Tiny objects in a power set

Question

I'm constructing a toy example in order to get into Cauchy completeness for categories. Suppose to have a set $X$ (=a discrete preorder) and compute its powerset $\mathcal{P}(X)$, which is its $\{0<1\}-$category of $\{0<1\}-$presheaves. Now, tiny objects in this category should be those subsets $P\in \mathcal{P}(X)$ such that their covariant representable functor preserves colimits, i.e. for any indexing $A_i$, $P\subseteq\bigcup A_i$ iff $\exists i. P\subseteq A_i$. Singletons (=representables) do satisfy this condition, but conversely this condition seems to me to be satisfied precisely by singletons and the empty set. Obviously $\{0<1\}-$presheaves on the preorder of such tiny objects is not my original $\mathcal{P}(X)$, so I think I am missing something. Does the equivalence $Tiny([X^{op}, \{0<1\}])\simeq \overline{X}$ break in this specific base of enrichment? Or is there some way to prove that a tiny object in this sense is not empty?

Answer 1

The empty set doesn't satisfy this. $\varnothing \subseteq \bigcup A_i$ is always true because the empty set is contained in any set. But there might not exist any $i$ such that $\varnothing \subseteq A_i$, even though for all $i$, $\varnothing \subseteq A_i$. Can you see why?