# Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions.

Is there a process of 'overtification' corresponding to the process of compactification? Since all topological spaces are overt, this would have to be defined over locales instead.

Edit: I've very new to this topic, so I am not well versed. Most of my understanding comes from this Math Overflow answer by Andrej Bauer. I'll mostly reproduce that here, probably more for my benefit than anyone else's.

To define overtness, first we must suitable define compactness in a way that can be nicely dualized.

If $X$ is a space, let $\mathcal{O}(X)$ be its topology (as a poset) equipped with the Scott topology. Let $\Sigma = \{0,1\}$ be the Sierpinski space, with topology $\{\emptyset, \{1\}, \{0,1\}\}$, which also happens to be the Scott topology on the poset $0 \leq 1$. The Sierpinski space acts as an 'open-set' classifier, meaning that for any open subspace $A \subseteq X$, $\chi^{-1}_A(1) = A$, where $\chi_A$ is the characteristic function $$\chi_A(x) = \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{if }x\not\in A \end{cases}$$

Anyway, let's look at the function $\forall_X: \mathcal{O}(X) \rightarrow \Sigma$ defined as $$\forall_X(U) = \begin{cases} 1 & \text{if } U = X \\ 0 & \text{if } U \neq X \end{cases}$$

This map is the universal quantifier over $X$ (hence the $\forall$ symbol), because its value on a subset $U$ is the truth value of the statement $\forall x \in X .(x \in U)$. It is not too hard to see that this is the right adjoint to the frame map $\mathcal{O}(1) \rightarrow \mathcal{O}(X)$, which is the map induced by the unique map $X \rightarrow 1$. (Note that $\mathcal{O}(1) = \Sigma$)

We then have:

$\textbf{Theorem. }$ A space $X$ is compact if and only if $\forall_X$ is continuous.(This is not obvious to me, but I haven't spent too much time thinking about it)

Anyway, now we are ready to dualize. We just have to look for a $\textit{left}$ adjoint to $\Sigma \rightarrow \mathcal{O}(X)$, which we find in $$\exists_X(U) = \begin{cases} 0 & \text{if } U = \emptyset \\ 1 & \text{else} \end{cases}$$

The notation is again inspired by the fact that on a subset $U$ the value of $\exists_X$ is the truth value of the statement $\exists x \in X.(x \in U)$.

Now we define:

$\textbf{Definition.}$ $X$ is overt if and only if $\exists_X$ is continuous.

'Overtification' would then be a nice or natural way to take a frame/locale and 'make it overt' in the way that compactification can take a space and 'make it compact'.

If anyone has an further reading suggestions in this area I would gladly welcome them! Thanks.

• Have you tried to construct an adjoint functor for the forgetful functor that forgets overt locales and treat them as locales? – user40276 Jun 24 '14 at 2:37
• Please define "overtness" and "overtification" so that more people can understand + answer the question. – Martin Brandenburg Jun 24 '14 at 12:20
• I join @MartinBrandenburg in asking more overtness in this definition :-) – magma Jun 24 '14 at 13:22
• David: any duality principle relies on the logical structure of the axioms involved. So it relies on the quantifiers and logical connectives forming the axioms. This is no exception. – magma Jun 24 '14 at 13:24
• @MartinBrandenburg Hi, sorry I've been travelling I will do that right now. – David Myers Jun 25 '14 at 2:04