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.

Given a category with a stable system of monics, one can form the corresponding category of partial maps. To each map in this category there is, on the domain of the map, an associated idempotent, which measures the degree of partiality. This structure is captured abstractly by the notion of a restriction category, in which every arrow is required to have such an associated idempotent. Categori...