The Clowder Project

Let $X$ be a set.

1. Co/Completeness. The (posetal) category (associated to) $\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )$ is complete and cocomplete:
   1. Products. The products in $\mathcal{P}\webleft (X\webright )$ are given by intersection of subsets.
   2. Coproducts. The coproducts in $\mathcal{P}\webleft (X\webright )$ are given by union of subsets.
   3. Co/Equalisers. Being a posetal category, $\mathcal{P}\webleft (X\webright )$ only has at most one morphisms between any two objects, so co/equalisers are trivial.
2. Cartesian Closedness. The category $\mathcal{P}\webleft (X\webright )$ is Cartesian closed with internal Hom
   \[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (-_{1},-_{2}\webright )\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]

   given by^[1]

   \[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus U\webright )\cup V \]

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