Rob Bensinger comments on Controllables and Observables, Revisited

[Rob Bensinger]

To get an intuition for morphisms, I tried listing out every frame that has a morphism going to a simple 2x2 frame

$C_0=\begin{array}{cc}& \begin{array}{cc}{f}_0& f_1\end{array}\\ \begin{array}{c}{b}_0\\ b_1\end{array}& \left(\begin{array}{cc}{w}_0& w_1\\ w_2& w_3\end{array}\right)\end{array}$ .

Are any of the following wrong? And, am I missing any?

Frames I think have a morphism going to $C_0$:

$C_0$

$C_0^{\ast }=\begin{array}{cc}& \begin{array}{cc}{e}_0& e_1\end{array}\\ \begin{array}{c}{a}_0\\ a_1\end{array}& \left(\begin{array}{cc}{w}_0& w_2\\ w_1& w_3\end{array}\right)\end{array}$

${\perp }_{\left\{\right\}}=0=\begin{array}{cc}& \begin{array}{c}{e}_0\end{array}\\ \begin{array}{}\end{array}& \left(\begin{array}{c}\end{array}\right)\end{array}$

$1_{\{w_0,w_1\}}=\begin{array}{cc}& \begin{array}{cc}{e}_0& e_1\end{array}\\ \begin{array}{c}{a}_0\end{array}& \left(\begin{array}{cc}{w}_0& w_1\end{array}\right)\end{array}$

$1_{\{w_2,w_3\}}=\begin{array}{cc}& \begin{array}{cc}{e}_0& e_1\end{array}\\ \begin{array}{c}{a}_0\end{array}& \left(\begin{array}{cc}{w}_2& w_3\end{array}\right)\end{array}$

Every frame that looks like a frame on this list (other than $0$), but with extra columns added — regardless of what’s in those columns. (As a special case, this includes the five $1_S$ frames corresponding to the other five ensurables that can be deduced from the matrix: $1_{\{w_0,w_1,w_2\}}$, $1_{\{w_0,w_1,w_3\}}$, $1_{\{w_0,w_2,w_3\}}$, $1_{\{w_1,w_2,w_3\}}$, $1_{\{w_0,w_1,w_2,w_3\}}$. If $W$ has more than four elements, then there will be additional $1_S$ frames /​ additional ensurables beyond these seven.)

Every frame biextensionally equivalent to one of the frames on this list.

[Scott Garrabrant]

$C_0^{\ast }$ is wrong. You can see it has Ensurables that $C_0$ does not have.

[Scott Garrabrant]

You can also duplicate rows in $C_0$, and then add columns, so you can get things like $\left(\begin{array}{ccc}{w}_0& w_1& w_2\\ w_0& w_1& w_3\\ w_2& w_3& w_0\end{array}\right)$. There are infinitely many biextensional Cartesian frames over $\{w_0,w_1,w_2,w_3\}$ with morphism to $C_0$, with arbitrarily large dimensions.