so consider the derived k-algebroid of the abelian k-algebroid of fd vector spaces over a field k...

i guess that we should specify whether the chain complexes that we want to consider here should be "finitary" or "bounded" or something... probably yes... i'll try proceeding for the moment without worrying about this too much yet...

so it seems like the derived k-algebroid here is "semi-simple" with basis z... ??or something like that??...

??try next the case of reps of the walking arrow quiver ... ??also the walking loop quiver?? ...