Motivated by the categorification of cluster algebras, Buan–Marsh–Reineke–Reiten–Todorov introduced a theory of mutation for cluster-tilting objects in certain 2-Calabi–Yau triangulated categories. This lead to many variations or generalisations, such as tau-tilting, 2-term silting or relative tilting.
The point-of-view of extriangulated categories, introduced in collaboration with Hiroyuki Nakaoka, turns out to be relevant for the study of mutations. Indeed, most mutation theories arising in representation theory can be related to the existence of certain “good” extriangulated structures. This is the point that I will try and make in this talk, by introducing the notion of a 0-Auslander extriangulated category.
This is based on joint works with Mikhail Gorsky, Hiroyuki Nakaoka, Arnau Padrol, Vincent Pilaud and Pierre-Guy Plamondon.