The Clowder Project

Let $X$ be a set.

1. Functoriality I. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
   \[ \mathcal{P}_{*}\colon \mathsf{Sets}\to \mathsf{Sets}, \]

   where

   • Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
     \[ \mathcal{P}_{*}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \]

   • Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on morphisms
     \[ \mathcal{P}_{*|A,B}\colon \mathsf{Sets}\webleft (A,B\webright )\to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ) \]

     of $\mathcal{P}_{*}$ at $\webleft (A,B\webright )$ is the map defined by by sending a map of sets $f\colon A\to B$ to the map

     \[ \mathcal{P}_{*}\webleft (f\webright )\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]

     defined by

     \[ \mathcal{P}_{*}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{*}, \]

     as in Definition 2.4.4.1.1.

2. Functoriality II. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
   \[ \mathcal{P}^{-1}\colon \mathsf{Sets}^{\mathsf{op}}\to \mathsf{Sets}, \]

   where

   • Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
     \[ \mathcal{P}^{-1}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \]

   • Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on morphisms
     \[ \mathcal{P}^{-1}_{A,B}\colon \mathsf{Sets}\webleft (A,B\webright )\to \mathsf{Sets}\webleft (\mathcal{P}\webleft (B\webright ),\mathcal{P}\webleft (A\webright )\webright ) \]

     of $\mathcal{P}^{-1}$ at $\webleft (A,B\webright )$ is the map defined by by sending a map of sets $f\colon A\to B$ to the map

     \[ \mathcal{P}^{-1}\webleft (f\webright )\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]

     defined by

     \[ \mathcal{P}^{-1}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f^{-1}, \]

     as in Definition 2.4.5.1.1.

3. Functoriality III. The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ defines a functor
   \[ \mathcal{P}_{!}\colon \mathsf{Sets}\to \mathsf{Sets}, \]

   where

   • Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
     \[ \mathcal{P}_{!}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}\webleft (A\webright ). \]

   • Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on morphisms
     \[ \mathcal{P}_{!|A,B}\colon \mathsf{Sets}\webleft (A,B\webright )\to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ) \]

     of $\mathcal{P}_{!}$ at $\webleft (A,B\webright )$ is the map defined by by sending a map of sets $f\colon A\to B$ to the map

     \[ \mathcal{P}_{!}\webleft (f\webright )\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]

     defined by

     \[ \mathcal{P}_{!}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}, \]

     as in Definition 2.4.6.1.1.

4. Adjointness I. We have an adjunction
   
   witnessed by a bijection
   \[ \underbrace{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (A\webright ),B\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mkern 5mu\mathsf{Sets}\webleft (B,\mathcal{P}\webleft (A\webright )\webright )} \cong \mathsf{Sets}\webleft (A,\mathcal{P}\webleft (B\webright )\webright ), \]

   natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $B\in \text{Obj}\webleft (\mathsf{Sets}^{\mathsf{op}}\webright )$.

5. Adjointness II. We have an adjunction
   
   witnessed by a bijection of sets
   \[ \mathrm{Rel}\webleft (\text{Gr}\webleft (A\webright ),B\webright ) \cong \mathsf{Sets}\webleft (A,\mathcal{P}\webleft (B\webright )\webright ) \]

   natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $B\in \text{Obj}\webleft (\mathrm{Rel}\webright )$, where $\text{Gr}$ is the graph functor of Chapter 6: Constructions With Relations, Item 1 of Proposition 6.3.1.1.2 and $\mathcal{P}_{*}$ is the functor of Chapter 6: Constructions With Relations, Proposition 6.4.5.1.1.