so what about the idea of "abelian (infinity,1)-category"? seems fairly clear how it ought to go... htpy kernel and htpy cokernel being inverse to each other or something... so probably has already been developed...

??so what about "derived category of representations of quiver q" vs "representations of quiver q (or something) into the (infinity,1)-category of chain complexes of abelian groups"?? ... and so forth ... ???.... ??also something about representations of q into the abelian (infinity,1)-category of spectrums... something about equivalence of spectrum-enriched (infinity,1)-categories ... ??and so forth...

what about model category approach to "abelian (infinity,1)-category" ?? ...
??what about "abelian (infinity,1)-cat" as possible improvement on concept of "triangulated category" (_without_ t-structure ... ?? ...) ? ...

??what _about_ grothendieck's "derivator" concept? ... os... ??....

??about the idea of "improvement" here... if the emphasis is on how a triangulated category can carry many t-structures, then i think that "abelian (infinity,1)-category" might qualify as an improved version of the concept... ?? ... whereas if the emphasis is on how a triangulated category is a mere category rather than higher-dimensional such, then of course "abelian (infinity,1)-category probably doesn't qualify... ??... ??or something...