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Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects
I asked for the possibility of constructing an isomorphism via the order-preserving bijection between the corresponding posets of subobjects. In the case of the category of sets just work with the singletons to define the bijection - I'm working with classical NGB set theory -
Now, an interesting counterexample has been exhibited in the aforementioned post. However, I've another question. Well, which are the categories sharing the same property as set (i.e. to give rise to an isomorphism between the corresponding posets of subobjects). I was thinking to toposes. Might it be a good way or are there also counterexamples coming from toposes?
Actualy, there exists a paper by Barr and Diaconescu entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf). Well, I think that what I'm searching for is related to the fact that the subobject lattice is atomistic, i.e. every element may be expressed as the join of its atoms. It is property related to my problem?
Topos in itself does not suffice.
Consider the presheaf topos on $\{0,1\}$ (as discrete category) and the two presheaves $(\emptyset,\ast)$ and $(\ast,\emptyset)$ (a presheaf here just amounts to choosing two sets). You can manually list all subobjects of both and the subobject lattices are isomorphic. However the two presheaves are not isomorphic, since an isomorphism of presheaves is componentwise.