# Mattia Ornaghi (U Milano): Dg enhancements via A∞-categories

**Research Project:**

The notion of triangulated category was developed in the 90s by Jean-Louis Verdier and Alexander Grothendieck and nowadays plays an important role in algebraic geometry. The main example of triangulated category is the bounded derived category of coherent sheaves and the study of such category has many applications concerning the geometry of moduli spaces or some problems in birational geometry. However triangulated categories have some serious drawbacks, for example the non-functoriality of the mapping cone or the non-existence of homotopy colimits and homotopy limits. These technical problems suggest the definition of "pretriangulated dg-category” i.e. a dg-category whose homotopy category is triangulated. Roughly speaking the pretriangulated dg-categories “enhance” the triangulated categories. The natural question is if such of an enhancement is unique. During my stay at ESI, I will work on this problem using the technique of A∞-categories I studied in my PhD thesis. Applications concern Homological Mirror Symmetry and Noncommutative Motives.