The Clowder Project

The internal Hom of $\mathcal{P}(X)$ from $U$ to $V$ is the subset $[U,V]_{X}$¹ of $X$ given by

where $U^{\textsf{c}}$ is the complement of $U$ of Definition 4.3.11.1.1.

Let $X$ be a set.

1. 1.

   Functoriality. The assignments $U,V,(U,V)\mapsto \mathbf{Hom}_{\mathcal{P}(X)}$ define functors

   \[ \begin{array}{ccc} {[U,-]_{X}}\colon \mkern -15mu & {(\mathcal{P}(X),\supset )} \mkern -17.5mu& {}\mathbin {\to }{(\mathcal{P}(X),\subset )},\\ {[-,V]_{X}}\colon \mkern -15mu & {(\mathcal{P}(X),\subset )} \mkern -17.5mu& {}\mathbin {\to }{(\mathcal{P}(X),\subset )},\\ {[-_{1},-_{2}]_{X}}\colon \mkern -15mu & {(\mathcal{P}(X)\times \mathcal{P}(X),\subset \times \supset )} \mkern -17.5mu& {}\mathbin {\to }{(\mathcal{P}(X),\subset )}. \end{array} \]

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

   1. (a)

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

   2. (b)

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

   3. (c)

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

2. 2.

   Adjointness. We have adjunctions

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

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

   1. (a)

      The following conditions are equivalent:

      1. (i)

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

      2. (ii)

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

   2. (b)

      The following conditions are equivalent:

      1. (i)

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

      2. (ii)

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

3. 3.

   Interaction With the Empty Set I. We have

   \begin{align*} [U,\text{Ø}]_{X} & = U^{\textsf{c}},\\ [\text{Ø},V]_{X} & = X, \end{align*}

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

4. 4.

   Interaction With $X$. We have

   \begin{align*} [U,X]_{X} & = X,\\ [X,V]_{X} & = V, \end{align*}

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

5. 5.

   Interaction With the Empty Set II. The functor

   \[ D_{X} \colon \mathcal{P}(X)^{\mathsf{op}}\to \mathcal{P}(X) \]

   defined by

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

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

   1. (a)

      The diagram

      
      commutes, i.e. we have
      \[ \underbrace{D_{X}(D_{X}(U))}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[[U,\text{Ø}]_{X},\text{Ø}]_{X}}=U \]

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

   2. (b)

      The diagram

      
      commutes, i.e. we have
      \[ \underbrace{D_{X}(U\cap D_{X}(V))}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[U\cap [V,\text{Ø}]_{X},\text{Ø}]_{X}}=[U,V]_{X} \]

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

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

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

   2. (b)

      Interaction With Inverse Images. The diagram

      
      commutes, i.e. we have
      \[ f^{-1}(D_{Y}(U))=D_{X}(f^{-1}(U)) \]

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

   3. (c)

      Interaction With Codirect Images. The diagram

      
      commutes, i.e. we have
      \[ f_{*}(D_{X}(U))=D_{Y}(f_{!}(U)) \]

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

7. 7.

   Interaction With Unions of Families of Subsets I. The diagram

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

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

8. 8.

   Interaction With Unions of Families of Subsets II. The diagram

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

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

9. 9.

   Interaction With Unions of Families of Subsets III. The diagram

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

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

10. 10.

    Interaction With Intersections of Families of Subsets I. The diagram

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

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

11. 11.

    Interaction With Intersections of Families of Subsets II. The diagram

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

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

12. 12.

    Interaction With Intersections of Families of Subsets III. The diagram

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

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

13. 13.

    Interaction With Binary Unions. We have equalities of sets

    \begin{align*} [U\cap V,W]_{X} & = [U,W]_{X}\cup [V,W]_{X},\\ [U,V\cap W]_{X} & = [U,V]_{X}\cap [U,W]_{X} \end{align*}

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

14. 14.

    Interaction With Binary Intersections. We have equalities of sets

    \begin{align*} [U\cup V,W]_{X} & = [U,W]_{X}\cap [V,W]_{X},\\ [U,V\cup W]_{X} & = [U,V]_{X}\cup [U,W]_{X} \end{align*}

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

15. 15.

    Interaction With Differences. We have equalities of sets

    \begin{align*} [U\setminus V,W]_{X} & = [U,W]_{X}\cup [V^{\textsf{c}},W]_{X}\\ & = [U,W]_{X}\cup [U,V]_{X},\\ [U,V\setminus W]_{X} & = [U,V]_{X}\setminus (U\cap W)\end{align*}

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

16. 16.

    Interaction With Complements. We have equalities of sets

    \begin{align*} [U^{\textsf{c}},V]_{X} & = U\cup V,\\ [U,V^{\textsf{c}}]_{X} & = U\cap V,\\ [U,V]^{\textsf{c}}_{X} & = U\setminus V \end{align*}

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

17. 17.

    Interaction With Characteristic Functions. We have

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

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

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

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

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

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

20. 20.

    Interaction With Codirect Images. Let $f\colon X\to Y$ be a function. We have a natural transformation

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

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