The Clowder Project

Let $X$ be a set.

1. 1.

   Functoriality. The assignment $\mathcal{U}\mapsto \bigcup _{U\in \mathcal{U}}U$ defines a functor

   \[ \bigcup \colon (\mathcal{P}(\mathcal{P}(X)),\subset )\to (\mathcal{P}(X),\subset ). \]

   In particular, for each $\mathcal{U},\mathcal{V}\in \mathcal{P}(\mathcal{P}(X))$, the following condition is satisfied:

   • (★)
   If $\mathcal{U}\subset \mathcal{V}$, then $\displaystyle \bigcup _{U\in \mathcal{U}}U\subset \bigcup _{V\in \mathcal{V}}V$.
2. 2.

   Associativity. The diagram

   
   commutes, i.e. we have
   \[ \bigcup _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcup _{A\in \mathcal{A}}(\bigcup _{U\in A}U) \]

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

3. 3.

   Left Unitality. The diagram

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

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

4. 4.

   Right Unitality. The diagram

   
   commutes, i.e. we have
   \[ \bigcup _{\left\{ u\right\} \in \chi _{X}(U)}\left\{ u\right\} =U \]

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

5. 5.

   Interaction With Unions I. The diagram

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

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

6. 6.

   Interaction With Unions II. The diagrams

   
   commute, i.e. we have
   \begin{align*} U\cup \left(\bigcup _{V\in \mathcal{V}}V\right) & = \bigcup _{V\in \mathcal{V}}(U\cup V),\\ \left(\bigcup _{U\in \mathcal{U}}U\right)\cup V & = \bigcup _{U\in \mathcal{U}}(U\cup V)\end{align*}

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

7. 7.

   Interaction With Intersections I. We have a natural transformation

   
   with components
   \[ \bigcup _{W\in \mathcal{U}\cap \mathcal{V}}W\subset \left(\bigcup _{U\in \mathcal{U}}U\right)\cap \left(\bigcup _{V\in \mathcal{V}}V\right) \]

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

8. 8.

   Interaction With Intersections II. The diagrams

   
   commute, i.e. we have
   \begin{align*} U\cap \left(\bigcup _{V\in \mathcal{V}}V\right) & = \bigcup _{V\in \mathcal{V}}(U\cap V),\\ \left(\bigcup _{U\in \mathcal{U}}U\right)\cap V & = \bigcup _{U\in \mathcal{U}}(U\cap V)\end{align*}

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

9. 9.

   Interaction With Differences. The diagram

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

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

10. 10.

    Interaction With Complements I. The diagram

    
    does not commute in general, i.e. we may have
    \[ \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U\neq \bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]

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

11. 11.

    Interaction With Complements II. The diagram

    
    commutes, i.e. we have
    \[ \left(\bigcup _{U\in \mathcal{U}}U\right)^{\textsf{c}}=\bigcap _{U\in \mathcal{U}}U^{\textsf{c}} \]

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

12. 12.

    Interaction With Complements III. The diagram

    
    commutes, i.e. we have
    \[ \left(\bigcap _{U\in \mathcal{U}}U\right)^{\textsf{c}}=\bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]

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

13. 13.

    Interaction With Symmetric Differences. The diagram

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

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

14. 14.

    Interaction With Internal Homs 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 $\mathcal{U}\in \mathcal{P}(\mathcal{P}(X))$.

15. 15.

    Interaction With Internal Homs 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 $V\in \mathcal{P}(X)$.

16. 16.

    Interaction With Internal Homs 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 $\mathcal{V}\in \mathcal{P}(\mathcal{P}(X))$.

17. 17.

    Interaction With Direct Images. Let $f\colon X\to Y$ be a map of sets. The diagram

    
    commutes, i.e. we have
    \[ \bigcup _{U\in \mathcal{U}}f_{!}(U)=\bigcup _{V\in f_{!}(\mathcal{U})}V \]

    for each $\mathcal{U}\in \mathcal{P}(X)$, where $f_{!}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f_{!})_{!}(\mathcal{U})$.

18. 18.

    Interaction With Inverse Images. Let $f\colon X\to Y$ be a map of sets. The diagram

    
    commutes, i.e. we have
    \[ \bigcup _{V\in \mathcal{V}}f^{-1}(V)=\bigcup _{U\in f^{-1}(\mathcal{U})}U \]

    for each $\mathcal{V}\in \mathcal{P}(Y)$, where $f^{-1}(\mathcal{V})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f^{-1})^{-1}(\mathcal{V})$.

19. 19.

    Interaction With Codirect Images. Let $f\colon X\to Y$ be a map of sets. The diagram

    
    commutes, i.e. we have
    \[ \bigcup _{U\in \mathcal{U}}f_{*}(U)=\bigcup _{V\in f_{*}(\mathcal{U})}V \]

    for each $\mathcal{U}\in \mathcal{P}(X)$, where $f_{*}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f_{*})_{*}(\mathcal{U})$.

20. 20.

    Interaction With Intersections of Families I. The diagram

    
    commutes, i.e. we have
    \[ \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcap _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right) \]

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

21. 21.

    Interaction With Intersections of Families II. Let $X$ be a set and consider the compositions

    
    given by
    \[ \begin{aligned} \mathcal{A} & \mapsto \bigcup _{U\in {\scriptsize \displaystyle \bigcap _{A\in \mathcal{A}}}A}U,\\ \mathcal{A} & \mapsto \bigcup _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right), \end{aligned} \quad \begin{aligned} \mathcal{A} & \mapsto \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U,\\ \mathcal{A} & \mapsto \bigcap _{A\in \mathcal{A}}\left(\bigcup _{U\in A}U\right) \end{aligned} \]

    for each $\mathcal{A}\in \mathcal{P}(\mathcal{P}(\mathcal{P}(X)))$. We have the following inclusions:

    
    All other possible inclusions fail to hold in general.