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Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function.

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).

Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that "collaps the cardinal $\kappa$ to $\omega$" I mean by that, if $p:B_\kappa \to 1$ denotes the unique map, then $p^*\kappa \simeq p^* \omega$ as sheaves over $B_\kappa$ . For example, $B_\kappa$ can be taken to be the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in this locale of injective function. (See details in edit below)

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself. THe following are very loose bound that shows it, though experts on forcing surely known much better bounds:

If $X$ is a non-degenerate locales, then the total number of locale section of $p^* \kappa$ is larger than $\kappa$ as every element of $\kappa$ gives a globale section, and it is smaller than the function space $\kappa^{\mathcal{O}(X) \times 2^{\mathcal{O}(X)}}$ as every locale section can be written as: you chose a cover of its domain of definition (a special subset of $\mathcal{O}(X)$) and then for each element of that subset, you chose an element of $\kappa$.

So for any locale $X$, picking a $\kappa$ to be larger than $\omega^{\mathcal{O}(X) \times 2^{\mathcal{O}(X)}}$ we get that $p^* \omega$ can't have as much section as $p^* \kappa$ and hence they can't be isomorphic, so $X$ can't collaps any cardinal bigger than this to $\omega$.

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps).

The problem is that there exists a family of non-trivial Boolean locales $B_\kappa$ indexed by infinite cardinal numbers $\kappa$, such that the only locale $X$ that has map to all the $B_\kappa$ is the empty locale

The map $B_\kappa \to 1$ are open surjections, so in particular they are effective descent map, stable regular epimorphism and triquotient map, so pretty much any class of epimorphism you might think about will contains these.

But if $X$ was projective with respect to any class containing these cover, then the unique map $X \to 1$ would lift to maps $X \to B_\kappa$ for all $\kappa$ which contradict our claim above.

Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function.

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps).

The problem is that there exists a family of non-trivial Boolean locales $B_\kappa$ indexed by infinite cardinal numbers $\kappa$, such that the only locale $X$ that has map to all the $B_\kappa$ is the empty locale

The map $B_\kappa \to 1$ are open surjections, so in particular they are effective descent map, stable regular epimorphism and triquotient map, so pretty much any class of epimorphism you might think about will contains these.

But if $X$ was projective with respect to any class containing these cover, then the unique map $X \to 1$ would lift to maps $X \to B_\kappa$ for all $\kappa$ which contradict our claim above.

Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function.

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).

Here is some clarification on the construction of locales $B_\kappa$. This is a fairly standard observation, but I'm strugling to find a reference, so given that it is fairly simple, I'll write the details.

We fixe $\kappa$ some infinite cardinal number.

We start $I_\kappa$ the locale that classifies injection $i:\omega \to \kappa$, so that a map $X \to I_\kappa$ is the same as the data of an injective map $p^* \omega \to p^* \kappa$.

It is easy to write a propositional geometric theory of such injections (it has base proposition $R_{x,y}$ for $x \in \omega$ and $y \in \kappa$ which is interpreted as $i(x)=y$ and all the axioms that make this into an "injective" functional relation)

$I_\kappa$ is non-trivial because it has plenty of points.

Now, consider the (open) sublocale $V_y \subset I_\kappa$ for $y \in \kappa$ that clasifies these injection that further satisfies $\exists x \in \omega, i(x)=y$.

$V_y$ is dense: indeed, the basic open of $I_\kappa$ are the finite intersection of $R_{x,z}$ and for any finite intersection $\cap R_{x_i,z_i}$ of these, if it is non-degenerate you can explicitely construct a point of it that is also in $V_y$ (take a function that send $x_i$ to $z_i$ and some other value to $y$, if its impossible it means the intersection is empty).

Now the intersection of all $V_y$ is hence a dense sublocale. (an intersection of a familly of dense sublocale is dense).

By definition this intersection classifies bijection from $\omega \to \kappa$. So this is exactly the $T_\kappa$ I mentioned in the comment.

Alternatively, you can define $B_\kappa$ to be the double negation sublocale of $I_\kappa$, which is hence included in all the $V_y$, so that $B_\kappa \subset T_\kappa$ also "collaps the cardinal $\kappa$ to $\omega$).

Both $B_\kappa$ and $T_\kappa$ are non-trivial because they are dense in $I_\kappa$ which is non-trivial.

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps, where the $\beta S$ for $S$ a set are I think the projective).

The problem is that there exists a family of non-trivial Boolean locales $B_\kappa$ indexed by infinite cardinal numbers $\kappa$, such that the only locale $X$ that has map to all the $B_\kappa$ is the empty locale

The map $B_\kappa \to 1$ are open surjections, so in particular they are effective descent map, stable regular epimorphism and triquotient map, so pretty much any class of epimorphism you might think about will contains these.

But if $X$ was projective with respect to any class containing these cover, then the unique map $X \to 1$ would lift to maps $X \to B_\kappa$ for all $\kappa$ which contradict our claim above.

Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function.

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps).

The problem is that there exists a family of non-trivial Boolean locales $B_\kappa$ indexed by infinite cardinal numbers $\kappa$, such that the only locale $X$ that has map to all the $B_\kappa$ is the empty locale

The map $B_\kappa \to 1$ are open surjections, so in particular they are effective descent map, stable regular epimorphism and triquotient map, so pretty much any class of epimorphism you might think about will contains these.

But if $X$ was projective with respect to any class containing these cover, then the unique map $X \to 1$ would lift to maps $X \to B_\kappa$ for all $\kappa$ which contradict our claim above.

Explicitely, $B_\kappa$ can be taken to be any (non-trivial) Boolean locale that collaps the cardinal $\kappa$ to $\omega$. For example, $B_\kappa$ is the double negation sublocale of the locale of injective functions $\omega \to \kappa$. The fact that $B_\kappa$ is non trivial then follows from the fact that it is dense in the locale of injective function.

And a locale $X$ having functions to all the $B_\kappa$ would collapse all infinite cardinals at the same time (in the sense that all $p^*\kappa$ for infinite cardinal $\kappa$ would be isomorphic), which is impossible as a locale can't collapse to $\omega$ cardinal much larger than itself (I don't have the precise obstruction/bound in mind for this, if some one know them, feel free to edit or comment with some details).