The Clowder Project

Let $X$ be a set.

1. 1.

   Functoriality. The assignment $U\mapsto U^{\textsf{c}}$ defines a functor

   \[ (-)^{\textsf{c}}\colon \mathcal{P}(X)^{\mathsf{op}}\to \mathcal{P}(X). \]

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

   • (★)
   If $U\subset V$, then $V^{\textsf{c}}\subset U^{\textsf{c}}$.
2. 2.

   De Morgan’s Laws. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} (U\cup V)^{\textsf{c}} & = U^{\textsf{c}}\cap V^{\textsf{c}},\\ (U\cap V)^{\textsf{c}} & = U^{\textsf{c}}\cup V^{\textsf{c}} \end{align*}

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

3. 3.

   Involutority. The diagram

   
   commutes, i.e. we have
   \[ (U^{\textsf{c}})^{\textsf{c}}=U \]

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

4. 4.

   Interaction With Characteristic Functions. We have

   \[ \chi _{U^{\textsf{c}}}=1-\chi _{U} \]

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

5. 5.

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

   
   commutes, i.e. we have
   \[ f_{!}(U^{\textsf{c}})=f_{*}(U)^{\textsf{c}} \]

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

6. 6.

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

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

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

7. 7.

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

   
   commutes, i.e. we have
   \[ f_{*}(U^{\textsf{c}})=f_{!}(U)^{\textsf{c}} \]

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