Quantifiers as Adjoints in Generalized Logics

Question

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories.

In the continuous model theory of metric structures (see http://ptmat.fc.ul.pt/~alexus/papers/mtfms.pdf ) and more generally, the continuous model theory developed by Chang and Keisler (http://books.google.co.uk/books/about/Continuous_Model_Theory.html?id=uTGdPSI5rI4C&redir_esc=y), sets of truth values are (roughly) ordered, compact, Hausdorff spaces with two distinguished elements (0 and 1) that act like true and false. In this setting, the analogues of $\forall$ and $\exists$ are $\sup$ and $\inf$.

I have a two part question:

1) In the continuous model theory setting, how does one describe quantifiers in a similar categorical way?

2) Is there a natural analogue of quantification in the setting of truth values in a non-ordered, compact, (and Hausdorff if you really want it) topological space. I appreciate that if there is, it might be quite strange since I'm not requiring there be a ''true'' and ''false''

Answer 1

First, let's be clear about how we can view quantifiers as adjoint functors. Here's one way to look at it.

What is a predicate $P(x)$, with the variable $x$ coming from the sort $X$? Viewing $X$ as a set, we can view $P(x)$ as a subset of $X$. The (definable, if you like) subsets of $X$ come with a natural (partial order) category structure, where there is an arrow $P(x) \rightarrow Q(x)$ if and only if $P\subseteq Q$. Note that we can also view this category structure as the partial order of implications between predicates. Let's call this category $\text{Pred}(X)$.

Now there is a functor $D_y:\text{Pred}(X)\rightarrow \text{Pred}(X\times Y)$ corresponding to adding a dummy variable $y$ of type $Y$. So $D_y(P(x)) = \{(x,y)\in X\times Y\mid x\in P(x)\}$. Let's write this as $P(x,\dot{y})$, with the dot denoting that $y$ is a dummy variable. This is a functor, since if $P(x)\rightarrow Q(x)$, then $P(x,\dot{y})\rightarrow Q(x,\dot{y})$ (again, you can think of this as containment or implication).

There are also functors $\exists y:\text{Pred}(X\times Y)\rightarrow \text{Pred}(X)$ and $\forall y:\text{Pred}(X\times Y)\rightarrow \text{Pred}(X)$, which do what you expect: $\exists y\,P(x,y) = \{x\in X\mid \exists y\, (x,y)\in P(x,y)\}$, and $\forall y\,P(x,y) = \{x\in X\mid \forall y\, (x,y)\in P(x,y)\}$.

Now $\exists y\dashv D_y\dashv \forall y$. The adjunctions correspond to the following statements $$\exists y\, P(x,y) \rightarrow Q(x) \iff P(x,y)\rightarrow Q(x,\dot y)$$

$$P(x)\rightarrow \forall y\,Q(x,y) \iff P(x,\dot{y})\rightarrow Q(x,y)$$

Okay, but now instead of viewing a predicate $P(x)$ as a subset of $X$, we can turn the picture around and view it as a (definable, if you like) map $X\rightarrow \{0,1\}$. What's the corresponding partial order structure on maps $X\rightarrow \{0,1\}$? Well, $P(x) \rightarrow Q(x)$ if and only if for all $x\in X$, $P(x) \leq Q(x)$. That is, the partial order is pointwise comparison of functions.

Now the functor $D_y$ just takes a map $X\rightarrow \{0,1\}$ and composes with the projection $X\times Y \rightarrow X$, and its adjoints are given by $$(\exists y\, P(x,y))(x) = \begin{cases} 1\text{ if }\exists y\,P(x,y) = 1\\ 0\text{ otherwise}\end{cases} = \text{max}_{y\in Y}P(x,y)$$

$$(\forall y\, P(x,y))(x) = \begin{cases} 1\text{ if }\forall y\,P(x,y) = 1\\ 0\text{ otherwise}\end{cases} = \text{min}_{y\in Y}P(x,y)$$

But now it's easy to guess (and to check) that you can replace $\{0,1\}$ with $[0,1]$, take the same category structure on predicates (pointwise $\leq$ in $[0,1]$), and get the continuous model theory quantifiers as adjoints to the "dummy variable functor", replacing max and min by sup and inf.

It may seem like I've gotten inf and sup mixed up, but that's just because in continuous model theory $0$ is true and $1$ is false, contrary to the convention when viewing a subset of $X$ as a map $X\rightarrow \{0,1\}$.

What about your second question? If the compact Hausdorff space isn't ordered, it's not clear to me what the right category structure should be, since the one in which the quantifiers appear naturally comes directly from the order. Of course, one can more easily generalize this situation replacing $\{0,1\}$ and $[0,1]$ with any complete lattice - this is what happens in topos theory.