In any category $\mathcal{C}$ with pullbacks, we can define the subobject fibration $cod:{\sf Sub}(\mathcal{C})\to\mathcal{C}$ and view the fibers ${\sf Sub}(\mathcal{C})_X$ as power-categories of $X$ — they are poset categories consisting of one member from each equivalence class of monomorphisms into $X$, with the containment relation given by arrows between monos in a slice. Taking $\mathcal{C}={\bf Set}$ yields ${\sf Sub}({\bf Set})_X\cong\mathcal{P}(X)$ as posets, justifying the terminology and suggesting we look for interpretations of various operations on subsets — for example, intersections correspond to pullbacks.
What are unions in the subobject lattice of an object in a category with pullbacks?
I would think that the union of subobjects $u:U\to X$, $v:V\to X$ would be a coequalizer of a coproduct by a pullback $$U\cup V=(U\coprod V)/(U\times_X V)$$ as suggested here, but it isn't clear to me how to carry out a construction like this if the ambient category only has pullbacks a-priori. If the fiber categories are posets with supremums then we can trivially identify supremums with coequalizers of coproducts and unions, but it isn't clear to me that the subobject lattices have supremums without additional ambient structure on the category. If we're working in a topos as suggested here the extra structure makes this construction easy to carry out; if extra structure on the category is required, do we need the full expressive power of a topos? Any assistance is appreciated.
Best Answer
Let me just quote the first sentence from here:
This is precisely the structure you want. Regularity and pullback-stability of the finite unions is not strictly necessary, but it's what makes these things logically reasonable: in particular, the subobject posets become distributive lattices for which pullback functors are lattice homomorphisms. Since images are nice in regular categories you can also "construct" the unions as the image of the coproduct of two subobjects, assuming those coproducts are there at all, in which case the category is extensive.
Distributive lattices are also, themselves, coherent categories so it makes sense to think about coherent categories as categorified distributive lattices in roughly the sense that toposes are categorified Heyting algebras. The nLab has some general examples of coherent categories; non-additive nice categories tend to be extensive even if they're not very close to toposes, such as categories of spaces or of categories.