Basically Stone duality or more general the duality between spatial locales and sober spaces is about enriching the set of morphisms $X \to \{0,1\}$ with an additional structure and then finding appropriate conditions for a map $\mathrm{Hom}(X,\{0,1\}) \to \mathrm{Hom}(Y,\{0,1\})$ to be the pullback of a unique morphism $Y \to X$. For example in the discrete case, we get the following basic set-theoretic observation:
Let $X,Y$ be sets. A map $\mathrm{Hom}(X,\{0,1\}) \to \mathrm{Hom}(Y,\{0,1\})$ is the pullback of a morphism iff it preserves arbitrary suprema and finite infima if we endow $\mathrm{Hom}(X,\{0,1\}) \cong \mathcal{P}(X)$ with the partial order given by inclusion.
Question. Is there a similar statement for the functor $\mathrm{Hom}(-,\{0,...,n\})$, where $n \geq 1$? That is can we endow these sets with additional structure such that we can find a criterion when a map $\mathrm{Hom}(X,\{0,...,n\}) \to \mathrm{Hom}(Y,\{0,...,n\})$ is the pullback of a map $Y \to X$?
First observe that there is a natural isomorphism between this functor and $\mathcal{P}^{[n]}$, which is defined by $\mathcal{P}^{[n]}(X) := \{(A_1,...,A_n) : A_1 \subseteq ... \subseteq A_n \subseteq X\}$ on objects and by preimages on morphisms. It is a partial order (inclusion in each entry) with suprema and infima. But it also has various operations (endofunctors) coming from automorphisms of $\{0,...,n\}$. For example, the permutation $i \mapsto n - i$ yields the operation $K : (A_1,....,A_n) \mapsto (A_n^c,...,A_1^c)$ and the cycle $(0 ~ 1)$ yields the operation $H : (A_1,...,A_n) \to (A_2 \setminus A_1,A_2,...,A_n)$.
For $n = 2$ I can show that a map $\mathcal{P}^{[n]}(X) \to \mathcal{P}^{[n]}(Y)$ is a pullback iff it preserves suprema, finite infima and commutes with $H$ and $K$ (for $n=1$ this is also true, we don't even need $H$ or $K$). Remark that $H^2=K^2=(HK)^3=1$ and that $K$ transforms suprema into infima and vice versa. Is this a known structure? If not, it is possible to endow $\mathrm{Hom}(X,3)$ with a more well-known structure?
For $n>2$ I don't think that $H$ and $K$ suffice. But I guess that the whole set of endofunctors of $\mathcal{P}^{[n]}$ suffices.
