in this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. we focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of fomin and reading, and colored quiver mutation.

in this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. we focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of fomin and reading, and colored quiver mutation.

In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that definition is natural, it leads to powerful results, and throws light on old, familiar objects. In particular: a recent theorem says category $D_{\text{qc}}(X)$ approximable whenever $X$ quasicompact separated scheme. As corollaries (seemingly technical) statement one can pro...