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Racky Dichminky (Sep 28 2022 at 18:44):Tobias Fritz said:
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.
oh I just appreciate that someone tried with me!! I am too can't get an example yet unfortunately
Tobias Fritz (Sep 28 2022 at 18:59):Well, it's been a fun question to think about to refresh my memory on triangulated categories :sweat_smile:
How about the category of even-dimensional vector spaces as a subcategory of all (say finite-dimensional) vector spaces? Given , its mapping cone is the direct sum of the kernel and the cokernel of , and if both and are even-dimensional, then the usual dimension formulas from linear algebra should imply that also its mapping cone is even-dimensional. Therefore we have a triangulated subcategory, but clearly it's not saturated.
Racky Dichminky (Sep 28 2022 at 19:11):Tobias Fritz said:
Well, it's been a fun question to think about to refresh my memory on triangulated categories :sweat_smile:
How about the category of even-dimensional vector spaces as a subcategory of all (say finite-dimensional) vector spaces? Given , its mapping cone is the direct sum of the kernel and the cokernel of , and if both and are even-dimensional, then the usual dimension formulas from linear algebra should imply that also its mapping cone is even-dimensional. Therefore we have a triangulated subcategory, but clearly it's not saturated.
What's the triangulated structure here? Do you want graded vector spaces?
I think if you take graded vector spades which are even dimensionsal in every degree that works, thanks!
Racky Dichminky (Sep 28 2022 at 19:11):Oh are you just thinking take the identity for Σ or something?
Tobias Fritz (Sep 28 2022 at 19:12):Hmm, I don't see how a grading would be relevant; I just had the usual triangulated structure on a semisimple abelian category in mind.
Tobias Fritz (Sep 28 2022 at 19:12):Right, is the identity, I just had started with the most trivial type of example.
Racky Dichminky (Sep 28 2022 at 19:14):Tobias Fritz said:
Hmm, I don't see how a grading would be relevant; I just had the usual triangulated structure on a semisimple abelian category in mind.
Gotcha, I was thinking of the grading because the derived category D(k) is (triangulated) equivalent to the category of graded vector spaces
Tobias Fritz (Sep 28 2022 at 19:14):I forgot what the canonical literature is on this, so forgive me for pointing to Example 8.1 of my own old notes on triangulated categories :wink:
Tobias Fritz (Sep 28 2022 at 19:15):Ah, right! Then that's a more interesting example I guess.
Racky Dichminky (Sep 28 2022 at 19:24):Tobias Fritz said:
Ah, right! Then that's a more interesting example I guess.
Both are good!