subobject classifier for partial orders Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.)
If not, what is a category which is "close" to the category of partial orders (e.g. it should consists of special order-theoretic constructs) and has a subobject classifier? Bonus question: Is there also such an elementary topos? Notice that the category of partial orders has all limits, colimits and it is cartesian closed.

The fact that all internal co-categories in a coherent category are necessarily co-equivalence relations [see Peter Lumsdaine's TAC article A small observation on co-categories] provides a telltale sign that the category of posets fails to be a topos. 
For the inclusion functor $\textbf{Poset} \to \textbf{Cat}$ is represented by the internal co-category whose (co-?)nerve is the inclusion of the non-empty finite ordinals $\Delta \to \textbf{Poset}$, which is evidently not a co-equivalence relation (since not all posets are equivalence relations).
