# Union-Closed Families form a subcategory of a functor-structured category: can we describe it through categorical closure operators?

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?