Let us consider the functor-structured category $S(\mathcal{P})$ induced by the powerset functor. As it is well-known it is the category having the pairs $(X,\mathcal{F})$ (with $\mathcal{F}$ any set system on $X$) as objects and with structure-preserving maps as arrows.

One full subcategory of $S(\mathcal{P})$ is given for example by the category of union-closed families. Observe now that the objects $(X,\mathcal{P}(X)$, for any set $X$, are always union-closed, and arbitrary intersections of union-closed families on the same ground set is again a union-closed family on such a ground set.

Therefore, given $(X,\mathcal{F}) \in Obj(S(\mathcal{P}))$, we can always find the smallest union-closed family on $X$ generated by $\mathcal{F}$. In other terms, we're defining a closure operator on the lattice $\mathcal{P}(X)$.

Recent researches on categorical closure operators define a closure operator on a concrete category $(\mathcal{C},\mathcal{X})$, with $\mathcal{X}$ finitely complete and with suitable $(E,M)$-factorizations ($E \subseteq Epi(\mathcal{C})$ and $M \subseteq Mono(\mathcal{C})$). Given an $\mathcal{X}$-object $C$, we will always mean a subobject of $U(C)$ (here $U: \mathcal{C} \longrightarrow \mathcal{X}$ stands for the forgetful functor). Recall that if $X \in Obj(\mathcal{X})$, each $M$-morphism with codomain $X$ is called a $M$-subobject of $X$. A closure operator on $\mathcal{C}$ is a family of maps $c=(c_C: $M$-SubC \longrightarrow $M$-SubC)_{C \in \mathcal{C}}$ satisfying some suitable conditions (it is a "pre-closure" operator, in general idempotence it is not satisfied).

Well, my question is: it is possible to use categorical closure operators to describe the category of union-closed families?

Similarly, if I consider the functor-structured category $S(Q_2)$ (with $Q_2(\Omega):=\Omega \times \Omega$) and the category of preorders, it is possible to do the same thing as above?


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