By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it $\mathbf{GPos}$. Okay, now let $J$ denote the grounded poset with two elements. Explicitly: $$J = \{1,0\}, \qquad 1_J > 0_J$$
Then $J$ satisfies the following "universal" property: for all grounded posets $X$, there is a maximum arrow $X \rightarrow J$. In particular, this is the mapping that takes every element of $X$ to $1_J$, except for $0_X$ which is mapped to $0_J$.
Unfortunately, this "universal" property doesn't characterize $J$, not even close, because every grounded poset with a maximum element satisfies this same property! So what I'm looking for is a strengthening of the aforementioned "universal" property. It should:
- characterize $J$ as an object of $\mathbf{GPos}.$
- make sense in any $\mathbf{Pos}$-enriched category, or at least in any $\mathbf{GPos}$-enriched category.
- determine objects up to isomorphism; meaning that if $\mathbf{C}$ is a $\mathbf{GPos}$-enriched category, then it should be provable that any two objects of $\mathbf{C}$ satisfying this property are isomorphic.
Given a $\mathbf{Pos}$-enriched category $\mathcal{C}$ and an object $Y$ in $\mathcal{C}$, a downward-closed subobject of $Y$ is a monomorphism $m : X \to Y$ in $\mathcal{C}$ such that, for every object $T$ in $\mathcal{C}$, the induced map $$\mathcal{C} (T, m) : \mathcal{C} (T, X) \to \mathcal{C} (T, Y)$$ is the embedding of a downward-closed subset.
In the case where $\mathcal{C} = \mathbf{Pos}$ or $\mathcal{C} = \mathbf{GPos}$, this coincides with the usual notion, and it is easy to see that $J$ classifies downward-closed subobjects in those cases. This is a universal property that characterises $J$ up to isomorphism.