The Clowder Project

Let $X$ be a set.

1. 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. (a)

      If $U\subset A$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,V\webright ]_{X}$.

   2. (b)

      If $V\subset B$, then $\webleft [U,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.

   3. (c)

      If $U\subset A$ and $V\subset B$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.

2. 2.

   Adjointness. We have adjunctions

   
   witnessed by bijections
   \begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\webleft [V,W\webright ]_{X}\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{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. (a)

      The following conditions are equivalent:

      1. (i)

         We have $U\cap V\subset W$.

      2. (ii)

         We have $U\subset \webleft [V,W\webright ]_{X}$.

   2. (b)

      The following conditions are equivalent:

      1. (i)

         We have $U\cap V\subset W$.

      2. (ii)

         We have $V\subset \webleft [U,W\webright ]_{X}$.

3. 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. 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. 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. (a)

      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. (b)

      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. 6.

   Interaction With the Empty Set III. Let $f\colon X\to Y$ be a function.

   1. (a)

      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. (b)

      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. (c)

      Interaction With Codirect 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 $U\in \mathcal{P}\webleft (X\webright )$.

7. 7.

   Interaction With Unions of Families of Subsets I. The diagram

   
   does not commute in general, i.e. we may have
   \[ \bigcup _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \left[\bigcup _{U\in \mathcal{U}}U,\bigcup _{V\in \mathcal{V}}V\right]_{X} \]

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

8. 8.

   Interaction With Unions of Families of Subsets II. The diagram

   
   commutes, i.e. we have
   \[ \left[\bigcup _{U\in \mathcal{U}}U,V\right]_{X}= \bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

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

9. 9.

   Interaction With Unions of Families of Subsets III. The diagram

   
   commutes, i.e. we have
   \[ \left[U,\bigcup _{V\in \mathcal{V}}V\right]_{X}= \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

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

10. 10.

    Interaction With Intersections of Families of Subsets I. The diagram

    
    does not commute in general, i.e. we may have
    \[ \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \left[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\right]_{X} \]

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

11. 11.

    Interaction With Intersections of Families of Subsets II. The diagram

    
    commutes, i.e. we have
    \[ \left[\bigcap _{U\in \mathcal{U}}U,V\right]_{X}= \bigcup _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

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

12. 12.

    Interaction With Intersections of Families of Subsets III. The diagram

    
    commutes, i.e. we have
    \[ \left[U,\bigcap _{V\in \mathcal{V}}V\right]_{X}= \bigcap _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

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

13. 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. 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. 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. 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. 17.

    Interaction With Characteristic Functions. We have

    \[ \chi _{\webleft [U,V\webright ]_{\mathcal{P}\webleft (X\webright )}}\webleft (x\webright )=\operatorname*{\operatorname {\mathrm{max}}}\webleft (1-\chi _{U}\ (\mathrm{mod}\ 2),\chi _{V}\webright ) \]

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

18. 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. 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. 20.

    Interaction With Codirect Images. 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 $U,V\in \mathcal{P}\webleft (X\webright )$.