The Clowder Project

Let $X$ be a set.

1. Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}$ define functors
   \[ \begin{array}{ccc} {\webleft [U,-\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright ),\supset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )},\\ {\webleft [-,V\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )},\\ {\webleft [-_{1},-_{2}\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )}. \end{array} \]

   In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:

   1. If $U\subset A$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,V\webright ]_{X}$.
   2. If $V\subset B$, then $\webleft [U,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.
   3. If $U\subset A$ and $V\subset B$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.
2. Adjointness. We have adjunctions
   
   witnessed by bijections
   \begin{align*} \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\webleft [V,W\webright ]_{X}\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\webleft [U,W\webright ]_{X}\webright ). \end{align*}

   In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:

   1. The following conditions are equivalent:
      1. We have $U\cap V\subset W$.
      2. We have $U\subset \webleft [V,W\webright ]_{X}$.
   2. The following conditions are equivalent:
      1. We have $U\cap V\subset W$.
      2. We have $V\subset \webleft [U,W\webright ]_{X}$.
3. Interaction With the Empty Set I. We have
   \begin{align*} \webleft [U,\text{Ø}\webright ]_{X} & = U^{\textsf{c}},\\ \webleft [\text{Ø},V\webright ]_{X} & = X, \end{align*}

   natural in $U,V\in \mathcal{P}\webleft (X\webright )$.

4. Interaction With $X$. We have
   \begin{align*} \webleft [U,X\webright ]_{X} & = X,\\ \webleft [X,V\webright ]_{X} & = V, \end{align*}

   natural in $U,V\in \mathcal{P}\webleft (X\webright )$.

5. Interaction With the Empty Set II. The functor
   \[ D_{X} \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ) \]

   defined by

   \begin{align*} D_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [-,\text{Ø}\webright ]_{X}\\ & = \webleft (-\webright )^{\textsf{c}}\end{align*}

   is an involutory isomorphism of categories, making $\text{Ø}$ into a dualising object for $\webleft (\mathcal{P}\webleft (X\webright ),\cap ,X,\webleft [-,-\webright ]_{X}\webright )$ in the sense of . In particular:

   1. The diagram
      

      commutes, i.e. we have

      \[ \underbrace{D_{X}\webleft (D_{X}\webleft (U\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft [U,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=U \]

      for each $U\in \mathcal{P}\webleft (X\webright )$.

   2. The diagram
      

      commutes, i.e. we have

      \[ \underbrace{D_{X}\webleft (U\cap D_{X}\webleft (V\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U\cap \webleft [V,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=\webleft [U,V\webright ]_{X} \]

      for each $U,V\in \mathcal{P}\webleft (X\webright )$.

6. Interaction With the Empty Set III. Let $f\colon X\to Y$ be a function.
   1. Interaction With Direct Images. The diagram
      

      commutes, i.e. we have

      \[ f_{*}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{!}\webleft (U\webright )\webright ) \]

      for each $U\in \mathcal{P}\webleft (X\webright )$.

   2. Interaction With Inverse Images. The diagram
      

      commutes, i.e. we have

      \[ f^{-1}\webleft (D_{Y}\webleft (U\webright )\webright )=D_{X}\webleft (f^{-1}\webleft (U\webright )\webright ) \]

      for each $U\in \mathcal{P}\webleft (X\webright )$.

   3. Interaction With Direct Images With Compact Support. The diagram
      

      commutes, i.e. we have

      \[ f_{!}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{*}\webleft (U\webright )\webright ) \]

      for each $U\in \mathcal{P}\webleft (X\webright )$.

7. Interaction With Unions of Families of Subsets I. The diagram
   

   does not commute in general, i.e. we may have

   \[ \bigcup _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \webleft[\bigcup _{U\in \mathcal{U}}U,\bigcup _{V\in \mathcal{V}}V\webright]_{X} \]

   in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

8. Interaction With Unions of Families of Subsets II. The diagram
   

   commutes, i.e. we have

   \[ \webleft[\bigcup _{U\in \mathcal{U}}U,V\webright]_{X}= \bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

   for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.

