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Racky Dichminky (Sep 25 2022 at 17:10):Hi!! I am really curious if there is an example of a triangulated category and a triangulated subcategory that isn't closed under retracts.. I am sort of totally blanking about it!!
Racky Dichminky (Sep 26 2022 at 19:35):I hope I'll ever find someone who's capable of answering this question soon
Matteo Capucci (he/him) (Sep 26 2022 at 23:50):You might try asking on mathoverlflow.com too
Morgan Rogers (he/him) (Sep 27 2022 at 08:38):There hasn't been a lot of discussion of triangulated categories here, so Matteo's suggestion is a good one. But if you find an example and want to discuss it further here, please do!
Tobias Fritz (Sep 27 2022 at 09:04):If I understand correctly, the question is about finding a triangulated subcategory that isn't saturated, yes? Since being a retract in a triangulated category is equivalent to being a direct summand.
Then how about just taking a semisimple abelian category, considered as a triangulated category in the usual way, together with a full abelian subcategory that is not closed under direct summands? I think you should be able to take for a field , and consider the full subcategory of all pairs of the form . Does this work?
Tobias Fritz (Sep 27 2022 at 10:36):Ugh, scratch that example! I got confused about the subcategory being full vs only containing morphisms of the form . It should be the former, but then my subcategory isn't triangulated since it's not closed under kernels and cokernels. Then I don't know of an example offhand.