The Clowder Project

Let $X$ be a set.

1. 1.

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

   \[ \bigcap \colon \webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright ),\supset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \]

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

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

   Oplax Associativity. We have a natural transformation

   
   with components
   \[ \bigcap _{A\in \mathcal{A}}\left(\bigcap _{U\in A}U\right)\subset \bigcap _{U\in \bigcap _{A\in \mathcal{A}}A}U \]

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

3. 3.

   Left Unitality. The diagram

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

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

4. 4.

   Oplax Right Unitality. The diagram

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

   in general, where $U\in \mathcal{P}\webleft (X\webright )$. However, when $U$ is nonempty, we have

   \[ \bigcap _{\left\{ x\right\} \in \chi _{X}\webleft (U\webright )}\left\{ x\right\} \subset U. \]
5. 5.

   Interaction With Unions I. The diagram

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

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

6. 6.

   Interaction With Unions II. The diagram

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

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

7. 7.

   Interaction With Intersections I. We have a natural transformation

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

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

8. 8.

   Interaction With Intersections II. The diagrams

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

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

9. 9.

   Interaction With Differences. The diagram

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

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

10. 10.

    Interaction With Complements I. The diagram

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

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

11. 11.

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

12. 12.

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

13. 13.

    Interaction With Symmetric Differences. The diagram

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

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

14. 14.

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

15. 15.

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

16. 16.

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

17. 17.

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

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

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

18. 18.

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

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

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

19. 19.

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

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

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

20. 20.

    Interaction With Unions 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 $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

21. 21.

    Interaction With Unions 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 $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\webright )$. We have the following inclusions:

    
    All other possible inclusions fail to hold in general.