A simple application of a 2-categorical covering theory to a construction of triangulated orbit categories.

Abstract:

Abstract: Throughout this talk k is a commutative ring. We first review some justifications of the definition of the orbit category C/F of a k-category C by an auto-equivalence F. Many authors do not pay enough attention on its definition and usually satisfy themselves by giving the definition only in the case that F is an automorphism, which sometimes leads us to an error. A 2-categorical covering theory (developed in arXiv:0807.4706, 0905.3884, 1111.2239, 1204.0196 and their journal paper versions) gives a simple account on equivalences between those justifications and enables us to have strict arguments on coverings of k-categories in wide range (not only for cyclic group actions but also for category lax actions), which sometimes makes the argument clear and simple as the present application shows. We are interested in when the orbit category of a triangulated category by an auto-equivalence is again triangulated. An answer was given by Keller, which was practical in many cases such as in defining cluster categories. Here we give a special constructive theorem to have this phenomenon, which gives an alternative simple unified proof of the key statement of a result due to Grimeland--Jacobsen (arXiv:1508.02970).