The Clowder Project

Functoriality II. The assignment $X\mapsto \mathcal{P}(X)$ defines a functor

where

• •

  Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have

  \[ \mathcal{P}^{-1}(A)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(A). \]
• •

  Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, the action on morphisms

  \[ \mathcal{P}^{-1}_{A,B}\colon \mathsf{Sets}(A,B)\to \mathsf{Sets}(\mathcal{P}(B),\mathcal{P}(A)) \]

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

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

  defined by

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

  as in Definition 4.6.2.1.1.

Functoriality III. The assignment $X\mapsto \mathcal{P}(X)$ defines a functor

where

• •

  Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have

  \[ \mathcal{P}_{*}(A)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(A). \]
• •

  Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, the action on morphisms

  \[ \mathcal{P}_{!|A,B}\colon \mathsf{Sets}(A,B)\to \mathsf{Sets}(\mathcal{P}(A),\mathcal{P}(B)) \]

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

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

  defined by

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

  as in Definition 4.6.3.1.1.