• Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
    \begin{gather*} \begin{aligned} U\setminus - & \colon \webleft (\mathcal{P}\webleft (X\webright ),\supset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\setminus V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\setminus -_{2} \colon \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ), \end{gather*}

    where $-_{1}\setminus -_{2}$ is the functor where

    • Action on Objects. For each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, we have

      \[ \webleft [-_{1}\setminus -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\setminus V. \]

    • Action on Morphisms. For each pair of morphisms

      \begin{align*} \iota _{A} & \colon A\hookrightarrow B,\\ \iota _{U} & \colon U\hookrightarrow V \end{align*}

      of $\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, the image

      \[ \iota _{U}\setminus \iota _{V}\colon A\setminus V\hookrightarrow B\setminus U \]

      of $\webleft (\iota _{U},\iota _{V}\webright )$ by $\setminus $ is the inclusion

      \[ A\setminus V\subset B\setminus U \]

      i.e. where we have

      • If $A\subset B$ and $U\subset V$, then $A\setminus V\subset B\setminus U$.
    and where $U\setminus -$ and $-\setminus V$ are the partial functors of $-_{1}\setminus -_{2}$ at $U,V\in \mathcal{P}\webleft (X\webright )$.


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