What are projective locales / injective frames?

Question

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.

Answer 1

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 $\omega$" 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 $\omega$.

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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.

Answer 2

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.