??... ??"recognizing a triangulated category as a derived category" ... ??"core of a t-structure" ... ??in situations where a derived category is "shared" (??carries multiple t-structures or something?? ??also something about extent to which a t-structure breaks symmetry... that is is really extra structure...??... ??something about distinct t-structures with the same core, or something??...) to what extent can that sharing typically or generally be lifted to the chain complex (chain-complex-enriched) categories?? or something??...

??what about triangulated categories vs some sort of categories enriched over the derived category of the category of abelian groups, or something like that?? ... and so forth...