[Added: This is a follow-up of an earlier post.]
Consider the following "reconstruction puzzle", stated informally:
Given a concrete poset, e.g. the poset of undirected unlabeled finite graphs without isolated vertices, ordered by embeddability (arrow heads, identities and composition omitted in the diagram):
Now forget about the inner structure of the objects and consider only the corresponding abstract poset:
The "reconstruction puzzle" is to reconstruct the inner structure of the objects unambiguously from their "positions" in the poset.
It's obvious what a solution of this puzzle is and that it can be solved "by hand" for at least some of the smaller objects, just carefully considering the in- and out-arrows.
Question #1: How can such a puzzle be stated formally?
Question #2: How can it be solved "algorithmically"?
Question #3: What's the mathematics behind this kind of puzzle?
I suppose it's not category theory, since category theory is not concerned with the inner structure of objects.
Question #4: Can it be shown - at least for this concrete example - that every object is reconstructible up to isomorphism?
Since for this concrete example there are no isomorphic objects we could have omitted "up to isomorphism".
Question #5: What's an obvious way to generalize this kind of puzzles (from posets to what?)