Tuesday, March 8, 14:00, 7.527
A simple application of a 2-categorical covering theory to a
construction of triangulated orbit categories.
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,
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
(not only for cyclic group actions but also for category lax actions),
which sometimes makes the argument clear and simple as the present
We are interested in when the orbit category of a triangulated category by
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
due to Grimeland--Jacobsen (arXiv:1508.02970).