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...

In line with the development of a theme of a classical paper by D. K. Faddeev, motivic extensions of certain additive categories are studied.