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:



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> $\diamondsuit\_F$: there exists a transfinite sequence $\langle \mu\_\alpha:\alpha\in\omega\_1\rangle$ such that
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> $\bullet$ $\mu\_\alpha\in F(2^{\alpha})$ for every $\alpha\in\omega\_1$;
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> $\bullet$ for every $\mu\in F(2^{\omega\_1})$ the set $\{\alpha\in\omega\_1:\mu\_\alpha=F\pi\_\alpha(\mu)\}$ is stationary in $\omega\_1$.
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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.



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> **Problem.** Does $\diamondsuit\_P$ follow from the Jensen diamond principle? Or it is a stronger statement (still holding in the Constructible Universe)?
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The diamond principle for functors
The answer to this question is affirmative and follows from a more general 



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> **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 $\omega\_1$. 
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*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 $\omega\_1$.


