An extension of Stone duality

Question

First let me recall Stone duality in terms of propositional logic.

Let $L$ and $K$ be propositional signatures (i.e., sets of propositional variables). Let $T$ be a propositional theory over $L$ and $S$ a propositional theory over $K$. An interpretation of $T$ in $S$ is a map $I\colon L\to \{\text{$K$-sentences}\}$ (which extends to a map $\{\text{$L$-sentences}\}\to \{\text{$K$-sentences}\}$, which we also denote by $I$) such that $S\models I(\phi)$ for all $\phi\in T$. Two interpretations $I$ and $I'$ of $T$ in $S$ are homotopic if $S\models I(p)\leftrightarrow I'(p)$ for all $p\in L$. Let me denote the category of all propositional theories (over all propositional signatures) and homotopy classes of interpretations by $\mathrm{PropTh}$.

There is a contravariant functor $\mathrm{Mod}\colon \mathrm{PropTh}\to \mathrm{Top}$, which maps a propositional theory $T$ over a propositional signature $L$ to the set $\mathrm{Mod}(T)$ of all models $M\colon L\to \{0,1\}$ of $T$ equipped with the topology generated by the sets of the form $$\{M\colon L\to \{0,1\}\mid M\models T\cup \{\phi\}\}$$ for each $L$-sentence $\phi$. Stone's duality theorem states two things:

• This functor is fully faithful, i.e., for all propositional theories $T$ and $S$, the canonical map from the set of all homotopy classes of interpretations of $T$ in $S$ to the set of all continuous maps $\mathrm{Mod}(S)\to \mathrm{Mod}(T)$ is bijective.

• A topological space lies in the essential image of this functor if and only if it is compact, totally disconnected, and Hausdorff.

I wonder what happens if we replace the category $\mathrm{PropTh}$ by the larger category $\mathrm{PropClass}$ of propositional classes:

A propositional class is a pair $(L, W)$, where $L$ is a propositional signature and $W\subseteq \{0,1\}^L$ is a set of $L$-structures. An interpretation between propositional classes $(L, W)$ and $(K, V)$ is a map $I\colon L\to \{\text{$K$-sentences}\}$ such that for each $K$-structure $M\colon K\to \{0,1\}$ in $V$, the induced $L$-structure $IM:=M \circ I\colon L\to \{0,1\}$ is in $W$. Two interpretations $I$ and $I'$ between $(L, W)$ and $(K, V)$ are homotopic if for any $K$-structure $M$ in $V$ and all $p\in L$, $IM\models p$ if and only if $I'M\models p$ (i.e., $IM=I'M$). Denote the category of all propositional classes and homotopy classes of interpretations by $\mathrm{PropClass}$.

Note that there is a fully faithful functor from $\mathrm{PropTh}$ to $\mathrm{PropClass}$, sending a propositional theory $T$ over $L$ to the propositional class $(L,\{\text{models of $T$}\})$. In this sense, propositional classes generalize propositional theories.

There is a canonical contravariant functor from $\mathrm{PropClass}$ to $\mathrm{Top}$, sending a propositional class $(L, W)$ to the set $W$ equipped with the topology generated by the sets of the form $$\{M\in W\mid M\models \phi\}$$ for each $L$-sentence $\phi$.

Questions:

1. Is that functor fully faithful? If not, is there a way to make it fully faithful, maybe by replacing the category $\mathrm{Top}$?

2. Can one describe the essential image of that functor?

Answer 1

Let $(L,W)$ be a propositional class, so $W\subseteq 2^L$. The topology you assign to $W$ is exactly the subspace topology inherited from $2^L$, where $2$ gets the discrete topology and $2^L$ gets the product topology. So the essential image of your functor is just the subcategory of $\mathsf{Top}$ consisting of all spaces which are subspaces of Cantor cubes. These are exactly the Hausdorff zero-dimensional spaces (where zero-dimensional means that there is a basis of clopen sets).

I believe the functor $(L,W)\mapsto W$ is faithful, but it is not full. For a counterexample, let $L = \{p\}$, and let $K = \{q_n\mid n\in \omega\}$. Let $W$ be the set containing both $L$-structures, so as a topological space it is a discrete space with $2$ points. Let $V$ be the set of $K$-structures in which exactly one variable $q_n$ is true. As a topological space it is a countably infinite discrete space. Then there are continuum-many continuous maps $V\to W$, but only countably many interpretations of $W$ in $V$ (one for each $K$-sentence).

I think the natural extension of Stone duality here should be a contravariant equivalence between $\mathsf{PropClass}$ and the category $\mathsf{Stone}_D$ of Stone spaces (compact, Hausdorff, zero-dimensional) with a distinguished dense set. An arrow $(X,D)\to (Y,E)$ is a continuous map $f\colon X\to Y$ such that $f(D)\subseteq E$. The functor maps $(L,W)$ to $(\mathrm{Mod}(\mathrm{Th}(W)),W)$, where $\mathrm{Th}(W)$ is the set of all propositional sentences true in every structure in $W$.

The idea is simple: An interpretation of $(L,W)$ in $(K,V)$ is exactly the same data as an interpretation of $\mathrm{Th}(W)$ in $\mathrm{Th}(V)$, with the additional requirement that $I(V)\subseteq W$. By ordinary Stone duality, it corresponds to a continuous map $\mathrm{Mod}(\mathrm{Th}(V))\to \mathrm{Mod}(\mathrm{Th}(W))$. And $V$ is dense in $\mathrm{Mod}(\mathrm{Th}(V))$ (indeed, the latter is the closure of $V$ in $2^L$) and similarly for $W$. To see that the essential image is exactly $\mathsf{Stone}_D$, note that every Stone space $X$ is homeomorphic to $\mathrm{Mod}(T)$ for some $L$-theory $T$. The homeomorphism maps a dense set $D$ in $X$ to a set $W$ of models of $T$, and $T = \mathrm{Th}(W)$ by density.