A derived Gabriel-Popescu theorem for t-structures

The Gabriel-Popescu theorem exhibits any Grothendieck abelian category as an exact localization of a category of modules over a suitable ring. Generalizing to the derived framework, we replace abelian categories with (enhanced) triangulated categories endowed with a t-structure. Such categories, under appropriate “Grothendieck-like” assumptions, can be exhibited as t-exact quotients of derived categories of suitable dg-algebras concentrated in nonpositive degrees, hence yielding a “derived Gabriel-Popescu theorem”. In this talk, we describe a proof of this result which exploits the underlying philosophy that “(enhanced) triangulated categories with t-structures really behave like abelian categories”. We shall encounter suitably defined “derived epi-mono factorizations” and derived injective objects. This is joint work with Julia Ramos González.