Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums.


A triangulated category $\mathcal{T}$ is called compactly generated if there is a set $\mathcal{S}$ of objects of $\mathcal{T}$ which consists of compact objects and satisfies
$$
\text{Hom}(G[n],X)=0, ~\forall G\in \mathcal{S}, \forall n\in \mathbb{Z} \text{ implies } X=0.
$$


In particular, an object $E$ is a compact generator of $\mathcal{T}$ if $\{E\}$could play the role of $\mathcal{S}$ in the above definition.


It is well-known that for a quasi-compact, separated scheme $X$, the derived category of complexes of quasi-coherent $\mathcal{O}\_X$-modules $D(X)$ has a compact generator.


**My question** is: is there a triangulated category $\mathcal{T}$ which is compactly generated but does not have a compact generator?



Is there a compact generated triangulated category which does not have a compact generator?
Examples can be found amongst the derived categories of algebraic stacks, see [Hall--Rydh: Algebraic groups and compact generation of their derived categories of representations](http://arxiv.org/abs/1405.1890v3), in particular theorem A. The easiest example should be $\mathrm{B}\mathbb{G}\_{\mathrm{m}}$.


