TITLE: What are projective locales / injective frames?
QUESTION [9 upvotes]: Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ and $\neg(x\land y)\to(\neg x\lor\neg y)$. On the other hand, existence of locales without points shows that the terminal locale is not regular projective. I am convinced somebody should already have found out which locales are regular projective, but who?
The same question about projectivity with respect to arbitrary epimorphisms of locales is probably easier but less interesting. Still, I don't know anything about that either.
PS As Simon Henry pointed out, I should rather pick some pullback stable class of locale epimorphisms. I guess I don't know an answer about any of them (except maybe the proper ones), so please just  choose an as large as possible class of nicely behaved epimorphisms of your choice - say, quotients, or of effective descent, or triquotients, etc.

REPLY [2 votes]: I am convinced by the answer of Simon Henry completely. This is just an addendum to it, mainly for myself: I want to look at these $I_\kappa$, $T_\kappa$ and $B_\kappa$ in as much detail as possible. In fact I will be just more or less repeating parts of what Simon wrote in the language more familiar to me.
So, let first $F_\kappa$ be the free frame on generators $u_{n\alpha}$, $n\in\omega$, $\alpha\in\kappa$. Note that each element of $F_\kappa$ is a join of finite meets of generators $u_{n_1\alpha_1}\land\cdots\land u_{n_m\alpha_m}$. Then the frame of opens of $I_\kappa$ is the quotient of $F_\kappa$ by the obvious relations which ensure that locale maps $X\to I_\kappa$ are in one-to-one correspondence with families $U_{n\alpha}$ of opens of $X$ satisfying
$$
\begin{aligned}
\bullet\ &\bigvee_\alpha U_{n\alpha}=\top\text{ for all $n$};\\
\bullet\ &U_{n\alpha}\wedge U_{n\beta}=\bot\text{ for all $n$ and all $\alpha\ne\beta$};\\
\bullet\ &U_{m\alpha}\wedge U_{n\alpha}=\bot\text{ for all $m\ne n$ and all $\alpha$}.
\end{aligned}
$$
In particular, each embedding $i:\omega\hookrightarrow\kappa$ determines a map  $X\to I_\kappa$ for any $X$, by declaring $U_{n\,i(n)}=\top$ for all $n$ and $U_{n\alpha}=\bot$ for all $n$, $\alpha$ with $\alpha\ne i(n)$. This implies that $u_{n_1\alpha_1}\land\cdots\land u_{n_m\alpha_m}=\bot$ if and only if there are either some $n_i=n_j$ with $\alpha_i\ne\alpha_j$ or some $\alpha_i=\alpha_j$ with $n_i\ne n_j$.
For each $\alpha\in\kappa$ let $v_\alpha=\bigvee_nu_{n\alpha}$. Then by what we just said, $v_\alpha\land u_{n_1\alpha_1}\land\cdots\land u_{n_m\alpha_m}\ne\bot$ for any $u_{n_1\alpha_1}\land\cdots\land u_{n_m\alpha_m}\ne\bot$, hence $\neg v_\alpha=\bot$. It follows that imposing further relations $v_\alpha=\top$ for all $\alpha$ gives a nontrivial dense sublocale $T_\kappa$ of $I_\kappa$; in particular $T_\kappa$ contains $B_\kappa:=I_\kappa^{\neg\neg}$.
Finally, given any map $X\to B_\kappa$ (or to $T_\kappa$ as well), all these relations imply that in the topos $\operatorname{Sh}(X)$ of sheaves on $X$, projections of the subobject
$$
\coprod_{(n,\alpha)\in\omega\times\kappa}U_{n\alpha}\rightarrowtail\coprod_{\omega\times\kappa}1
$$
both to $\coprod_\omega1$ and to $\coprod_\kappa1$ are isomorphisms.
And clearly also in fact such maps are in one-to-one correspondence with isomorphisms $\coprod_\omega1\cong\coprod_\kappa1$ in $\operatorname{Sh}(X)$.
It follows that for each such map, if $X$ is nontrivial then there is an embedding $\kappa\hookrightarrow\hom_{\operatorname{Sh}(X)}(1,\coprod_\omega1)$. Hence for each nontrivial $X$ there is no such map for $\kappa$ exceeding the cardinality of $\hom_{\operatorname{Sh}(X)}(1,\coprod_\omega1)$, i. e. of the set of all countable partitions of $X$ into clopens.