-   ${{\text{\tt{CONE}}}}$ of convex cones for $\mathbb R_+$,

-   ${\text{$\mathbb Q$-{\text{\tt{VEC}}}}}$ and ${\text{$\mathbb R$-{\text{\tt{VEC}}}}}$ of vector spaces over the field of rational and real numbers, respectively, for $\mathbb Q$ and $\mathbb R$.

We consider the following functors, where $A$ is a fixed finite alphabet. Recall that we use the term *cubic functor* for the functor $T1\times(-)^A$ where $T$ is a monad on ${{\text{\tt{Set}}}}$. We chose the name since $T1\times(-)^A$ assigns to objects $X$ a full direct product, i.e., a full cube.

-   The *cubic functor* ${{\ensuremath{F_{\,{\mathbb S}}}}}$ on ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$, i.e., the functor acting as $$\begin{aligned}
            & {{\ensuremath{F_{\,{\mathbb S}}}}}X=\mathbb S\times X^A\text{ for }X\text{ object of }{\text{$\mathbb S$-{\text{\tt{SMOD}}}}},
            \\
            & {{\ensuremath{F_{\,{\mathbb S}}}}}f=\operatorname{id}_{\mathbb S}\times(f\circ -)
            \text{ for }f\colon X\to Y\text{ morphism of }{\text{$\mathbb S$-{\text{\tt{SMOD}}}}}.
        \end{aligned}$$ The underlying ${{\text{\tt{Set}}}}$ functors of cubic functors are also sometimes called deterministic-automata functors, see e.g. [@jacobs.silva.sokolova:2015], as their coalgebras are deterministic weighted automata with output in the semiring.

-   The *cubic functor* ${{\ensuremath{F_{\,{[0,1]}}}}}$ on ${{\text{\tt{PCA}}}}$, i.e., the functor ${{\ensuremath{F_{\,{[0,1]}}}}}X=[0,1]\times X^A$ and ${{\ensuremath{F_{\,{[0,1]}}}}}f=\operatorname{id}_{[0,1]}\times(f\circ -)$.

-   A *subcubic convex functor* ${{\ensuremath{\widehat F}}}$ on  whose action will be introduced in Definition \[Ghat-def\].[^2]  The name originates from the fact that ${{\ensuremath{\widehat F}}}X$ is a certain convex subset of ${{\ensuremath{F_{\,{[0,1]}}}}}X$ and that 