$\newcommand\Mod{\mathrm{Mod}}\DeclareMathOperator\Fun{Fun}$If $k$ is a field, a persistent $k$-module is a functor $\mathbb{R}\to \Mod_k$ where $\mathbb{R}$ is a poset under the natural ordering of $\mathbb{R}$. A persistent $k$-module $X$ is said to be of finite type when each $X_t$ is finite dimensional and the maps $X_s \to X_t$ are isomorphisms for all but a finite set of pairs $s<t$. A persistent $k$-module $X$ of finite type admits an ‘interval decomposition’, ie. a decomposition as a direct sum $$
X\cong\bigoplus_{i} I(s_i, t_i)
$$
of interval modules, where for $s<t$ the interval module $I(s,t)$ consists of the ground field $k$ in the interval $[s,t]$ and zero otherwise, the structure maps being either identities or zero.

I have two questions about these objects:

- It is clear that the persistent $k$-modules of finite type are compact objects in the category $\Fun(\mathbb{R}, \Mod_k)$ of persistent modules. Are they *all* the compact objects in this category?

- What is an example of a persistent $k$-module that *does not* admit an interval decomposition?

Any answer to these questions, even when the characteristic of $k$ is zero, would be of great interest to me.