Speaker: Rosanna Laking
Title: Definability and approximation theory in triangulated categories

Abstract: Approximation theory allows one to study the category of R-modules Mod(R) via subcategories whose objects “approximate” arbitrary modules in Mod(R) i.e. precovering and preenveloping subcategories.  This fruitful theory has made many important contributions towards classification problems in Representation Theory.  

In this talk I will report on joint work with Jorge Vitòria, in which we develop a parallel techniques for algebraic compactly generated triangulated categories (e.g. derived module categories).  In this setting there are some obstructions to the development of the theory of approximations.  The main problem is that precovers and preenvelopes in module categories are typically constructed using limits or colimits, which rarely exist in triangulated categories.  Our approach is to make use of the abelian and exact categories underlying the construction of algebraic compactly generated triangulated categories to transfer the classical results to the triangulated setting.  In particular, our techniques allow us to show that definable subcategories have good approximation properties.