcal{P}(X) \to \mathcal{P}(Y),\\ \mathcal{P}_{-1}(R) & \colon \mathcal{P}(Y) \to \mathcal{P}(X),\\ \mathcal{P}^{-1}(R) & \colon \mathcal{P}(Y) \to \mathcal{P}(X),\\ \mathcal{P}_{*}(R) & \colon \mathcal{P}(X) \to \mathcal{P}(Y)\end{align*}

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

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

  as in Definition 8.7.1.1.1, Definition 8.7.2.1.1, Definition 8.7.3.1.1, and Definition 8.7.4.1.1.