# Finite triangulated categories and DG-enhancements

We say that a triangulated category over a field is finite if it is idempotent complete, its $$\operatorname{hom}$$-sets are finite-dimensional, and there are finitely many indecomposables. Such categories arise in different contexts and have been widely studied by Xiao and Zhu (2005), Amiot (2007), Krause (2012), Keller (2018), Hanihara (2020), and others.

Most triangulated categories arise as the derived category of something, often called enhancement, such as a DG-algebra. That enhancement is essentially unique in several cases of interest, as established by Franke (1996), Schwede (2007), Lunts and Orlov (2010), Canonaco and Stellari (2018), Antieau (2018), etc. In this talk, we will consider the existence and uniqueness of enhancements for finite triangulated categories.