9. Interaction With Unions of Families of Subsets III. The diagram
   

   commutes, i.e. we have

   \[ \webleft[U,\bigcup _{V\in \mathcal{V}}V\webright]_{X}= \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

   for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

10. Interaction With Intersections of Families of Subsets I. The diagram
    

    does not commute in general, i.e. we may have

    \[ \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \webleft[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\webright]_{X} \]

    in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

11. Interaction With Intersections of Families of Subsets II. The diagram
    

    commutes, i.e. we have

    \[ \webleft[\bigcap _{U\in \mathcal{U}}U,V\webright]_{X}= \bigcup _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

    for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.

12. Interaction With Intersections of Families of Subsets III. The diagram
    

    commutes, i.e. we have

    \[ \webleft[U,\bigcap _{V\in \mathcal{V}}V\webright]_{X}= \bigcap _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

    for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

13. Interaction With Binary Unions. We have equalities of sets
    \begin{align*} \webleft [U\cap V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cup \webleft [V,W\webright ]_{X},\\ \webleft [U,V\cap W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\cap \webleft [U,W\webright ]_{X} \end{align*}

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

14. Interaction With Binary Intersections. We have equalities of sets
    \begin{align*} \webleft [U\cup V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cap \webleft [V,W\webright ]_{X},\\ \webleft [U,V\cup W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\cup \webleft [U,W\webright ]_{X} \end{align*}

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

15. Interaction With Differences. We have equalities of sets
    \begin{align*} \webleft [U\setminus V,W\webright ]_{X} & = \webleft [U,W\webright ]_{X}\cup \webleft [V^{\textsf{c}},W\webright ]_{X}\\ & = \webleft [U,W\webright ]_{X}\cup \webleft [U,V\webright ]_{X},\\ \webleft [U,V\setminus W\webright ]_{X} & = \webleft [U,V\webright ]_{X}\setminus \webleft (U\cap W\webright )\end{align*}

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

16. Interaction With Complements. We have equalities of sets
    \begin{align*} \webleft [U^{\textsf{c}},V\webright ]_{X} & = U\cup V,\\ \webleft [U,V^{\textsf{c}}\webright ]_{X} & = U\cap V,\\ \webleft [U,V\webright ]^{\textsf{c}}_{X} & = U\setminus V \end{align*}

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

17. Interaction With Characteristic Functions. We have
    \[ \chi _{\webleft [U,V\webright ]_{\mathcal{P}\webleft (X\webright )}}\webleft (x\webright )=\operatorname*{\text{max}}\webleft (1-\chi _{U}\ (\mathrm{mod}\ 2),\chi _{V}\webright ) \]

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

18. Interaction With Direct Images. Let $f\colon X\to Y$ be a function. The diagram
    

    commutes, i.e. we have an equality of sets

    \[ f_{*}\webleft (\webleft [U,V\webright ]_{X}\webright )=\webleft [f_{!}\webleft (U\webright ),f_{*}\webleft (V\webright )\webright ]_{Y}, \]

    natural in $U,V\in \mathcal{P}\webleft (X\webright )$.

19. Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. The diagram
    

    commutes, i.e. we have an equality of sets

    \[ f^{-1}\webleft (\webleft [U,V\webright ]_{Y}\webright )=\webleft [f^{-1}\webleft (U\webright ),f^{-1}\webleft (V\webright )\webright ]_{X}, \]

    natural in $U,V\in \mathcal{P}\webleft (X\webright )$.

20. Interaction With Direct Images With Compact Support. Let $f\colon X\to Y$ be a function. We have a natural transformation
    

    with components

    \[ \webleft [f_{*}\webleft (U\webright ),f_{!}\webleft (V\webright )\webright ]_{Y}\subset f_{!}\webleft (\webleft [U,V\webright ]_{X}\webright ) \]

    indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.