The diamond principle for functors

Question

Let $F:\mathbf{Comp}\to\mathbf{Set}$ be a continuous functor from the category of compact Hausdorff spaces to the category of sets such that $|Fn|\le\mathfrak c$ for any finite ordinal $n$. The continuity of $F$ means that $F$ preserves limits of inverse spectra.

Typical examples of such a functor $F$ are the functors of countable power, of the hyperspace, or of spaces of probability measures.

I am interested in validity of the following functorial version of the Jensen Diamond principle:

$\diamondsuit_F$: there exists a transfinite sequence $\langle \mu_\alpha:\alpha\in\omega_1\rangle$ such that

$\bullet$ $\mu_\alpha\in F(2^{\alpha})$ for every $\alpha\in\omega_1$;

$\bullet$ for every $\mu\in F(2^{\omega_1})$ the set $\{\alpha\in\omega_1:\mu_\alpha=F\pi_\alpha(\mu)\}$ is stationary in $\omega_1$.

Here $\pi_\alpha:2^{\omega_1}\to 2^\alpha$, $\pi_\alpha:f\mapsto f{\restriction}\alpha$, is the projection onto the $\alpha$th face of $2^{\omega_1}$.

Observation. The classical Jensen Diamond Principle is just $\diamondsuit_{Id}$ for the identity functor $Id$. It is equivalent to the Principle $\diamondsuit_{Id^\omega}$ for the functor $Id^\omega$ of countable power. Using the Parovichenko Theorem, it can be shown that the Jensen Diamond Principle is equivalent to $\diamondsuit_{\exp}$ for the functor $\exp$ of hyperspace.

I am interested in $\diamondsuit_P$ for the functor $P$ of probability measures.

Problem. Does $\diamondsuit_P$ follow from the Jensen diamond principle? Or it is a stronger statement (still holding in the Constructible Universe)?

Answer 1

The answer to this question is affirmative and follows from a more general

Theorem. Let $X_{\omega_1}$ be the limit of a continuous well-ordered spectrum $\langle X_\alpha,p_\alpha^\beta:\alpha\le \beta<\omega_1\rangle$ in the category of sets such that each set $X_\alpha$, $\alpha<\omega_1$ has cardinality $\le\mathfrak c$. The Jensen Diamond Principle $\diamond$ implies that there exists a transfinite sequence $(\mu_\alpha)_{\alpha\in\omega_1}\in\prod_{\alpha\in\omega_1}X_\alpha$ such that for any $\mu\in X_{\omega_1}$ the set $\{\alpha\in\omega_1:\mu_\alpha=p^{\omega_1}_\alpha(\mu)\}$ is stationary in $\omega_1$.

Proof. Let $C=2^\omega$ be the Cantor cube. For ordinals $\alpha\le\beta\le\omega_1$ let $\pi_\alpha^\beta:C^{\beta}\to C^\alpha$, $\pi_\alpha:x\mapsto x{\restriction}\alpha$, the projection onto the $\alpha$th face of the cube $C^\beta$.

It is well-known that $\diamondsuit$ implies the existence of a transfinite sequence $(x_\alpha)_{\alpha\in\omega_1}\in C^{\omega_1}$ such that for any $x\in C^{\omega_1}$ the set $\{\alpha\in\omega_1:x_\alpha=\pi^{\omega_1}_\alpha(x)\}$ is stationary in $\omega_1$.

Taking into acount that the spectrum $\langle X_\alpha,p_\alpha^\beta:\alpha\le \beta<\omega_1\rangle$ is continuous and consists of sets of cardinality $\le\mathfrak c=|C|$, it is possible to construct inductively a transfinite sequence of injective maps $(f_\alpha:X_\alpha\to C^\alpha)_{\alpha\le\omega_1}$ such that for any $\alpha<\beta\le\omega_1$ we have the equality $f_\alpha\circ p_\alpha^\beta=\pi^\beta_\alpha\circ f_\beta$.

Let $\Omega=\{\alpha\in\omega_1:x_\alpha\in f_\alpha(X_\alpha)\}$. Let $(\mu_\alpha)_{\alpha\in\omega_1}\in\prod_{\alpha\in\omega_1}X_\alpha$ be any transfinite sequence such that $f_\alpha(\mu_\alpha)=x_\alpha$ for any $\alpha\in\Omega$. We claim that this sequence has the required property.

Given any $\mu\in X_{\omega_1}$ consider the element $x=f_{\omega_1}(\mu)\in C^{\omega_1}$. The choice of the transfinite sequence $(x_\alpha)_{\alpha\in\omega_1}$ ensures that the set $S=\{\alpha\in\omega_1:x_\alpha=x{\restriction}\alpha\}$ is stationary in $\omega_1$. For every $\alpha\in S$ we have $$x_\alpha=x{\restriction}\alpha=\pi_\alpha^{\omega_1}(x)=\pi_\alpha^{\omega_1}\circ f_{\omega_1}(\mu)=f_\alpha\circ p_\alpha^{\omega_1}(\mu)\in f_\alpha(X_\alpha)$$ and hence $\alpha\in\Omega$ and $f_\alpha(\mu_\alpha)=x_\alpha=f_\alpha\circ p_\alpha^{\omega_1}(\mu)$. Now the injectivity of the map $f_\alpha$ ensures that $\mu_\alpha=p_\alpha^{\omega_1}(\mu)$.

Therefore, the set $\{\alpha\in\omega_1: \mu_\alpha=p_\alpha^{\omega_1}(\mu)\}\supset S$ is stationary in $\omega_1$.